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0	https://www.slideshare.net/slideshow/5207871-2/250042359	5207871 (2) \| PPT Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Cookie Policy Download free for 30 days Sign in UploadLanguage (EN)Support BusinessMobileSocial MediaMarketingTechnologyArt & PhotosCareerDesignEducationPresentations & Public SpeakingGovernment & NonprofitHealthcareInternetLawLeadership & ManagementAutomotiveEngineeringSoftwareRecruiting & HRRetailSalesServicesScienceSmall Business & EntrepreneurshipFoodEnvironmentEconomy & FinanceData & AnalyticsInvestor RelationsSportsSpiritualNews & PoliticsTravelSelf ImprovementReal EstateEntertainment & HumorHealth & MedicineDevices & HardwareLifestyle Change Language Language English Español Português Français Deutsche Cancel Save Submit search EN Uploaded byAmerWagdyGerges1 PPT, PDF 562 views 5207871 (2) AI-enhanced description The document discusses accounting for leases. It begins by explaining the nature of lease transactions and the advantages of leasing. It then describes the accounting criteria and procedures for classifying leases as capital or operating from the perspective of the lessee. Specifically, it outlines the four criteria that must be met for a lease to be classified as a capital lease by the lessee. The document also provides examples of accounting entries for capital and operating leases. Health & Medicine◦ Read more 0 Save Share Embed Download Download to read offline 1 / 87 2 / 87 3 / 87 4 / 87 5 / 87 6 / 87 7 / 87 8 / 87 9 / 87 10 / 87 11 / 87 12 / 87 13 / 87 14 / 87 15 / 87 16 / 87 17 / 87 18 / 87 19 / 87 20 / 87 21 / 87 22 / 87 23 / 87 24 / 87 25 / 87 26 / 87 27 / 87 28 / 87 29 / 87 30 / 87 31 / 87 32 / 87 33 / 87 34 / 87 35 / 87 36 / 87 37 / 87 38 / 87 39 / 87 40 / 87 41 / 87 42 / 87 43 / 87 44 / 87 45 / 87 46 / 87 47 / 87 48 / 87 49 / 87 50 / 87 51 / 87 52 / 87 53 / 87 54 / 87 55 / 87 56 / 87 57 / 87 58 / 87 59 / 87 60 / 87 61 / 87 62 / 87 63 / 87 64 / 87 65 / 87 66 / 87 67 / 87 68 / 87 69 / 87 70 / 87 71 / 87 72 / 87 73 / 87 74 / 87 75 / 87 76 / 87 77 / 87 78 / 87 79 / 87 80 / 87 81 / 87 82 / 87 83 / 87 84 / 87 85 / 87 86 / 87 87 / 87 ![Image 111: Chapter 21-24 E21-1: Journal entries for Adams through Jan. 1, 2012. LO 2 Describe the accounting criteria and procedures for capitalizing leases by the lessee. Accounting by the Lessee Depreciation Expense 8,313 Accumulated Depreciation—Capital Leases 8,313 ($41,565 ÷ 5 = $8,313) Interest Expense 3,160 Interest Payable 3,160 ($41,565 – $9,968) X .10] 12/31/11 ]( Ad Recommended PPT Accounting Chapter 10.ppt bymayada34 PPTX Ch11 byESHETIE MEKONENE AMARE PPT Ch13 byhelal682621 PPTX Capital Structure bypremarhea PPTX Chapter3 bykim rae KI PPTX Ias 37 byESHETIE MEKONENE AMARE PDF PDF Kieso Intermediate Accounting chapter 12 bySafa444074 PPTX Cash Flow Statement PPT byWe Learn - A Continuous Learning Forum from Welingkar's Distance Learning Program. 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Explain thenature, economic substance, and advantages of lease transactions. 2. Describe the accounting criteria and procedures for capitalizing leases by the lessee. 3. Contrast the operating and capitalization methods of recording leases. 4. Identify the classifications of leases for the lessor. 5. Describe the lessor’s accounting for direct-financing leases. 6. Identify special features of lease arrangements that cause unique accounting problems. 7. Describe the effect of residual values, guaranteed and unguaranteed, on lease accounting. 8. Describe the lessor’s accounting for sales-type leases. 9. List the disclosure requirements for leases. Learning Objectives Chapter 21-4 Leasing Environment Who are players? Advantages of leasing Conceptual natureof a lease Accounting by Lessee Accounting by Lessor Special Accounting Problems Capitalization criteria Accounting differences Capital lease method Operating method Comparison Residual values Sales-type leases Bargain- purchase option Initial direct costs Current versus noncurrent Disclosure Unresolved problems Economics of leasing Classification Direct-financing method Operating method Accounting for Leases Chapter 21-5 Largest group ofleased equipment involves: Information technology Transportation (trucks, aircraft, rail) Construction Agriculture LO 1 Explain the nature, economic substance, and advantages of lease transactions. A lease is a contractual agreement between a lessor and a lessee, that gives the lessee the right to use specific property, owned by the lessor, for a specified period of time. The Leasing Environment Chapter 21-6 Three general categories: Banks. Captiveleasing companies. Independents. LO 1 Explain the nature, economic substance, and advantages of lease transactions. Who Are the Players? The Leasing Environment Chapter 21-7 1. 100% Financingat Fixed Rates. 2. Protection Against Obsolescence. 3. Flexibility. 4. Less Costly Financing. 5. Tax Advantages. 6. Off-Balance-Sheet Financing. The Leasing Environment LO 1 Explain the nature, economic substance, and advantages of lease transactions. Advantages of Leasing Chapter 21-8 Capitalize a leasethat transfers substantially all of the benefits and risks of property ownership, provided the lease is noncancelable. Leases that do not transfer substantially all the benefits and risks of ownership are operating leases. The Leasing Environment LO 1 Explain the nature, economic substance, and advantages of lease transactions. Conceptual Nature of a Lease Chapter 21-9 Operating Lease CapitalLease Journal Entry: Rent expense xxx Cash xxx Journal Entry: Leased equipment xxx Lease liability xxx The issue of how to report leases is the case of substance versus form. Although technically legal title may not pass, the benefits from the use of the property do. A lease that transfers substantially all of the benefits and risks of property ownership should be capitalized (only noncancellable leases may be capitalized). The Leasing Environment LO 1 Explain the nature, economic substance, and advantages of lease transactions. Chapter 21-10 If the lesseecapitalizes a lease, the lessee records an asset and a liability generally equal to the present value of the rental payments. Records depreciation on the leased asset. Treats the lease payments as consisting of interest and principal. Accounting by the Lessee LO 2 Describe the accounting criteria and procedures for capitalizing leases by the lessee. Typical Journal Entries for Capitalized Lease Illustration 21-2 Chapter 21-11 To record alease as a capital lease, the lease must be noncancelable. One or more of four criteria must be met: 1. Transfers ownership to the lessee. 2. Contains a bargain-purchase option. 3. Lease term is equal to or greater than 75 percent of the estimated economic life of the leased property. 4. The present value of the minimum lease payments (excluding executory costs) equals or exceeds 90 percent of the fair value of the leased property. Accounting by the Lessee LO 2 Describe the accounting criteria and procedures for capitalizing leases by the lessee. Chapter 21-12 LO 2 Describethe accounting criteria and procedures for capitalizing leases by the lessee. Lease Agreement Leases that DO NOT meet any of the four criteria are accounted for as Operating Leases. Accounting by the Lessee Illustration 21-4 Chapter 21-13 Capitalization Criteria LO 2Describe the accounting criteria and procedures for capitalizing leases by the lessee. Accounting by the Lessee Transfer of Ownership Test Not controversial and easily implemented. Bargain-Purchase Option Test At the inception of the lease, the difference between the option price and the expected fair market value must be large enough to make exercise of the option reasonably assured. Chapter 21-14 Capitalization Criteria LO 2Describe the accounting criteria and procedures for capitalizing leases by the lessee. Accounting by the Lessee Economic Life Test (75% Test) Lease term is generally considered to be the fixed, noncancelable term of the lease. Bargain-renewal option can extend this period. At the inception of the lease, the difference between the renewal rental and the expected fair rental must be great enough to make exercise of the option to renew reasonably assured. Chapter 21-15 Recovery of InvestmentTest (90% Test) LO 2 Accounting by the Lessee Minimum Lease Payments:  Minimum rental payment  Guaranteed residual value  Penalty for failure to renew  Bargain-purchase option Executory Costs:  Insurance  Maintenance  Taxes Exclude from PV of Minimum Lease Payment Calculation Capitalization Criteria Chapter 21-16 Accounting by theLessee Discount Rate Lessee computes the present value of the minimum lease payments using its incremental borrowing rate, with one exception.  If the lessee knows the implicit interest rate computed by the lessor and it is less than the lessee’s incremental borrowing rate, then lessee must use the lessor’s rate. Recovery of Investment Test (90% Test) Capitalization Criteria LO 2 Chapter 21-17 Asset and LiabilityRecorded at the lower of: 1. present value of the minimum lease payments (excluding executory costs) or 2. fair-market value of the leased asset. Asset and Liability Accounted for Differently LO 2 Describe the accounting criteria and procedures for capitalizing leases by the lessee. Accounting by the Lessee Chapter 21-18 LO 2 Describethe accounting criteria and procedures for capitalizing leases by the lessee. Accounting by the Lessee Depreciation Period If lease transfers ownership, depreciate asset over the economic life of the asset. If lease does not transfer ownership, depreciate over the term of the lease. Asset and Liability Accounted for Differently Chapter 21-19 LO 2 Describethe accounting criteria and procedures for capitalizing leases by the lessee. Accounting by the Lessee Effective-Interest Method The effective-interest method is used to allocate each lease payment between principal and interest. Asset and Liability Accounted for Differently Depreciation Concept Depreciation and the discharge of the obligation are independent accounting processes. Chapter 21-20 E21-1 (Capital Leasewith Unguaranteed Residual Value): On January 1, 2011, Adams Corporation signed a 5-year noncancelable lease for a machine. The terms of the lease called for Adams to make annual payments of $9,968 at the beginning of each year, starting January 1, 2011. The machine has an estimated useful life of 6 years and a $5,000 unguaranteed residual value. Adams uses the straight-line method of depreciation for all of its plant assets. Adams’s incremental borrowing rate is 10%, and the Lessor’s implicit rate is unknown. LO 2 Accounting by the Lessee Instructions (a) What type of lease is this? Explain. (b) Compute the present value of the minimum lease payments. (c) Prepare all necessary journal entries for Adams for this lease through January 1, 2012. Chapter 21-21 E21-1: What typeof lease is this? Explain. LO 2 Describe the accounting criteria and procedures for capitalizing leases by the lessee. Accounting by the Lessee Capitalization Criteria: 1. Transfer of ownership 2. Bargain purchase option 3. Lease term => 75% of economic life of leased property 4. Present value of minimum lease payments => 90% of FMV of property NO NO Lease term 5 yrs. Economic life 6 yrs. YES 83.3% FMV of leased property is unknown. Capital Lease, #3 Chapter 21-22 E21-1: Compute presentvalue of the minimum lease payments. LO 2 Describe the accounting criteria and procedures for capitalizing leases by the lessee. Accounting by the Lessee Payment $ 9,968 Present value factor (i=10%,n=5) 4.16986 PV of minimum lease payments $41,565 Leased Machine Under Capital Leases 41,565 Lease Liability 41,565 Lease Liability 9,968 Cash 9,968 1/1/11 Journal Entries: Chapter 21-23 E21-1: Lease AmortizationSchedule LO 2 Describe the accounting criteria and procedures for capitalizing leases by the lessee. Accounting by the Lessee 10% Lease Interest Reduction Lease Date Payment Expense in Liability Liability 1/1/11 41,565 $ 1/1/11 9,968 $ 9,968 $ 31,597 12/31/11 9,968 3,160 6,808 24,789 12/31/12 9,968 2,479 7,489 17,300 12/31/13 9,968 1,730 8,238 9,062 12/31/14 9,968 906 9,062 0 Chapter 21-24 E21-1: Journal entriesfor Adams through Jan. 1, 2012. LO 2 Describe the accounting criteria and procedures for capitalizing leases by the lessee. Accounting by the Lessee Depreciation Expense 8,313 Accumulated Depreciation—Capital Leases 8,313 ($41,565 ÷ 5 = $8,313) Interest Expense 3,160 Interest Payable 3,160 ($41,565 – $9,968) X .10] 12/31/11 Chapter 21-25 E21-1: Journal entriesfor Adams through Jan. 1, 2012. LO 2 Describe the accounting criteria and procedures for capitalizing leases by the lessee. Accounting by the Lessee Lease Liability 6,808 Interest Payable 3,160 Cash 9,968 1/1/12 Chapter 21-26 LO 3Contrast the operating and capitalization methods of recording leases. Accounting by the Lessee Operating Method The lessee assigns rent to the periods benefiting from the use of the asset and ignores, in the accounting, any commitments to make future payments. Illustration: Assume Adams accounts for it as an operating lease. Adams records this payment on January 1, 2011, as follows. Rent Expense 9,968 Cash 9,968 Chapter 21-27 E21-1: Comparison ofCapital Lease with Operating Lease LO 3 Contrast the operating and capitalization methods of recording leases. Accounting by the Lessee E21-1 Capital Lease Operating Depreciation Interest Lease Date Expense Expense Total Expense Diff. 2011 8,313 $ 3,160 $ 11,473 $ 9,968 $ 1,505 $ 2012 8,313 2,479 10,792 9,968 824 2013 8,313 1,730 10,043 9,968 75 2014 8,313 906 9,219 9,968 (749) 2015 8,313 8,313 9,968 (1,655) 41,565 $ 8,275 $ 49,840 $ 49,840 $ 0 Chapter 21-28 1. Interest Revenue. 2.Tax Incentives. 3. High Residual Value. Accounting by the Lessor Benefits to the Lessor LO 4 Identify the classifications of leases for the lessor. Chapter 21-29 A lessor determinesthe amount of the rental, based on the rate of return needed to justify leasing the asset. If a residual value is involved (whether guaranteed or not), the company would not have to recover as much from the lease payments Economics of Leasing Accounting by the Lessor LO 4 Identify the classifications of leases for the lessor. Chapter 21-30 E21-10 (Computation ofRental): Fieval Leasing Company signs an agreement on January 1, 2010, to lease equipment to Reid Company. The following information relates to this agreement. 1. The term of the noncancelable lease is 6 years with no renewal option. The equipment has an estimated economic life of 6 years. 2. The cost of the asset to the lessor is $343,000. The fair value of the asset at January 1, 2010, is $343,000. 3. The asset will revert to the lessor at the end of the lease term at which time the asset is expected to have a residual value of $61,071, none of which is guaranteed. 4. The agreement requires annual rental payments, beg. Jan. 1, 2010. 5. Collectibility of the lease payments is reasonably predictable. There are no important uncertainties surrounding the amount of costs yet to be incurred by the lessor. Accounting by the Lessor LO 4 Identify the classifications of leases for the lessor. Chapter 21-31 Accounting by theLessor LO 4 Identify the classifications of leases for the lessor. Residual value 61,071 $ PV of single sum (i=10%, n=6) 0.56447 PV of residual value 34,473 $ Fair market value of leased equipment 343,000 $ Present value of residual value (34,473) Amount to be recovered through lease payment 308,527 PV factor of annunity due (i=10%, n=6) 4.79079 Annual payment required 64,400 $ E21-10 (Computation of Rental): Assuming the lessor desires a 10% rate of return on its investment, calculate the amount of the annual rental payment required. ÷ x - Chapter 21-32 a. Operating leases. b.Direct-financing leases. c. Sales-type leases. Classification of Leases by the Lessor Accounting by the Lessor LO 4 Identify the classifications of leases for the lessor. Chapter 21-33 Classification of Leasesby the Lessor Accounting by the Lessor LO 4 Identify the classifications of leases for the lessor. A sales-type lease involves a manufacturer’s or dealer’s profit, and a direct-financing lease does not. Illustration 21-10 Chapter 21-34 Classification of Leasesby the Lessor Accounting by the Lessor LO 4 Identify the classifications of leases for the lessor. A lessor may classify a lease as an operating lease but the lessee may classify the same lease as a capital lease. Illustration 21-11 Chapter 21-35 In substance thefinancing of an asset purchase by the lessee. Direct-Financing Method (Lessor) Accounting by the Lessor LO 5 Describe the lessor’s accounting for direct-financing leases. Chapter 21-36 Accounting by theLessor E21-10: Prepare an amortization schedule that would be suitable for the lessor. LO 5 Describe the lessor’s accounting for direct-financing leases. Chapter 21-37 Accounting by theLessor E21-10: Prepare all of the journal entries for the lessor for 2010 and 2011. LO 5 Describe the lessor’s accounting for direct-financing leases. 1/1/10 Lease Receivable 343,000 Equipment 343,000 1/1/10 Cash 64,400 Lease Receivable 64,400 12/31/10 Interest Receivable 27,860 Interest Revenue 27,860 Chapter 21-38 Accounting by theLessor E21-10: Prepare all of the journal entries for the lessor for 2010 and 2011. LO 5 Describe the lessor’s accounting for direct-financing leases. 1/1/11 Cash 64,400 Lease Receivable 36,540 Interest Receivable 27,860 12/31/11 Interest Receivable 24,206 Interest Revenue 24,206 Chapter 21-39 Records each rentalreceipt as rental revenue. Depreciates the leased asset in the normal manner. Operating Method (Lessor) Accounting by the Lessor LO 5 Describe the lessor’s accounting for direct-financing leases. Chapter 21-40 Illustration: Assume Fievalaccounts for the lease as an operating lease. It records the cash rental receipt as follows: Operating Method (Lessor) Accounting by the Lessor LO 5 Describe the lessor’s accounting for direct-financing leases. Cash 64,400 Rental Revenue 64,400 Depreciation is recorded as follows: Depreciation Expense 57,167 Accumulated Depreciation 57,167 $343,000 / 6 years = 57,167 Chapter 21-41 1. Residual values. 2.Sales-type leases (lessor). 3. Bargain-purchase options. 4. Initial direct costs. 5. Current versus noncurrent classification. 6. Disclosure. Special Accounting Problems LO 6 Identify special features of lease arrangements that cause unique accounting problems. Chapter 21-42 Meaning of ResidualValue - Estimated fair value of the leased asset at the end of the lease term. Guaranteed Residual Value – Lessee agrees to make up any deficiency below a stated amount that the lessor realizes in residual value at the end of the lease term. Residual Values Special Accounting Problems LO 7 Describe the effect of residual values, guaranteed and unguaranteed, on lease accounting. Chapter 21-43 Lessee Accounting forResidual Value The accounting consequence is that the minimum lease payments, include the guaranteed residual value but excludes the unguaranteed residual value. Residual Values Special Accounting Problems LO 7 Describe the effect of residual values, guaranteed and unguaranteed, on lease accounting. Chapter 21-44 Illustration (Guaranteed ResidualValue – Lessee Accounting): Caterpillar Financial Services Corp. (a subsidiary of Caterpillar) and Sterling Construction Corp. sign a lease agreement dated January 1, 2011, that calls for Caterpillar to lease a front-end loader to Sterling beginning January 1, 2011. The terms and provisions of the lease agreement, and other pertinent data, are as follows. The term of the lease is five years. The lease agreement is noncancelable, requiring equal rental payments at the beginning of each year (annuity due basis). The loader has a fair value at the inception of the lease of $100,000, an estimated economic life of five years, and no residual value. Special Accounting Problems LO 7 Describe the effect of residual values, guaranteed and unguaranteed, on lease accounting. Chapter 21-45 Illustration (Guaranteed ResidualValue – Lessee Accounting): Sterling pays all of the executory costs directly to third parties except for the property taxes of $2,000 per year, which is included as part of its annual payments to Caterpillar. The lease contains no renewal options. The loader reverts to Caterpillar at the termination of the lease. Sterling’s incremental borrowing rate is 11 percent per year. Sterling depreciates on a straight-line basis. Caterpillar sets the annual rental to earn a rate of return on its investment of 10 percent per year; Sterling knows this fact. Caterpillar estimates a residual value of $5,000 a the end of the lease. Special Accounting Problems LO 7 Describe the effect of residual values, guaranteed and unguaranteed, on lease accounting. Chapter 21-46 Illustration (Guaranteed ResidualValue – Lessee Accounting): Caterpillar would compute the amount of the lease payments as follows: Special Accounting Problems LO 7 Describe the effect of residual values, guaranteed and unguaranteed, on lease accounting. Illustration 21-16 NOTE: For the Lessee, the minimum lease payment includes the guaranteed residual value but excludes the unguaranteed residual value. Solution on notes page Chapter 21-47 Illustration 21-17 Illustration (GuaranteedResidual Value – Lessee Accounting): Computation of Lessee’s capitalized amount Special Accounting Problems LO 7 Describe the effect of residual values, guaranteed and unguaranteed, on lease accounting. Solution on notes page Chapter 21-48 Illustration (Guaranteed ResidualValue – Lessee Accounting): Computation of Lease Amortization Schedule Illustration 21-18 Special Accounting Problems LO 7 Chapter 21-49 Illustration (Guaranteed ResidualValue – Lessee Accounting): At the end of the lease term, before the lessee transfers the asset to Caterpillar, the lease asset and liability accounts have the following balances. Illustration 21-19 Special Accounting Problems LO 7 Describe the effect of residual values, guaranteed and unguaranteed, on lease accounting. Chapter 21-50 Illustration (Guaranteed ResidualValue – Lessee Accounting): Assume that Sterling depreciated the leased asset down to its residual value of $5,000 but that the fair market value of the residual value at December 31, 2015, was $3,000. Sterling would make the following journal entry. Special Accounting Problems LO 7 Describe the effect of residual values, guaranteed and unguaranteed, on lease accounting. Loss on Capital Lease 2,000.00 Interest Expense (or Interest Payable) 454.76 Lease Liability 4,545.24 Accumulated Depreciation—Capital Leases 95,000.00 Leased Equipment under Capital Leases 100,000.00 Cash 2,000.00 Chapter 21-51 Illustration (Unguaranteed ResidualValue – Lessee Accounting): Assume the same facts as those above except that the $5,000 residual value is unguaranteed instead of guaranteed. Caterpillar would compute the amount of the lease payments as follows: Special Accounting Problems LO 7 Describe the effect of residual values, guaranteed and unguaranteed, on lease accounting. Illustration 21-20 Solution on notes page Chapter 21-52 Illustration (Unguaranteed ResidualValue – Lessee Accounting): Computation of Lease Amortization Schedule Illustration 21-21 Special Accounting Problems LO 7 Describe the effect of residual values, guaranteed and unguaranteed, on lease accounting. Chapter 21-53 Illustration (Unguaranteed ResidualValue – Lessee Accounting): At the end of the lease term, before Sterling transfers the asset to Caterpillar, the lease asset and liability accounts have the following balances. Illustration 21-22 Special Accounting Problems LO 7 Describe the effect of residual values, guaranteed and unguaranteed, on lease accounting. Chapter 21-54 Special Accounting Problems Illustration21-23 Comparative Entries, Lessee Company Chapter 21-55 Special Accounting Problems Illustration:Assume a direct-financing lease with a residual value (either guaranteed or unguaranteed) of $5,000. Caterpillar determines the payments as follows. LO 7 Describe the effect of residual values, guaranteed and unguaranteed, on lease accounting. Lessor Accounting for Residual Value The lessor works on the assumption that it will realize the residual value at the end of the lease term whether guaranteed or unguaranteed. Illustration 21-24 Chapter 21-56 Special Accounting Problems LessorAccounting for Residual Value Illustration: Lease Amortization Schedule, for Lessor— Guaranteed or Unguaranteed Residual Value Illustration 21-25 Chapter 21-57 LO 7 Describethe effect of residual values, guaranteed and unguaranteed, on lease accounting. Special Accounting Problems Lessor Accounting for Residual Value Illustration: Caterpillar would make the following entries for this direct-financing lease in the first year. Illustration 21-26 Solution on notes page Chapter 21-58 Primary difference betweena direct-financing lease and a sales-type lease is the manufacturer’s or dealer’s gross profit (or loss). Lessor records the sale price of the asset, the cost of goods sold and related inventory reduction, and the lease receivable. Difference in accounting for guaranteed and unguaranteed residual values. Sales-Type Leases (Lessor) Special Accounting Problems LO 8 Describe the lessor’s accounting for sales-type leases. Chapter 21-59 Sales-Type Leases (Lessor) SpecialAccounting Problems LO 8 Describe the lessor’s accounting for sales-type leases. Illustration: To illustrate a sales-type lease with a guaranteed residual value and with an unguaranteed residual value, assume the same facts as in the preceding direct- financing lease situation. The estimated residual value is $5,000 (the present value of which is $3,104.60), and the leased equipment has an $85,000 cost to the dealer, Caterpillar. Assume that the fair market value of the residual value is $3,000 at the end of the lease term. Chapter 21-60 Sales-Type Leases (Lessor) SpecialAccounting Problems LO 8 Describe the lessor’s accounting for sales-type leases. Illustration: Computation of Lease Amounts by Caterpillar Financial—Sales-Type Lease Illustration 21-28 Chapter 21-61 Sales-Type Leases (Lessor) SpecialAccounting Problems LO 8 Describe the lessor’s accounting for sales-type leases. Illustration: Caterpillar makes the following entries to record this transaction on January 1, 2011, and the receipt of the residual value at the end of the lease term. Illustration 21-29 Chapter 21-62 Sales-Type Leases (Lessor) SpecialAccounting Problems LO 8 Describe the lessor’s accounting for sales-type leases. Illustration: Caterpillar makes the following entries to record this transaction on January 1, 2011, and the receipt of the residual value at the end of the lease term. Illustration 21-29 (January 1, 2012) Chapter 21-63 Sales-Type Leases (Lessor) SpecialAccounting Problems LO 8 Describe the lessor’s accounting for sales-type leases. Illustration: Caterpillar makes the following entries to record this transaction on January 1, 2011, and the receipt of the residual value at the end of the lease term. Illustration 21-29 Chapter 21-64 Present value ofthe minimum lease payments must include the present value of the option. Only difference between the accounting treatment for a bargain-purchase option and a guaranteed residual value of identical amounts is in the computation of the annual depreciation. Bargain Purchase Option (Lessee) Special Accounting Problems LO 8 Describe the lessor’s accounting for sales-type leases. Chapter 21-65 The accounting forinitial direct costs: For operating leases, the lessor should defer initial direct costs. For sales-type leases, the lessor expenses the initial direct costs. For a direct-financing lease, the lessor adds initial direct costs to the net investment. Initial Direct Costs (Lessor) Special Accounting Problems LO 8 Describe the lessor’s accounting for sales-type leases. Chapter 21-66 FASB Statement No.13 does not indicate how to measure the current and noncurrent amounts. It requires that for the lessee the “obligations shall be separately identified on the balance sheet as obligations under capital leases and shall be subject to the same considerations as other obligations in classifying them with current and noncurrent liabilities in classified balance sheets.” Current versus Noncurrent Special Accounting Problems LO 8 Describe the lessor’s accounting for sales-type leases. Chapter 21-67 1. General descriptionof the nature of the lease. 2. Nature, timing and amount of cash inflows and outflows associated with leases, including payments for each of the five succeeding years. 3. Amount of lease revenues and expenses reported in the income statement each period. 4. Description and amounts of leased assets by major balance sheet classification and related liabilities. 5. Amounts receivable and unearned revenues under lease. Disclosing Lease Data Special Accounting Problems LO 9 List the disclosure requirements for leases. Chapter 21-68  Leasing wason the FASB’s initial agenda in 1973 and SFAS No. 13 was issued in 1976 (before the conceptual framework was developed). SFAS No. 13 has been the subject of more than 30 interpretations since its issuance.  The iGAAP leasing standard is IAS 17, first issued in 1982. This standard is the subject of only three interpretations. One reason for this small number of interpretations is that iGAAP does not specifically address a number of leasing transactions that are covered by U.S. GAAP. Examples include lease agreements for natural resources, sale-leasebacks, real estate leases, and leveraged leases. Chapter 21-69  Both U.S.GAAP and iGAAP share the same objective of recording leases by lessees and lessors according to their economic substance—that is, according to the definitions of assets and liabilities.  U.S. GAAP for leases in much more “rule-based” with specific bright-line criteria to determine if a lease arrangement transfers the risks and rewards of ownership; iGAAP is more general in its provisions. Chapter 21-70 LO 10 Chapter 21-71 LO 10Understand and apply lease accounting concepts to various lease arrangements. Solution on notes page Illustration 21A-2 Chapter 21-72 Solution on notes page Chapter 21-73 Illustration 21A-3 LO 10Understand and apply lease accounting concepts to various lease arrangements. Chapter 21-74 LO 10Understand and apply lease accounting concepts to various lease arrangements. Chapter 21-75 Illustration 21A-4 LO 10Understand and apply lease accounting concepts to various lease arrangements. Chapter 21-76 LO 10Understand and apply lease accounting concepts to various lease arrangements. Chapter 21-77 Illustration 21A-5 LO 10Understand and apply lease accounting concepts to various lease arrangements. Chapter 21-78 LO 11Describe the lessee’s accounting for sale-leaseback transactions. The term sale-leaseback describes a transaction in which the owner of the property (seller-lessee) sells the property to another and simultaneously leases it back from the new owner. Advantages: 1. May allow seller to refinance at lower rates. 2. May provide another source of working capital, particularly when liquidity is tight. 3. By selling the property, the seller-lessee may deduct the entire lease payment, which is not subject to alternative minimum tax considerations. Chapter 21-79 LO 11Describe the lessee’s accounting for sale-leaseback transactions. Determining Asset Use To the extent the seller-lessee continues to use the asset after the sale, the sale-leaseback is really a form of financing.  Lessor should not recognize a gain or loss on the transaction. If the seller-lessee gives up the right to the use of the asset, the transaction is in substance a sale.  Gain or loss recognition is appropriate. Chapter 21-80 LO 11Describe the lessee’s accounting for sale-leaseback transactions. Lessee If the lease meets one of the four criteria for treatment as a capital lease, the seller-lessee should  Account for the transaction as a sale and the lease as a capital lease.  Defer any profit or loss it experiences from the sale of the assets that are leased back under a capital lease.  Amortize profit over the lease term . Chapter 21-81 LO 11Describe the lessee’s accounting for sale-leaseback transactions. Lessee If none of the capital lease criteria are satisfied, the seller-lessee accounts for the transaction as a sale and the lease as an operating lease.  Lessee defers such profit or loss and amortizes it in proportion to the rental payments over the period when it expects to use the assets. Exceptions:  Losses Recognized and Minor Leaseback Chapter 21-82 LO 11Describe the lessee’s accounting for sale-leaseback transactions. Lessor If the lease meets one of the criteria in Group I and both of the criteria in Group II, the purchaser-lessor records the transaction as a purchase and a direct- financing lease. If the lease does not meet the criteria, the purchaser- lessor records the transaction as a purchase and an operating lease. Chapter 21-83 LO 11Describe the lessee’s accounting for sale-leaseback transactions. Sale-Leaseback Example American Airlines on January 1, 2011, sells a used Boeing 757 having a carrying amount on its books of $75,500,000 to CitiCapital for $80,000,000. American immediately leases the aircraft back under the following conditions: 1. The term of the lease is 15 years, noncancelable, and requires equal rental payments of $10,487,443 at the beginning of each year. 2. The aircraft has a fair value of $80,000,000 on January 1, 2011, and an estimated economic life of 15 years. 3. American pays all executory costs. 4. American depreciates similar aircraft that it owns on a straight-line basis over 15 years. 5. The annual payments assure the lessor a 12 percent return. 6. American’s incremental borrowing rate is 12 percent. Chapter 21-84 LO 11Describe the lessee’s accounting for sale-leaseback transactions. Sale-Leaseback Example This lease is a capital lease to American because  lease term exceeds 75 percent of the estimated life of the aircraft and  present value of the lease payments exceeds 90 percent of the fair value of the aircraft to CitiCapital. Assuming that collectibility of the lease payments is reasonably predictable and that no important uncertainties exist in relation to unreimbursable costs yet to be incurred by CitiCapital, it should classify this lease as a direct-financing lease. Chapter 21-85 LO 11Describe the lessee’s accounting for sale-leaseback transactions. Sale-Leaseback Example Illustration 21B-1 Chapter 21-86 LO 11Describe the lessee’s accounting for sale-leaseback transactions. Sale-Leaseback Example Chapter 21-87 Copyright © 2009John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. 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1	https://math.fudan.edu.cn/_upload/article/files/8a/c7/b6bc60b64933acee041a351adda9/d0bb8522-a61d-4d6c-ab50-0051ba33a403.pdf	教案平面和直线教学内容平面和直线是几何学中最基本的研究对象，是一些向量空间和几何空间中某些对象的最基本原型。由于曲线在局部可以用它的切线来近似，曲面在局部可以用它的切平面来近似，所以平面和直线也是几何和分析中“以直代曲”的最基本元素。因此学习空间解析几何中处理平面与直线的方法非常重要，而且是必须要掌握的数学工具。在本节中主要讲解以下几方面的内容：（1）平面和直线的代数表示，即它们的方程的形式如何？以及如何用向量的运算方法来建立这些方程；（2）平面与平面，平面与直线、直线与直线，以及点与这些对象的位置关系，如距离、夹角等，以及如何利用代数方法来处理。教学思路和要求（1）首先讲解平面的方程、直线的方程、以及建立这些方程的方法。（2）讲解点与平面的距离、点与直线的距离公式的建立和运用。（3）讲解平面与平面，平面与直线、直线与直线的夹角概念与计算方法；（4）进一步，利用这些工具讲解一些具有某些特性的直线或平面的方程的具体计算方法。教学安排一．平面方程的几种形式在 3 R 中，给定了与平面垂直的方向和平面上的一个点，就可以唯一决定这个平面。与平面垂直的方向称为这个平面的法向量。设所求平面的法向量为 ) , , ( C B A n ，而且平面过点 0 P ( 0 0 0 , , z y x )。对平面上的任何一点 ) , , ( z y x P ，P 与 0 P 的连线依然在平面上，因而 ) , , ( 0 0 0 0 z z y y x x P P     与n 垂直，即 n P P 0 = 0。用分量表示，就是 0 ) ( ) ( ) ( 0 0 0       z z C y y B x x A ，这个关系式称为平面的点法式方程。记常数 ) ( 0 0 0 Cz By Ax D     ，则上述方程可以写成 0     D Cz By Ax ，这个关系式称为平面的一般方程。例6.2.1 求过原点 ) 0 , 0 , 0 ( O 和点 ) 2 , 3 , 6 (  P ，且与平面 8 2 4    z y x 垂直的平面方程。解记所求平面为。因为过原点 ) 0 , 0 , 0 ( O 和点 ) 2 , 3 , 6 (  P ，所以其法向量n应与 ) 2 , 3 , 6 (   OP 垂直。又垂直于平面 8 2 4    z y x ，所以n应与向量 z n P0 P  O y x 图6.2.1 ) 2 , 1 , 4 ( 1   n 垂直。故可取 k j i k j i n n 6 4 4 2 1 4 2 3 6 1         OP 。利用平面的点法式方程，所求平面方程为 0 6 4 4     z y x ，即 0 3 2 2    z y x 。确定平面的另一类条件是，不在一条直线上的3 个点唯一决定一张平面。设平面所过的3 个点为 0 P ( 0 0 0 , , z y x )， ) , , ( 1 1 1 1 z y x P ， ) , , ( 2 2 2 2 z y x P ，因此 1 0P P 和 2 0P P 与平面的法向量n 垂直，即可以取法向量 2 0 1 0 P P P P   n ，设P(x，y，z)是平面上的任何一点，则有 0 ) ( 0 2 0 1 0 0      P P P P P P P P n 。这称为平面的三点式方程。容易看出，它正好就是四点共面的条件 0 0 2 0 2 0 2 0 1 0 1 0 1 0 0 0           z z y y x x z z y y x x z z y y x x 。由行列式的性质，上式展开后一定是 0     D Cz By Ax 形式，因此实际计算时不必死记公式，可以将 0 P ， 1 P ， 2 P 的坐标直接代入平面的一般式方程，用待定系数法解出A，B，C，D。特别地，将平面所过的3 个点取为过坐标轴的点 0 P (a，0，0)， 1 P (0，b，0 )， 2 P (0，0，c )，代入三点式方程，就有 c a b a z y a x 0 0    = 0，展开整理后，得（1）当a、b、c 均不为0 时，平面方程为 1    c z b y a x ，它称为平面的截距式方程（见图6.2.2），其中a，b，c 依次称为该平面在x，y， z 轴上的截距。此时平面的法向为        c b a 1 , 1 , 1 n 。（2）当a，b，c 中只有一个为0 时，所决定的平面是坐标平面， . 0 ; 0 ; 0 0 0 0       z y x Oxy c Ozx b Oyz a z (0,0, c ) n O y (0,b,0) (a,0,0) x 图6.2.2 例6.2.2 求过点 ) 1 , 2 , 3 ( ， ) 0 , 1 , 0 ( ， ) 2 , 0 , 1 ( 的平面方程。解将三点的坐标代入平面的一般方程 0     D Cz By Ax ，得到关于A、B、C、D 的方程组                . 0 2 , 0 , 0 2 3 D C A D B D C B A 它的一组解为A= 3，B = -7，C= -2，D =7（此方程组有无穷多个解，只取一个解就可以了），于是所求的平面方程为 3x -7y -2z + 7 = 0。二．直线方程的几种形式与平面类似，要确定空间中的一条直线，主要条件也有两类。一类是确定直线的方向和直线上的一个点，另一类是确定直线上的两个点。设直线L 的方向向量为 ) , , ( n m l v ，它过点 0 P ( 0 0 0 , , z y x )。于是，直线L 上任何一点 ) , , ( z y x P 与 0 P 的连线与v 平行，即 v // 0P P （见图6.2.3），按分量写开，就是 n z z m y y l x x 0 0 0      ，它称为直线的对称式方程或点向式方程。注意，若l，m，n 中有等于0 的，例如，当 0  l ， 0 ,  n m 时，则应将上述方程理解为          . , 0 0 0 n z z m y y x x 当 0  m l ， 0  n 时，则应将上述方程理解为      . , 0 0 y y x x 例6.2.3 求过点 ) 4 , 1 , 2 (  ，方向向量为 ) 1 , 1 , 3 (  的直线方程。解直接代入直线的对称式方程，便得所求的直线方程为 1 4 1 1 3 2       z y x 。若给定了直线上的两个点 0 P ( 0 0 0 , , z y x )和 ) , , ( 1 1 1 1 z y x P ，则 1 0P P 的方向就是v 的方向向量（见图6.2.4），代入直线的对称式方程，即得到直线的两点式方程 z v P0 O P y x 图6.2.3 z v P1 P0 O y P x 图6.2.4 0 1 0 0 1 0 0 1 0 z z z z y y y y x x x x         。若在直线的对称式方程中记 t n z z m y y l x x       0 0 0 ，将等式写开，便得到            , , , 0 0 0 n t z z m t y y l t x x 它称为直线的参数方程，其中t 是参数。参数方程对于求解某些具体问题很有效。例6.2.4 求直线 2 4 1 3 1 2      z y x 与平面 0 6 2     z y x 的交点。解这就是求方程 2 4 1 3 1 2      z y x 与 0 6 2     z y x 的公共解。将直线方程写成参数方程            , 2 4 , 3 , 2 t z t y t x 其中t 是参数。代入平面的方程，便得到 2 (2+ t) + (3+ t) + (4 + 2 t) - 6 = 0，解得t = -1。代入直线的参数方程，得到 2 , 2 , 1    z y x 。即，交点为(1, 2, 2)。另外，如果给定了空间中两张互不平行的平面 1 ： 0 1 1 1 1     D z C y B x A 和 2 ： 0 2 2 2 2     D z C y B x A ，那么这两张平面交于一条直线。也就是说，这两个平面的联立方程            0 , 0 2 2 2 2 1 1 1 1 D z C y B x A D z C y B x A 同样表示一条直线，它称为直线的一般方程。直线的一般方程看起来不太直观，用起来有时也不太方便。由于 1 的法向量为 1 n ( 1 1 1 , , C B A )， 2 的法向量为 2 n ( 2 2 2 , , C B A )，由立体几何知识，直线的方向向量v 与 1 n 和 2 n 都垂直（见图6.2.5），因此，可以取   1 n v 2 n ，再从联立方程中求出一组解( 0 0 0 , , z y x )，也就是直线上的一个定点的坐标，这样就可以将它化成对称式方程了。例6.2.5 将直线的一般方程           0 4 2 , 0 1 2 z x z y x 化成对称式方程。解取直线的方向向量为 v n1 n2 2 1 图6.2.5    2 1 n n v 2 0 1 1 1 2  k j i = k j i  5 2 ，再在平面上任意取一个公共点，如令 0 0  x ，代入方程，         0 4 2 , 0 1 z z y 则可以解出 3 0   y ， 2 0   z 。于是，直线的对称式方程为 1 2 5 3 2       z y x 。例6.2.6 求过点 ) 2 , 0 , 0 ( 0  M ，与直线 1 L ： 1 2 3 4 1 z y x      相交，而且平行于平面 1 ： 0 1 2 3     z y x 的直线方程。解设所求直线为L 。其方向向量为 ) , , ( Z Y X v 。显然 ) 0 , 3 , 1 ( 1 M 是直线 1 L 上的点， ) 1 , 2 , 4 ( 1  v 是 1 L 的方向向量。由于L 过点 ) 2 , 0 , 0 ( 0  M ，且与直线 1 L 相交，因此向量 1 0M M ， 1 v 和v 共面，因此 0 ) ( 1 1 0    v v M M ，这就是说 0 1 2 4 ) 2 ( 0 0 3 0 1       Z Y X ，即 0 2    Z Y X 。又因为L 与平面 1 平行，所以 ) , , ( Z Y X v 与 1 的法向量 ) 2 , 1 , 3 (  n 垂直，因此 0  n v ，即 0 2 3    Z Y X 。联立上述各方程可解得 Z Y X 2 , 0   ，取 1  Z 得 2  Y 。因此直线L 的方程为 1 2 2 0    z y x ，即       . 0 , 4 2 x z y 本例也可先求过 0 M 且平行于 1 的平面 2 的方程，再求 2 与直线 1 L 的交点 1 M ，最后求出过 0 M 和 1 M 的直线方程。读者不妨自行计算。三．平面束空间直线L 的一般方程为            . 0 , 0 2 2 2 2 1 1 1 1 D z C y B x A D z C y B x A 对于任意一组不同时为零的常数  , ，方程 0 ) ( ) ( 2 2 2 2 1 1 1 1         D z C y B x A D z C y B x A   表示一张平面   。显然，满足L 的一般方程的点 ) , , ( z y x 一定满足平面   的方程，所以平面   通过直线L 。于是，对于不同的不同时为零的数对  , ， 0 ) ( ) ( 2 2 2 2 1 1 1 1         D z C y B x A D z C y B x A   就确定了一族通过L 的平面，它称为通过L 的平面的平面束，以上方程也称为通过L 的平面束方程。显然确定平面束中一张平面，只要确定与的比值，因此也常将通过L 的平面束方程写成 0 ) ( 2 2 2 2 1 1 1 1         D z C y B x A k D z C y B x A （注意这个束中不包含平面 0 2 2 2 2     D z C y B x A ），或 0 ) ( 2 2 2 2 1 1 1 1         D z C y B x A D z C y B x A k （注意这个束中不包含平面 0 1 1 1 1     D z C y B x A ）。另外，方程 0      Cz By Ax 确定一张平面  ，而当取不同值时，就得到一族相互平行的平面的方程。因此上式也称为平行平面束方程。例6.2.7 过点 ) 1 , 1 , 1 ( 和直线L ：            0 5 3 2 , 0 2 2 3 z y x z y x 的平面方程。解设所求通过L 的平面方程为 0 ) 5 3 2 ( 2 2 3         z y x k z y x 。它通过 ) 1 , 1 , 1 ( 点,所以将该点的坐标代入上式得 0 3 6  k 。所以 2  k 。于是，所求的平面方程为 0 ) 5 3 2 ( 2 2 2 3         z y x z y x ，即 0 8 8 5 5     z y x 。四．点到平面、直线的距离平面解析几何中讨论了某一平面上的点到直线的距离问题，现在我们将它推广到空间。先考虑点到平面的距离。设平面方程为 0     D Cz By Ax 。过空间一已知点 ) z , , ( y x P 作平面的垂线，显然，垂线的方向就是平面的法向量 ) , , ( C B A n 。取平面上一定点 0 P ( 0 0 0 , , z y x )，联结P 与 0 P ，则 ) z , , ( y x P 到平面的距离d 就是 P P 0 在n 方向的投影长度（见图6.2.6）。记平面的单位法向量 \|\| \|\| 0 n n n  ，由内积的定义，得 \| \| 0 0 n   P P d ， n P d P0 图6.2.6 用分量表示，就是 2 2 2 0 0 0 \| ) ( ) ( ) ( \| C B A z z C y y B x x A d         ，由于 ) ( 0 0 0 Cz By Ax D     ，因此 d = 2 2 2 \| \| C B A D Cz By Ax      ，这就是点 ) z , , ( y x P 到平面 0     D Cz By Ax 的距离的计算公式。当 ) z , , ( y x P 在该平面上时，显然有d = 0 。例6.2.8 求点 ) 1 , 1 , 2 (  与平面2x -3y -6z + 1 = 0 的距离。解这时 ) 1 , 1 , 2 (   P ， ) 6 , 3 , 2 ( ) , , (    C B A ，由点到平面距离的计算公式得 d = 2 2 2 ) 6 ( ) 3 ( 2 \| 1 ) 1 ( ) 6 ( 1 ) 3 ( 2 2 \|              = 7 8 。再考虑点到直线的距离。设直线L 的方程为 n z z m y y l x x 0 0 0      ，连接空间一已知点 ) z , , ( y x P 和直线L 上的点 ) , , ( 0 0 0 0 z y x P 。直线的方向向量为 ) , , ( n m l v ，因此单位方向向量为  0 v \|\| \|\| v v 。由图6.2.7 中可以看出，点P 到直线的距离d 是以 P P 0 和 0 v 为邻边的平行四边形的底边 0 v 上的高，由外积的几何意义，图6.2.7 中的平行四边形的面积 \|\| \|\| 0 v d S  \|\| \|\| 0 0 v   P P ，于是点 ) z , , ( y x P 到直线的距离公式为 \|\| \|\| 0 0 v   P P d 。例 6.2.9 求点 ) 3 , 2 , 5 (  与直线 3 2 1 1 1 z y x     的距离。解这时 ) 3 , 2 , 5 (   P ， ) 0 , 1 , 1 ( 0   P ，则 P P 0 = ) 3 , 1 , 4 ( ) 0 , 1 , 1 ( ) 3 , 2 , 5 (      。而 ) 3 , 2 , 1 (  v ，   \|\| \|\| 0 v v v 14 1 ) 3 , 2 , 1 ( ，因此   0 0 v P P 14 1 3 2 1 3 1 4  k j i 14 9   (1, 1, -1)，所以距离为 \|\|    \|\| \|\| 0 0 v P P d 14 3 9 。 P d P0 v0 图6.2.7 五．交角一．平面与平面的交角空间中两张平面的交角就是它们的法向量的交角（通常取 2 0    ）。因此，设两张平面的方程为 0 1 1 1 1     D z C y B x A 和 0 2 2 2 2     D z C y B x A ，那么它们的交角就是它们的法向量 ) , , ( 1 1 1 1 C B A n 和 ) , , ( 2 2 2 2 C B A n 的交角或补角，即  cos    \|\| \|\| \|\| \|\| \| \| 2 1 2 1 n n n n 2 2 2 2 2 2 2 1 2 1 2 1 2 1 2 1 2 1 \| \| C B A C B A C C B B A A        。特别地，（1）当 0 2 1 2 1 2 1    C C B B A A 时，这两张平面垂直；（2）当 2 1 2 1 2 1 2 1 D D C C B B A A    时，这两张平面平行；（3）当 2 1 2 1 2 1 2 1 D D C C B B A A    时，这两张平面重合。例6.2.10 求平面 0 3 2     z y x 和 0 1 2     z y x 的夹角。解此时 ) 2 , 1 , 1 ( 1   n ， ) 1 , 1 , 2 ( 2  n ，于是由平面与平面交角余弦计算公式得 2 1 1 1 2 2 ) 1 ( 1 \| 1 2 1 ) 1 ( 2 1 \| cos 2 2 2 2 2 2                ，因此，这两张平面的夹角为 3   。二．直线与直线的交角与平面与平面情况类似，空间中两条直线的交角就是它们的方向向量的交角 （通常取 2 0    ）。设两条直线的方程为 1 1 1 1 1 1 n z z m y y l x x      和 2 2 2 2 2 2 n z z m y y l x x      ，那么它们的交角（0≤≤2 ）满足 2 2 2 2 2 2 2 1 2 1 2 1 2 1 2 1 2 1 \| \| cos n m l n m l n n m m l l          。特别地，（1）当 0 2 1 2 1 2 1    n n m m l l 时，这两条直线垂直；（2）当 2 1 2 1 2 1 n n m m l l   时，这两条直线平行；（3）若 2 1 2 1 2 1 n n m m l l   ，且两条直线有一个公共点，则这两条直线重合。例6.2.11 求直线 1 1 4 2 1 1       z y x 和直线          0 2 2 , 0 1 z y y x 的夹角。解直线 1 1 4 2 1 1       z y x 的方向向量可取为 ) 1 , 4 , 1 ( 1   v ；直线          0 2 2 , 0 1 z y y x 的方向向量可取为 k j i k j i v       2 2 2 1 0 0 1 1 2 。利用直线与直线交角余弦计算公式得   cos 2 2 2 2 2 2 1 2 ) 2 ( 1 ) 4 ( 1 \| 1 1 2 ) 4 ( ) 2 ( 1 \|               = 2 1 ，因此，这两条直线的夹角为= 4 。三．平面与直线的交角直线与平面的交角是直线与它在平面上的垂直投影所夹的角（通常取 2 0    ，见图6.2.8）。设平面方程为 A x+ B y+ C z + D = 0，直线方程为 n z z m y y l x x 0 0 0      ，那么它们的交角（ 2 0    ）满足 2 2 2 2 2 2 \| \| \| cos \| sin n m l C B A Cn Bm l A            。特别地，（1）当 n C m B l A   时，平面与直线垂直；（2）当 0    Cn Bm Al 时，平面与直线平行；（3）若 0    Cn Bm Al ，且平面与直线有一个公共点，则直线属于平面。例6.2.12 求平面 0 3     z y x 和直线 4 3 1 2 3 2       z y x 的位置关系。解由于给定直线与平面的夹角的正弦为 0 ) 4 ( 1 3 1 1 1 \| ) 4 ( 1 1 1 3 1 \| sin 2 2 2 2 2 2                ， n v   图6.2.8 所以平面和直线平行。在直线 4 3 1 2 3 2       z y x 上任取一点，可以就取为 ) 3 , 2 , 2 (  ，它也满足平面方程 0 3     z y x ，所以 ) 3 , 2 , 2 (  也在平面上。因此，直线在平面上。六．习题 1，3．（1）、（3），4．（2），5．（1）、（3），6，7，9，11，13，15．（1）、（2）、（3）， 16。
2	https://www.scribd.com/document/704706131/Environmental-Engineering-by-Peavyrowe-Pdfdrive	Environmental Engineering by Peavyrowe Pdfdrive \| PDF \| Sewage Treatment \| Secondary Sector Of The Economy Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Close suggestions Search Search en Change Language Upload Sign in Sign in Download free for 30 days 0 ratings 0% found this document useful (0 votes) 5K views 214 pages Environmental Engineering by Peavyrowe Pdfdrive This document provides an overview of the McGraw-Hill textbook "Environmental Engineering" by Howard S. Peavy, Donald R. Rowe, and George Tchobanoglous. The textbook covers topics related to… Full description Uploaded by shakil1800151 AI-enhanced description Go to previous items Go to next items Download Save Save Environmental Engineering by Peavyrowe Pdfdrive For Later Share 0%0% found this document useful, undefined 0%, undefined Print Embed Ask AI Report Download Save Environmental Engineering by Peavyrowe Pdfdrive For Later You are on page 1/ 214 Search Fullscreen McGraw-Hill Series in Water Resources and Environmental Engineering Rolf Eliassen, Paul H. King, a nd Ray K. Linsley Consulting Editors Bailey and Ollis: Biochemical Engineering Fundamentals Bisho p: Marine Pollution and Its Control Biswa s: Model sfor Water Quality Managemeni Boc kra th: Environmental Law for Engineers. Scientisls, and Managers Bouw er: Gro {ndwater Hydrol ogy Canter: En vironmental Impact Assessment Chanlett: Environmental Pr ote ction Gaudy and Gaudy: Microbiolog yfor Environmental Scientists and Engineers Haimes: Hi erar ch ical Analysis of Water Resources Systems: Modelling and Optimization of Large-Scale System s Hall and Dracup: Wat er Resources Systems Engineering Linsley and Franzini: Water Resources Engineering Lins ley, Kohle r, an d P aulhus: Hydrology for Engineers Metcalf & Eddy, Inc.: Wastewater Engineering.' Collec t ion and Pumping of Wastewater Metcalf & Eddy, Inc.: Was tewater Engineering. Trealment, Disposal. Reuse Peavy, Rowe, and Tch oban oglou s: Environmental Engineering Ric h: Low-Maint enanc e, Mechan ically-Simpl e Wa ste water Treatment Systems Sawyer and McCarty : Chemistry for Environmental Engineering Steel and McGhe e: Water Supply and Sewerage Tcho bano glou s, The isen, a nd Eliassen: Solid Wastes, Engineering Principles and Management Issues ENVIRONMENTAL ENGINEERING Howard S. Peavy Professor of Civil Engineering Montana Scate University Donald R. Rowe Professor of Civil Engineering King Saud UniverSity Saudi Arabia George Tchobanoglous Professor of Civil Engineering Universi ty of Califo rnia, Davis McGraw-Hill Book C ompany New York St. Louis San Francisco Auckland Bogota H amburg London Madrid Mexico Montreal New Dehli Panama Paris Siio Paulo Singapore Sydney Tokyo To r onto adDownload to read ad-free ENVIRONMENTAL INTERNATIONAL EDITION 1985 Exclusive rights by McGraw-Hili Book Co., Singapor e for manufacture and export. This book ca nnot be r e-export ed from the country to which it is con s ign ed by McGraw -HilI. 40 39 38 37 36 35 34 33 32 31 15 14 13 12 11 CTP BJE Copyright © 1985 by Mc Graw-Hili, In c. All rights reserved. No part of this pub lica tio n m ay be re produ ced or dis tribut ed in any form or by any means, or stored in a data base or a retrieval sy stem, without the prior written perm ission of the publisher. This book was set in Tim es Roman. The editors were Kiran Verma and David A. Damstra. The production supe rvisor was Leroy A. Youn g. ... Libra./},.()f.c;:()1" 9.ress in Publication Data Peavy, Howard S. Env iro nmen t al e ngin ee rin g. (MC G raw-Hili se ries in water res our ces and enviro n mental engineering) I nclud es biblio graphi cal references and indexes. Environmental engin eeri ng. I. Rowe, Donald R. II. Tchobanoglou s, Geor ge . II I. Title. IV. Se ri e s. TD145.P43 1985 628 8 4- 3854 ISBN 0-07-04 91 34-8 When ordering this title use ISBN 0-07 -1 00231-6 Printed in Singapore CONTENTS Preface Intrn clu C i on I-I ·The Em 'i r on lll c nt 1-2 T he Im[lact 01 Il ulllam upon the Enviro nm ent 1-3 The Im[lact or th e En\ironm ent upon Hum ans 1- 4 I mpr o\ 'c ment of Environme ntal Qua li ty 1- 5 The Rolc of the Envir on men t al E ng in eer R eI- ercnces Part 1 Wat e r 2 Water Qua lit y: Def ini ti ons, C har a deri s tic s; and Persp ect ive s 2 -1 The Hydraul ic Cycle and Water Qual ity PH Y SICAL WATER- QUA LITY PARAME TER S 2-2 Su spended S,1 Iid , 2- 3 Turbi dit y 2-4 Color 2-5 Ta ste and O do l 2-6 T emper ature C HE M I CAL \ATE R -QUA LI TY PARAME TER S 2- 7 Che mi s tn of S"lutions' 2-8 Total Di ss oh cd Solid, 2-9 Alkali nit\ 2· 10 Har dness :2-1 1 Fl uo ri de 2· 12 Metals 2 -1 ] Org a nH :" 2- 14 Nutllc nh XI/ I 2 4 6 7 I I 12 1 4 15 17 1 8 20 22 23 23 28 3 1 3 (, 37 38 -1 4 adDownload to read ad-free vi CONTENTS BIOLOGICAL WATER-QU-\UTY PARAMETERS 2-15 Pathogens 2-16 Pathogen Indicators WATER QUALITY REQUIREMENTS 2-17 In-Stream Standards 2-18 Potable Water Standards 2-19 Wastewater Effluent Standards Discussion Topics and Problem s References . 3 Water Purification Processes in Natural Systems PHYSICAL PROCESSES 3-1 Dilution 3-2 Sedimentation and Resuspension 3-3 Filtration 3-4 Gas Transfer 3-5 Heat Transfer CHEMICAL PROCESSES 3-6 Chemical Conversions BIOCHEMICAL PROCESSES 3-7 Metabolic Processes 3-8 Microorganisms in Natural Wat er Systems RESPONSE OF STREM .. S TO BIODEGRADABLE. ORGANIC WASTE 3-9 Dissolved-Oxygen Balance 3-10 Dissolved-Oxygen Model 3-.11 Organic Discharge and Stream Ecology APPLICATION OF NATURAL .pROCESSES IN ENGINEERED SYSTEMS 3-12 Physical Processes 3- i3 Chemical Processes 3-14 Biological Processes Di s cu ss ion T o pics and Problem s Refere.nc es 4 Engineered Systems for Water Purification 4-1· Historical Overview of Water Treatment 4-2 Water-Treatment Processes 4-3 4-4 4 -5 4 -(, 4 -7 WA TER-TRFA 1M ENT PRO CES SE S. THEORY AND .APPLICATION Aeraiillil Solids Se paration \j)\ - \IJ) Settlin g Operatioll s Cllag ulati o ll S'\ - \G1 o ft e nin g 46 47 50 54 54 55 56 57 62 63 64 64 65 66 66 71 73 73 74 75 79 83 ....... .... ?;l. _ ...... . .. ... ..... . g5 94 104 105 107 109 110 11 3 In 1.1 1 151 ·4-8 Filtration '4-9 Di si nf ec ti on OTHER WATER-TREATMENT PROCESSES 4-10 Dissolved Solids Removal DiSCUSSIOn Topics and Problems References 5 Engineered Systems for Wastewater Treatment and Disposal 5 -1 Wastewater Characteristi cs 5-2 Effluent Standards 5-3 Terminology in Wastewater Treatment PRIMARY TREATMENT 5-4 Screening 5-5 Comminuting 5-6 Grit Removal 5-7 Flow Measurement 5-8 Primary Sedimentation SECONDARY TREATMENT 5-9 Growth and Food Utilization 5-10 Suspended-Culture System s 5·11 Activated Sludge 5-12 Ponds and Lagoons 5·13 Attached-Culture Systems 5-14 Secondary Clarification CONTENTS vii , t· 165 182 190 190 '104 . 207 208 211 212 2Pl 217 220 221 224 224 . Effiw:ots ..................................................... . 229 230 234 234 248 255 268 277 278 279 281 285 292 SLUDGE TREATMENT AND DISPOSAL 5-16 Sludge Characteristics 5-17 Sludge Thickening 5-18 Sludge Digestion 5-19 Sludge Disposal ADVANCED WASTEWATER TREATMENT 5 -20 Nutrient Removal 5- 21 Solids Removal WASTEWATER DISPOSAL AND REUSE 5-22 Wastewater Di s posal 5-23 Wastewater Reuse Discussion Topics and Problems References 6 Environmental Engineering Hydraulics Design WATER DISTRIBUTION SYSTEMS 6-1 Method s of Distributing Water 6-2 Di s tribut ion Re serv oir s 6 -3 Di s tribution Systems 294 295 :' 301 302 303 306 .314 322 324 324 324 326 331 adDownload to read ad-free -: - - """' '""""' -----.., v ii CONTENT S 6-4 6-5 <-6 6-8 6-9 6-10 6-11 < 6-12 6-13 6-14 6-15 6-16 6-17 6-18 6-19 6-20 6-21 6-22 6-23 6-24 6-25 6-26 6-27 Distribution System Components Capacity and Pres s ure Requirement s Design of Distribution Systems Hydraulic Analy sis of Di s tribution Systems Cross-Connections in Di s tr ib ution Systems Constructi on of Water Di s tribution Systems Pumping Re quired for Water Supply Syst ems WASTEWATER CO LLEC TION Types of Co llection Systems Types of Se wer s Collection System Appurten ances Basic Consideration s in th e De sign of Sewers Design of Sanitar y Sewers Preparation of Contact Drawings and Spe cif ica ti ons Construction of S ewe rs Maintenan ce of Sewers Design of Storm wa ter Sewers WATER AND WASTEWATER PUMPIN G Pump s Pump Drive Unit s Pump Application Te rminology and Usag e Pump Operating C haracte ristics and C ur ves Analysis of Pump S ys tem s Pump Stations for Water and Wa s tewater HYDRAULIC ANALYSIS OF WATER AN D WASTEWATER TREATMENT Treatment Plant De si gn Preparation of Hydraulic FFOfiks< <" < <" << " " <" 0< < -" , ---.. - .. ------.. - Discussion Topic s and Problem s References Part 2 . Air '7 Air Quality : Definitions, Characteristics , and Perspectives AIR POLLUTION -PAS T, PRESENT , AND FUTURE 7-1 Historical Overvi e>.' 7-2 Global Impl i ca tions of Air P ollut i on 7-3 Units of Measur em ent 7-4 Sources of poJlutan ts CLASSIFICATION OF POLLUTA N TS 7-5 Particul ates 7-6 H y droc a rb ons 7-7 Carbon Monoxide 7 -8 Oxides o f Sulfur 333 335 33 7 338 346 347 348 348 3 49 3 49 3 53 3 54 360 369 371 37 1 37 1 372 372 375 378 384 3')0 397 397 397 ,--399, 406 412 417 41 8 418 420 424 426 429 431 442 445 44 9 7-9 Ox ides of N itr ogen 7-10 Phot ochemi cal Oxidant s 7 -1 1 I nd oo r Air Pollut i on 8 8-1 go: i)-} 8 -4 8-5 8-6 8-7 8-8 8-9 8 -1 0 8 -11 A IR- QUA LIT Y MA N AGEMENT C ONCEPTS Discussion To pics a nd Problems References Me t eoro logy and Natural Pur ifi cat i on Pr ocesses EL EME NT AL PR OPERT IES Of THE ATMOSP H ERE Scales of i\ l otill n Heat Press ur e Wlncl MOis tur e Rclative H umidit y IN FLUE N CE OF i\ I LTEOROLOG IC AL PHENOMENA ON AIR QUALITY Lapse Rat es and DI '> persion Press ur e Sy s tems and Dispersion Wind s and Dispersio n M OiS tur e and Dispersion Mo d el ing EFFECTS or AIR P OLLUT ION ON METEOROLOGICAL COND ITIONS 8-12 Change s on the M esos cale and Mic r os cale __ . __ 13 _ _ 0 .11 _ he_ M aC .f Qs eale Discussion Topics a nd P robl ems Referenc es 9 Engineered S ys tems for Air Pollution Control 9-1 Atm o sph eric Cle a nsing Processes 9-2 Approa ches to Co ntaminant Cont rol CON TR OL DEVICES rOR PARTICULATE CONTAM INA NTS 9 -3 Gra vit ati onal Se l1 ing Ch a mb ers 9-4 Ce ntr ifu gal 9 -5 Wct Colkctnr , '-)- (, Fa hr ic Fi lt er, (BaglHllls e Fi lte rs) 9-7 Electr os tat ic PrccipiL lt () fS ( ESP )' CONT I WL DEV I CES FOR GASEOUS CON T AM I NANTS ') -8 Ad so rpti(lll 9-9 ,\ O Sorptll )Jl ,-) · 10 Conlicns "tloll 9- 11 (' omo u '>1lt)n , \utt .lllIl1 t l\r C ONTENTS ix 455 461 463 464 477 480 483 483 484 486 491 493 495 495 495 496 498 498 499· 499 507 508 509 510 5 12 5 14 5 14 516 518 520 5 23 5 28 53 3 53 6 5 40 540 545 557 559 56J adDownload to read ad-free r CONTENTS xi x CONTENTS , 11-10 Dete rmination of Vehicle and Labor Requirements 607 Discu ssion Topic s and Problems 565 II-II Collection Routes 615 References 567 TRANSFER AND TRANSPORT 618 r 11-12 Transfer Sta tio n s 620 11-13 Location of Tr ansfer Station s 622 ... Part 3 Solid Waste 11-14 Transfe r Means and Method s 622 PROCESSING TECHNIQUES 626 t 10 Solid Waste: Defin itio ns, Char acteri stics, and r I-IS Mechan ical Volume Reducti on 627 Perspectives 573 11 - 16 Thermal Volume Redu c ti on 627 TYPES OF SOLID WASTES 573 11. - 17 Manual Co mp onent Se par a ti on 627 r 10-1 Municipal Wastes 574 ULTIMATE DISPOSAL 628 10-2 Indu strial Wa stes 574 11 -18 Landfi ll ing with Solid Wa stes 628 10- 3 Hazard ous Waste s 5 75 11-19 De sign and Operation of Landfills 638 SOUR C ES OF SOLID WASTES 5 75 11- 20 Landf arming 646 10-4 Sour ces of Municipal Wastes 575 11-21 Deep-Well Injection 647 10-5 Source s of Hazardou s Wastes 576 Di scussion Topic s and Pro bl ems 648 Refer ence s 652 PROPERTIES OF SOLID WASTES 576 10-6 Ph ysical C ompo sition 576 10- 7 Chemical Co mpo sition 582 10-8 Chan ges in Compo sition 5 88 . 12 Engineered Systems for Resou rce and SOLID-WASTE MANAGEMENT AN O VERVIEW 588 Energy Recovery 653 10-9 Materials Flow in S ociet y 5 88 10-10 Reduction in Raw Mat e ri als Usage 58 9 PRO C ESSING TECHNIQUES 653 10-11 Re ducti on in Sol id-W as te Qu antit ie s 590 12- i Mechanical Size Alter ati on 65.4 10-12 Reuse of Solid-Wa ste Mat erials 5 90 12 -2 Mechanical Compon ent Separation 656 10-13 .. I'vlil.t el: iil.l. s . .... 5 91 12-3 Ma gne ti c and Ele ctromec ha ni cal Sep ara tion 656-·-.. 10-14 En ergy Recovery 592 12-4 Drying and Dewatering 657 10-15 Day-to-Day Solid- Wa ste Managem ent 592 MA TERIALS-RECOVER Y SYSTEMS 657 Discu ssion Topics and Pro blems 5 92 Materials Speci ficati on s 657 Refer ences 59 3 1 2- 5 12 -6 Proce ssing and Rec ov er y Sy" stems 657 )2 -7 System De sign Layout 659 11 Engineered Systems for Solid-Waste Management 5 94 RE C OVERY OF BIOLOGICAL CONVERSION P·RODUCTS 659 II-I Functional Elements 5 94 12-8 Compo sting (Aerobic Conv ersion) 660 SOLID WASTE GENERATION 594 17.-9 Ana erobic Digestion .663 11-2 Typical Gene ration Rat es 595 RECOVERY OF THERMAL CONVERSION PRODUCTS 665 11-3 Es timat ion of Solid-Wa ste Qu antit ies 598 12 -10 Co mbu stion of Wa ste Mat erials 665 11-4 Fact ors Th at Affect Genera ti on Rat cs 5 98 12 -11 In cine ration with Heat Re cove ry 670 ON-SITE HANDLING. STORAG E. 12-12 Use of R efus e-D e ri v ed Fuel s (RDF) 671 t\ N D PRO CESS IN G 598 1 2- 13 Ga sification 671 11 -5 On -Site Handlin g 59 9 Pyrol ys is 672 11 -6 On- S it e Storage 599 1 2- 14 RE C OVERY OF ENE RGY FROM CO N VERSION PRODUCTS 672 11 -7 On -S it e Process in g of So lid Wastes 6 01 1 2- I 5 Ener gy -Reco ve ry S ys tems 673 C OLLE C TIO N OF SOLID \Vt\ ST ES 6 01 674 11 -8 Co ll ec ti on Services 6 01 1 2- 16 Efficiency Fact or s 675 11 -9 Types of Co ll ec ti o ll Syst cm, 1i05 1 2- 17 Dete rrnin ati on of t.ner g} Output and EtJi cicncy adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free adDownload to read ad-free 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3	https://www.youtube.com/watch?v=rBmLMUVNlEo	Cross Price Affect and Effect of Substitute Goods \| Microeconomics Consumer Equilibrium and Demand Scholarslearning Classes 30200 subscribers 38 likes Description 2466 views Posted: 24 Mar 2017 Class 12 Microeconomics-Consumer Equilibrium and Demand by Parul Madan Transcript: in this video we will understand the cross right effect now what is the cross price effect see the effect of change in the price of related goods on the quantity demanded of a commodity the new types are related with number one India substitute good what I am constitute so your substitute goods are those who that can be replaced for one another means your tea or coffee this little rf4 mine water these are into water gram right deep adversity could you can help you to lead on right people who do not happy what is in power to purchase week's loss is costlier than the low quality diet at all since these 24 K T right right so in these these are we no substitutes you will substitute we turn right into the substitute T or coffee you can substitute this layer of equipment to these are no particles these can be replaced for one another the shaking hands in your complimentary me you require the one hood to complete the other good one good complement or complete the other word let's take an example of this pink is been your lap top and motherhood your full octave are an engine so in order to date this would be required this is good complemented without this at work and animal vacuum about eight paint anonymous have to do without motherhood lack of cannot even capture so these are your compliment Rickles in this video mean understand detective substitute words or be quantity demanded in the next you another Santa complimentary bananas if that's not the quantity demanded substitute goods and it effect so let us take an example of tea and coffee here are some stupid we will be the effect of quantity demanded of tea do no change in the Kaiser call see e is rupees 10 per 100 okay now the price of coffee initial was rupees 10 but now you can some factors has increased to 315 per standard car when Li try the coffee was Ruby ten per hundred now this quantity demanded of tea was less than 150 units for one country like now the price of coffee has increased so if this type of copies increase the quantity demanded of casa might have decreased so where those people have shifted they might have shifted some coffee to tea so ultimately quantity demanded off he would have increased let's schedule $300 not plot this on the graph this is legal right this is the quantity we are taking the quantity of TN right Roxy T remains constant that is till 2 feet of our clamp as Wendy try the formula 10 repeat before Sigma of T 1 then take it on 50 100 150 200 250 300 115 300 the initial demand 50 was now the democracy has increased this is increased due to increase in the price of the substitute saddle pocket so this is how the substitute words that is coffee effective see through a quantity demanded of tea so the increase in this price of one commodity increases the quantity demanded of other commodity that they are being subsidy see we understood that with the rise in the price of the top to bottom topography how does it affect tea it's finally the market increases now maybe I understand it will decrease in the price of the substitute is coffee let's say the coffee prices decrease to reach 8 100 ground the users of P will shift there a man to copy the this quantity demanded reduces Clemson 102 degrees rise and decreasing beside us coffee let's see effect to graphically familiar quantity of tea this is the friend of t see the price of steel constant what is the affecting its demand the price of the its substitute right so the price of the party constant 1,500 150 when you find a copy 110 the quantity demanded for 200 now the coffee prices we do so people have shifted for me to coffee so ultimately reducing the demand of T this is the initiative Markinson and the price of the scale now there is no him the shifted marker there is decrease in the quantity month right so this is the effect of degrees in defense of the city to your commodity
4	https://www.reddit.com/r/AskPhysics/comments/1h3agfa/is_earths_rotation_slowing_down_gradually/	Is earths rotation slowing down gradually? : r/AskPhysics Skip to main contentIs earths rotation slowing down gradually? : r/AskPhysics Open menu Open navigationGo to Reddit Home r/AskPhysics A chip A close button Log InLog in to Reddit Expand user menu Open settings menu Go to AskPhysics r/AskPhysics r/AskPhysics 1.5M Members Online •9 mo. ago PuddingWise3116 Is earths rotation slowing down gradually? I understand that the motion is what prevents the earth from falling directly into the sun. I also understand that Earth moves through a vacuum, so there is very little resistance. Are there there any other forces at play? I am also wondering whether electromagnetism affects earths kinetic energy in any way since moving electric charges (I perhaps naively assume earths electric charge isn't exactly 0) should lose energy. EDIT: I want to clarify that by rotation, I meant orbit around the sun. I apologise for not being clearer about that. Read more Share Related Answers Section Related Answers Understanding Earth's rotation and its forces Factors affecting Earth's rotation speed How does time dilation affect GPS satellites? Explaining black holes in simple terms How does gravity affect light? New to Reddit? Create your account and connect with a world of communities. Continue with Email Continue With Phone Number By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy. Public Anyone can view, post, and comment to this community Top Posts Reddit reReddit: Top posts of November 30, 2024 Reddit reReddit: Top posts of November 2024 Reddit reReddit: Top posts of 2024 Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved. Expand Navigation Collapse Navigation
5	https://www.longdom.org/articles/qpochhammer-symbol-a-versatile-mathematical-tool-with-farreaching-applications-101786.html	q-Pochhammer Symbol: A Versatile Mathematical Tool with Far-Reach Home Journals A-Z JournalsBrowse By Subject Guidelines & Policies Editorial PoliciesOnline SubmissionInstructions to AuthorsPoliciesPublication ethics & malpractice statementReviewersTerms and Conditions AdvertisingConferencesContact us Powered by Translate Mathematica Eterna Open Access ISSN: 1314-3344 Journal HomeEditorial Board Instructions for Authors Instructions for AuthorsPublication ethics & malpractice statement Submit ManuscriptAims and ScopeArticles in processArchiveContact Awards Nomination25+ Million Readerbase Google Scholar citation report Citations : 1892 Mathematica Eterna received 1892 citations as per Google Scholar report Indexed In Google Scholar Gdansk University of Technology, Ministry Points 5 View More » Useful Links Covid-19 Journal Articles IssuesAim and ScopePeer Review ProcessOther CommentsAdvertisingCitations ReportIndexing and ArchivingTable of ContentsSubmit PaperTrack Your PaperFunded Work Share This Page ##### 25+ Million Website Visitors Journal Flyer Open Access Journals Agri and AquacultureBiochemistryBioinformatics & Systems BiologyBusiness & ManagementChemistryClinical SciencesEngineeringFood & NutritionGeneral ScienceGenetics & Molecular BiologyImmunology & MicrobiologyMedical SciencesNeuroscience & PsychologyNursing & Health CarePharmaceutical Sciences Commentary Article - (2023)Volume 13, Issue 2 View PDFDownload PDF q-Pochhammer Symbol: A Versatile Mathematical Tool with Far-Reaching Applications Stepen MarcCorrespondence: Stepen Marc, Department of Mathematics, University of Orléans, Orléans, France, Email: Author info » Description The q-pochhammer symbol, also known as the q-shifted factorial or the q-rising factorial, is a mathematical concept that finds its applications in diverse areas such as number theory, combinatorics, and quantum algebra. the symbol is named after the german mathematician h. pochhammer, who made significant contributions to the field. the q-pochhammer symbol generalizes the conventional pochhammer symbol, introducing a parameter q that adds an additional level of complexity and versatility to its applications. The q-pochhammer symbol is denoted as (a;q)n, where a and q are parameters, and n is a non-negative integer. (a;q)n = (1-a)(1-aq)(1-aq^2)...(1-aq^(n-1)) Where a and q are parameters, and n is a non-negative integer. It is important to note that the value of n determines the number of factors in the product. The q-pochhammer symbol arises in the theory of q-series, which is a generalization of power series where the terms are multiplied by certain factors involving the parameter q . these series have proven to be fundamental in the study of various mathematical objects, such as partitions, partitions with restricted parts, and mock theta functions. One of the key properties of the q-pochhammer symbol is its connection to basic hyper geometric series. A basic hyper geometric series, also known as a q-series, is a series that generalizes the classical hyper geometric series. The q-pochhammer symbol can be used to express basic hyper geometric series in a compact and elegant form. The q-pochhammer symbol also exhibits many interesting properties and identities. one such property is the q-binomial theorem, which states that for any complex number a and parameter q : (1-a)^{-1} = (a;q){_infty} Where (a;q){_infty} is the infinite product representation of the q-pochhammer symbol. This result provides a powerful tool for manipulating q-series and has applications in various branches of mathematics, such as number theory and combinatorics. Furthermore, the q-pochhammer symbol satisfies a q-analog of the addition theorem for binomial coefficients. this q-analog, known as the q-vandermonde identity, states that for any complex numbers a ,b , and parameter q : (a+b;q)n= _sum_{k= 0}^n(a;q)k(b;q){n-k}(q;q) _k Where (q;q) _k is the q-pochhammer symbol with a-1 and b=q^k . The q-pochhammer symbol also plays a fundamental role in the theory of quantum groups and quantum algebra. Quantum groups are noncommutative analogs of lie groups and lie algebras, and they arise naturally in the study of certain physical phenomena, such as quantum mechanics and statistical mechanics. the q-pochhammer symbol appears in the representation theory of quantum groups and provides a framework for constructing and classifying irreducible representations. The q-pochhammer symbol is a powerful mathematical tool with applications in diverse areas of mathematics, including number theory, combinatorics, and quantum algebra. Its connection to basic hyper geometric series, properties such as the q-binomial theorem and the q-vandermonde identity, and its role in the theory of quantum groups highlight its importance in various mathematical disciplines. The study of the q-pochhammer symbol continues to be an active area of research, with ongoing investigations into its properties and applications, ensuring its relevance in the future development of mathematics. Author Info Stepen Marc Department of Mathematics, University of Orléans, Orléans, France Citation: Marc S (2023) q-Pochhammer Symbol: A Versatile Mathematical Tool with Far-Reaching Applications. Math Eterna.13: 183. Received:22-May-2023, Manuscript No. ME- 23-24472; Editor assigned:25-May-2023, Pre QC No. ME- 23-24472 (PQ); Reviewed:09-Jun-2023, QC No. ME- 23-24472; Revised:16-Jun-2023, Manuscript No. ME- 23-24472 (R); Published: 23-Jun-2023 , DOI: 10.35248/1314-3344.23.13.183 Copyright:© 2023 Marc S. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited Content Links Tools Feedback Careers Privacy Policy Terms & Conditions Authors, Reviewers & Editors Contact Longdom Longdom Group SA Avenue Roger Vandendriessche, 18, 1150 Brussels, Belgium Phone: +442038085340 Email:info@longdom.org Connect FacebookLinkedinTwitterInstagram Copyright © 2025 Longdom Publishing. Top Original text Rate this translation Your feedback will be used to help improve Google Translate
6	https://www.nature.com/articles/s41415-022-4280-0	Osteoradionecrosis of the jaw: a window of opportunity \| British Dental Journal Your privacy, your choice We use essential cookies to make sure the site can function. We also use optional cookies for advertising, personalisation of content, usage analysis, and social media. By accepting optional cookies, you consent to the processing of your personal data - including transfers to third parties. Some third parties are outside of the European Economic Area, with varying standards of data protection. See our privacy policy for more information on the use of your personal data. Manage preferences for further information and to change your choices. Accept all cookies Skip to main content Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. Advertisement View all journals Search Search Search articles by subject, keyword or author Show results from Search Advanced search Quick links Explore articles by subject Find a job Guide to authors Editorial policies Log in Explore content Explore content Research articles Reviews & Analysis News & Comment Current issue Collections Follow us on Facebook Follow us on Twitter Subscribe Sign up for alerts RSS feed About the journal About the journal Journal Information BDJ Portfolio Call for Papers Open access publishing About the Editors About the BDA CPD BDJ Books For Advertisers Jobs Contact Publish with us Publish with us For Authors For Referees Language editing services Open access funding Submit manuscript Subscribe Sign up for alerts RSS feed nature british dental journal research insight article Research Insight Published: 13 May 2022 Osteoradionecrosis of the jaw: a window of opportunity Sven Butzin1 British Dental Journalvolume 232,page 628 (2022)Cite this article 400 Accesses 1 Citations Metrics details Access through your institution Buy or subscribe An audit into the timing of dental extractions pre-head and neck radiotherapy and the prevalence of osteoradionecrosis. Br Dent J 2022; To investigate the relative timings of dental extractions compared to the RCS guidelines, Ward et al. conducted an audit including 154 patients who received RTX for HNC at Queen Alexandra Hospital in Portsmouth between July 2016 and August 2017. The results revealed that only 125 (81.2%) patients were dentally assessed before the commencement of RTX; nevertheless 30.5% more compared to 2017. Patients who were not dentally assessed were mainly edentulous patients or those undergoing palliative treatment. In total, 102 patients (81.6%) required dental extractions which were carried out within a mean time of 5.7 days. Only one patient in the investigated cohort breached the recommended ten-day healing period due to urgency of treatment. In fact, 77.5% of cases requiring extractions were allowed more than 21 days for healing, and overall, only two patients developed ORNJ (1.3% ORNJ rate); one despite a healing time of 23 days and the other following extractions during RTX after the patient had initially refused pre-RTX dental treatment. This is a preview of subscription content, access via your institution Relevant articles Open Access articles citing this article. Valero’s host index is useful in predicting radiation-induced trismus and osteoradionecrosis of the jaw risks in locally advanced nasopharyngeal carcinoma patients Erkan Topkan , Efsun Somay …Ugur Selek BMC Cancer Open Access 12 July 2023 Access options Access through your institution Subscribe to this journal Receive 24 print issues and online access $259.00 per year only $10.79 per issue Learn more Buy this article Purchase on SpringerLink Instant access to full article PDF Buy now Prices may be subject to local taxes which are calculated during checkout Additional access options: Log in Learn about institutional subscriptions Read our FAQs Contact customer support Author information Authors and Affiliations Final Year Dental Student and Medical Doctor, University of Central Lancashire, UK Sven Butzin Authors 1. Sven ButzinView author publications Search author on:PubMedGoogle Scholar Rights and permissions Reprints and permissions About this article Cite this article Butzin, S. Osteoradionecrosis of the jaw: a window of opportunity. Br Dent J232, 628 (2022). Download citation Published: 13 May 2022 Issue Date: 13 May 2022 DOI: Share this article Anyone you share the following link with will be able to read this content: Get shareable link Sorry, a shareable link is not currently available for this article. Copy shareable link to clipboard Provided by the Springer Nature SharedIt content-sharing initiative This article is cited by Valero’s host index is useful in predicting radiation-induced trismus and osteoradionecrosis of the jaw risks in locally advanced nasopharyngeal carcinoma patients Erkan Topkan Efsun Somay Ugur Selek BMC Cancer (2023) Access through your institution Buy or subscribe Associated content Collection Research Insights Sections Relevant articles Author information Rights and permissions About this article This article is cited by Advertisement British Dental Journal (Br Dent J) ISSN 1476-5373 (online) ISSN 0007-0610 (print) nature.com sitemap About Nature Portfolio About us Press releases Press office Contact us Discover content Journals A-Z Articles by subject protocols.io Nature Index Publishing policies Nature portfolio policies Open access Author & Researcher services Reprints & permissions Research data Language editing Scientific editing Nature Masterclasses Research Solutions Libraries & institutions Librarian service & tools Librarian portal Open research Recommend to library Advertising & partnerships Advertising Partnerships & Services Media kits Branded content Professional development Nature Awards Nature Careers Nature Conferences Regional websites Nature Africa Nature China Nature India Nature Japan Nature Middle East Privacy Policy Use of cookies Your privacy choices/Manage cookies Legal notice Accessibility statement Terms & Conditions Your US state privacy rights © 2025 Springer Nature Limited
7	https://artofproblemsolving.com/wiki/index.php/Summation?srsltid=AfmBOooZZZ88EH_7Pc8GcypHBKSe-SCFLipvLp5mKV1_15Cum7L3Lk29	Art of Problem Solving Summation - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Summation Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Summation A summation is the sum of a number of terms (addends). Summations are often written using sigma notation . Contents [hide] 1 Definition 2 Identities 3 Problems 3.1 Introductory 3.2 Intermediate 3.3 Olympiad 4 See Also Definition For , and a set (or any other algebraic structure), . Here refers to the index of summation, is the lower bound, and is the upper bound. As an example, . Note that if , then the sum is . Quite often, sigma notation is used in a slightly different format to denote certain sums. For example, refers to a cyclic sum, and refers to all subsets which are in . Usually, the bottom of the sigma contains a logical condition, as in , where the sum only includes the terms which divide into . Identities , and in general , and in general Or See PaperMath’s sums if you want to look deeper into these identities Problems Introductory Evaluate the following sums: Intermediate The nine horizontal and nine vertical lines on an checkerboard form rectangles, of which are squares. The number can be written in the form where and are relatively prime positive integers. Find (Source) Olympiad See Also Cyclic sum Symmetric sum This article is a stub. Help us out by expanding it. Retrieved from " Categories: Definition Stubs Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
8	https://math.stackexchange.com/questions/340744/what-do-the-symbols-d-dx-and-dy-dx-mean	Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams What do the symbols d/dx and dy/dx mean? Ask Question Asked Modified 3 years, 9 months ago Viewed 188k times 31 $\begingroup$ Okay this may sound stupid but I need a little help... What do $\Large \frac{d}{dx}$ and $\Large \frac{dy}{dx}$ mean? I need a thorough explanation. Thanks. calculus notation Share edited Jun 27, 2019 at 16:30 Asaf Karagila♦ 407k4848 gold badges646646 silver badges1.1k1.1k bronze badges asked Mar 25, 2013 at 15:42 KamiKami 47511 gold badge44 silver badges55 bronze badges $\endgroup$ 1 3 $\begingroup$ This is not precalculus, it is calculus. These symbols are derivatives. Are you familiar with derivatives?\ $\endgroup$ Alex Becker – Alex Becker 2013-03-25 15:44:00 +00:00 Commented Mar 25, 2013 at 15:44 Add a comment \| 5 Answers 5 Reset to default 28 $\begingroup$ The symbol $$ \frac{dy}{dx} $$ means the derivative of $y$ with respect to $x$. If $y = f(x)$ is a function of $x$, then the symbol is defined as $$ \frac{dy}{dx} = \lim_{h\to 0}\frac{f(x+h) - f(x)}{h}. $$ and this is is (again) called the derivative of $y$ or the derivative of $f$. Note that it again is a function of $x$ in this case. Note that we do not here define this as $dy$ divided by $dx$. On their own $dy$ and $dx$ don't have any meaning (here). We take $\frac{dy}{dx}$ as a symbol on its own that can't be slit up into parts. The symbol $$ \frac{d}{dx} $$ you can consider as an operator. You can apply this operator to a (differentiable) function. And you get a new function. So if $f$ is a (differentiable) function that it makes sense to "apply" $\frac{d}{dx}$ to $f$ and write $$ \frac{d}{dx}f $$ If you write $y = f(x)$, then this is the same as $$ \frac{d}{dx}y = \frac{dy}{dx}. $$ Share edited Mar 25, 2013 at 15:55 answered Mar 25, 2013 at 15:49 ThomasThomas 44.6k1313 gold badges7676 silver badges139139 bronze badges $\endgroup$ 5 10 $\begingroup$ The confusion often arises from the fact that many writers call $dy/dx$ a "symbol" as if it were atomic, but then later start doing algebra with it. This leads to the question, "well, then what is $dy$ really?" $\endgroup$ Fixee – Fixee 2013-03-25 15:55:55 +00:00 Commented Mar 25, 2013 at 15:55 2 $\begingroup$ @Fixee: That is right. I often see this confusion which is why I always emphasize that $\frac{dy}{dx}$ is just a symbol. It is not a fraction. $\endgroup$ Thomas – Thomas 2013-03-25 15:57:40 +00:00 Commented Mar 25, 2013 at 15:57 $\begingroup$ @Thomas It's kinda a fraction. It's the limit of a fraction. $\endgroup$ Akiva Weinberger – Akiva Weinberger 2015-05-01 21:35:01 +00:00 Commented May 1, 2015 at 21:35 $\begingroup$ (Or the "standard part" of a fraction, if you're doing nonstandard analysis...) $\endgroup$ Akiva Weinberger – Akiva Weinberger 2015-05-01 21:35:50 +00:00 Commented May 1, 2015 at 21:35 $\begingroup$ @Fixee See my answer. It is nevertheless CAN be defined as a fraction of two functions, rather than an atomic object. $\endgroup$ porton – porton 2015-05-01 21:59:02 +00:00 Commented May 1, 2015 at 21:59 Add a comment \| 8 $\begingroup$ $\frac{d}{dx}$ means differentiate with respect to $x$. $\frac{dy}{dx}$ means differentiate $y$ with respect to $x$. Do you have any concrete examples for which you need to calculate these two? It would probably make it more easy to grasp for you if I could explain it in a few examples. Share edited Jun 27, 2019 at 4:53 nmasanta 9,6902525 gold badges2929 silver badges5858 bronze badges answered Mar 25, 2013 at 15:44 dreamerdreamer 3,4591010 gold badges5252 silver badges7979 bronze badges $\endgroup$ 1 $\begingroup$ Maybe calculating the flow rate of a tap based on how the container gets filled? $\endgroup$ The Empty String Photographer – The Empty String Photographer 2023-03-23 07:57:46 +00:00 Commented Mar 23, 2023 at 7:57 Add a comment \| 3 $\begingroup$ $d f$ means the differential of function $f$. By definition $(df)(x) = \lambda t\in\mathbb{R}:f'(x)\cdot t$. In other words, differential is the linear function (of an additional variable denoted $t$ here) whose tangent is the derivative of $f$. $d$ alone means the differential operator (a function of argument $f$). Exercise: Show that $\frac{df}{dx}=f'$. Share edited May 1, 2015 at 21:30 answered May 1, 2015 at 21:12 portonporton 5,23511 gold badge2020 silver badges4949 bronze badges $\endgroup$ 2 $\begingroup$ Note that $\frac{F}{G}$ is defined for two functions $F$ and $G$ as $\frac{F}{G}(x)=\frac{F(x)}{G(x)}$. Thus $\frac{df}{dx}$ makes sense. $\endgroup$ porton – porton 2015-05-01 21:16:52 +00:00 Commented May 1, 2015 at 21:16 $\begingroup$ It is described in en.wikipedia.org/wiki/Differential_of_a_function $\endgroup$ porton – porton 2015-05-02 19:33:53 +00:00 Commented May 2, 2015 at 19:33 Add a comment \| 1 $\begingroup$ I like to look at it this way: $dx$ and $dy$ are just representations of change in accordance to either $x$ or $y$ axis. If you take the the symbol for derivative $$\frac{dy}{dx}$$ and compare it to the formula for the slope: $$\frac{f(x_1) - f(x_2)}{x_1 - x_2}$$ we can clearly see that $dy$ and $dx$ depict change in $y$ and change in $x$ respectively. Share answered Dec 28, 2021 at 15:02 StanleyStanley 5399 bronze badges $\endgroup$ Add a comment \| 0 $\begingroup$ If $y=f(x)$ i.e., where $y$ is the equation ( the dependent variable) and $x$ is the independent variable. Meaning $x$ changes $y$. Now $\frac{dy}{dx}$ means differentiate the equation $y$ in respect to $x$. $\frac{d}{dx}$ means differentiate in respect to $x$. Same way $\log x$ means find the natural logarithm of $x$, $\frac{d}{dx} x$ means find the derivative of $x$. N.B. In an equation $k= h²+5 $, $\frac{dk}{dh}$ means differentiate the equation $k$ in respect to $h$. Its not always $\frac{dy}{dx}$. I hope you understand Share edited Jun 27, 2019 at 4:40 nmasanta 9,6902525 gold badges2929 silver badges5858 bronze badges answered Jun 27, 2019 at 3:36 aisien osasumwenaisien osasumwen 11 $\endgroup$ 2 $\begingroup$ Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, use MathJax. $\endgroup$ dantopa – dantopa 2019-06-27 03:57:18 +00:00 Commented Jun 27, 2019 at 3:57 $\begingroup$ See the answer given by Thomas $\endgroup$ nmasanta – nmasanta 2019-06-27 04:52:47 +00:00 Commented Jun 27, 2019 at 4:52 Add a comment \| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions calculus notation See similar questions with these tags. 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9	https://famous-trials.com/animalrights/2595-the-case-against-animal-personhood	Published Time: Sun, 07 Sep 2025 15:52:37 GMT The Case AGAINST Animal Personhood Home The Case AGAINST Animal Personhood Details The Case AGAINST Recognizing Animals as "Persons" for Legal Purposes The Case AGAINST Animal Rights Several noted legal scholars have challenged the notion that certain animals should be recognized as “persons” entitled to at least some basic rights. Among the critics and skeptics are Judge Richard A. Posner who worries that listening “too raptly to the siren song of animal rights” might harm the interests not only of humans, but animals as well. [“Animal Rights”, 110 Yale L.J. (2000)]. Richard Epstein sees it as folly to argue other species should ever have legal rights approaching those we grant to humans. As Epstein puts it, “The blunt point is that we have, and will continue to have, different moral obligations to our own conspecifics than we do to chimps or members of any other species.” [Richard A. Epstein, "Animals as Objects, or Subjects, of Rights"(2002)]. Law professor Richard L. Cupp (Are all animal rights opponents named Richard?) argues that granting rights to animals would open the “floodgates” of litigation and that a far better focus for those who care for animals would be to seek more legislative protection against mistreatment of animals. [“Animals as More than ‘Mere Things,’ but Still Property: A Call for Continuing Evolution of the Animal Welfare Paradigm,” 82 Cinn. L. Rev (2016)]. The most compelling arguments AGAINST extending rights to animal are summarized below. 1.Cognitive capacity is neither a necessary nor a sufficient condition for having legally enforceable rights. Richard Posner observes that cognitive capacity is “a precondition of some rights, such as the right to vote.” But, he argues, we give rights to some humans with extremely little cognitive function. Posner writes: “Most of us would think it downright offensive to give greater rights to monkeys than to retarded people, upon a showing that the monkey has a greater cognitive capacity than a profoundly retarded human being, unless perhaps the human being has no brain function at all above the autonomic level, that is, is in a vegetative state.” (RP, 532) Why should cognitive capacity be the basis for personhood?—there really is no good reason. Posner points out: “Most people would think it distinctly odd to proportion animal rights to animal intelligence, as Wise wishes to do, implying that dolphins, parrots, and ravens are entitled to more legal protection than horses (or most monkeys), and perhaps that the laws forbidding cruelty to animals should be limited to the most intelligent animals, inviting the crack ‘They don't have syntax, so we can eat them.’” (RP, 532) Posner also asks, if cognitive capacity is the basis for determining personhood, “What about computers?” Posner points out: “Someday, perhaps soon, there will be computers that have as many ‘neurons’ as chimpanzees, and the ‘neurons’ will be ‘wired’ similarly. Such computers may well be conscious. This will be a problem for Wise, for whom the essence of equality under law is that individuals with similar cognitive capacities should be treated alike regardless of their species.” (RP, 531) Posner also notes: “Nature is not valued by environmentalists for its mental attributes, and so the environmentalist is unlikely to want to give special protection to chimpanzees, dolphins, and other highly intelligent animals.” (RP, 535-36) Richard Epstein argues that cognitive capacity has never been the essential basis for granting human rights: “The movement for equal rights of all human beings must take into account the fact that all people do not have anything remotely like the same cognitive abilities.” He notes that lack of cognitive capacity has been a justification for granting extra protection to certain humans, not less: “What fate lies for adult human beings whose mental disabilities in fact preclude them from taking advantage of many of the rights they are afforded? Our standard position is to give them extra protection, not to exterminate them, and to do so because they are human beings, entitled to protection as such. It follows therefore we should resist any effort to bootstrap legal rights for animals on the change in legal rights of women and slaves. There is no next logical step to restore parity between animals on the one hand and women and slaves on the other.” (RE, 17) Epstein, like Posner, sees no good reason for using cognitive capacity as the measure of rights. He quotes the utilitarian philosopher Jeremy Bentham: “The question is not, Can they reason? Nor, Can they talk? but, Can they suffer?” But Epstein sees legal efforts directed towards reducing animal suffering as fraught with another set of problems: “Our intervention to prevent suffering is, however, usually confined to questions of how human beings ought to interact with animals, and there the problems are difficult enough.” (RE, 36) STEVEN WISE'S RESPONSE:We humans don't like to be in prison. Imprisonment destroys your liberty, autonomy, and self-determination, your ability to go where you want to go and do what we want to do and with whom. These are basic human attributes. If you lack legal rights there is nothing you can do when someone else deprives you of your liberty. You become their slave. Thus judges and legislators value your right to autonomy and self-determination; they value your liberty so much and have for so long that they developed something called "habeas corpus," which is Latin for "you have the body." Habeas corpus is so important that it has long been referred to as "The Great Writ." Its single purpose is to test the legality of a "person's" imprisonment. By "person" the law does not mean "human being," but any entity that has the capacity for any legal right. That can be a human being. But it can be many other entities, too. The important thing is that if you are not a "person" you are just a "thing," unable to have any legal rights. Personhood is not connected to biology. In some countries, rivers, national parks, a religious idol, a religion's holy books, even the Amazon rainforest have been designated as "persons" with certain legal rights. On the other hand, at times humans such as fetuses, slaves, and women were not allowed to use habeas corpus because the courts believed they weren't "persons."Until recently all nonhuman animals were automatically seen as the "things" they have been for two thousand years. There was nothing they could do when a human deprived them of their liberty or other things they cared about. That is finally changing. The first nonhuman animals are starting to be recognized as "persons" in some countries. Some US states have enacted Pet Trust Statutes by which some or all nonhuman animals are given the legal rights of trust beneficiaries. The Nonhuman Rights Project began showing one way to establish the personhood and right to liberty of at least those nonhuman animals who are autonomous and self-determining when it started filing test cases in 2013 demanding that their nonhuman animal clients - so far chimpanzees and elephants - be given the right to bodily liberty.The NhRP brings in the world's greatest experts to explain to the judge how extraordinarily cognitive complex and autonomous its clients actually are. It never argues that autonomy and self-determination are required for legal personhood, just that those qualities should qualify their holders for the right to bodily liberty that is protected by habeas corpus, whoever they might be. Judge Richard A. Posner 2. There is no principled way to draw a line between animals that should be recognized as “persons” (at least for habeas corpus purposes) and those that should not be given such recognition—and judges need to draw lines. Animal rights advocates such as Steven Wise argue that chimpanzees (and bonobos and other primates) should be recognized as “persons” entitled to their freedom. He would say the same thing about elephants and whales and dolphins. And, no doubt, he believes that petitions for habeas corpus should be available to an intelligent bird species such as the African Grey Parrot. The octopus is a highly intelligent (albeit possessed with an almost alien sort of intelligence) creature—maybe an octopus should be a person? And then what about border collies or pigs? You see the problem. No judge would order a border collie freed from its owner and no judge would liberate all pigs on an Iowa hog farm. Nor would many, if any, judges be ready to order parrots freed from their owners and sent to an aviary, no matter how wonderful that change might be for the parrot. Richard Posner puts the problem this way: “Wise wants judges, in good common-law fashion, to move step by step, and for the first step simply to declare that chimpanzees have legal rights. But judges asked to step onto a new path of doctrinal growth want to have some idea of where the path leads, even if it would be unreasonable to insist that the destination be clearly seen. Wise gives them no idea.” (RP, 532) Epstein makes essentially the same point: “[If] the line between humans and chimps is no longer decisive, then some other line has to be. Perhaps it is the line between chimps and great apes, or between both and horses and cows, or between horses and cows and snails and fish. Which of these lines are decisive and why?” (RE, 22) Richard Cupp worries that the lack of an obvious place to draw a line will lead to excessive litigation: “If the legal wall between animals and humans is broken through, [that will open] a floodgate of expansive litigation without a meaningful standard for determining how many of the billions of animals in the world are intelligent enough to merit personhood.” (Cupp brief, 22) Of course, even if there might be disagreement precisely where the line should be drawn, we instinctively understand that some animals deserve more legal protections than others. Epstein acknowledges that fact: “The more animals look and act like human beings, the greater the level of protection that we as humans are willing to afford them. Rights of bodily integrity do not have much of a future for mosquitoes. Second, the higher the species ranks on own tree of life, the stronger the justifications that must be advanced in order to harm members of that species. Cost aside, we would be wholly inappropriate to think that we should capture or breed chimpanzees for food, whatever our views on their use for medical experimentation.” (RE, 26) STEVEN WISE'S RESPONSE:The English poet, Alexander Pope, wrote in his 1734 “An Essay On Man” that “Whatever is, is right.” Law professor Christopher Stone noted in his 1972 book, “Should Trees Have Standing” that “(t)he fact is that each time there is a movement to confer rights onto some new 'entity,' the proposal is bound to soundodd or frightening, or laughable,” while Justice Anthony Kennedy of the United States Supreme Court reminded us in 2003 that “times can blind us to certain truths and later generations can see that laws once thought necessary and proper in fact serve only to oppress.” Not long ago in Western law, only white people had rights., not blacks or indigenous peoples. Only men had rights, not women. Only adults had rights, not children. The claim that those lines needed to be redrawn, that blacks should have rights or women or children, sounded “odd or frightening or laughable” to many. Each claim was stoutly resisted for decades, sometimes centuries, by those blind to the true meaning of such fundamental values that we nearly all agree with as liberty and equality by their times. Today many, including Posner, Epstein, and Cupp, are similarly blind to the claim that a nonhuman can have any rights by their times. Our times. The irony is they argue that any new line must be shown to be decisive, oblivious that the ancient line that exists between humans and the rest of the universe was based upon outdated and disproven biology and superseded moral values that long ago rendered it arbitrary and unjust. 3. Once we recognize some rights for animals, courts will be drawn into endless cases turning on whether animals should be given some other right—and some rights, almost everyone agrees, should be rights for humans only. Epstein makes this argument: “The natural cognitive and emotional limitations of animals, even the higher animals, preclude any creation of full parity [with humans]. What animal can be given the right to contract? To testify in court? To vote? To participate in political deliberation? To worship?” (RE, 18) Epstein also argues that granting rights to animals will raise questions as to whether we have a legal obligation to intervene in nature to protect them. As he puts it: “Do we train the lion to lie down with the lamb, or do we let the lion consume the lamb in order to maintain his traditional folkways? Do we ask chimpanzees to forgo eating monkeys. It is odd to intervene in nature to forestall some deadly encounters, especially if our enforced nonaggression could lead to extermination of predator species. But, if animals have rights, then how do we avoid making these second-tier judgments?” (RE, 19) Posner also argues that starting down the path towards animal rights will lead to a judicial quagmire, thicket, or worse: “But what is meant by liberating animals and giving them the rights of human beings of the same cognitive capacity? Does an animal's right to life place a duty on human beings to protect animals from being killed by other animals? Is capacity to feel pain sufficient cognitive capacity to entitle an animal to at least the most elementary human rights? What kinds of habitats must we create and maintain for all the rights-bearing animals in the United States? Does human convenience have any weight in deciding what rights an animal has? Can common-law courts actually work out a satisfactory regime of animal rights without the aid of legislatures? When human rights and animal rights collide, do human rights have priority, and if so, why? And what is to be done when animal rights collide with each other, as they do with laws that by protecting wolves endanger sheep? Must entire species of animals be ‘segregated’ from each other and from human beings, and, if so, what does ‘separate but equal’ mean in this context? May we ‘discriminate’ against animals, and if so, how much? Do species have "rights," or just individual animals, and if the latter, does this mean that according special legal protection for endangered species is a denial of equal protection? Is domestication a form of enslavement? Wise does not try to answer any of these questions. He is asking judges to set sail on an uncharted sea without a compass.” (RP, 533) STEVEN WISE'S RESPONSE: These are all “straw man” arguments in that no one actually makes them or is likely to make them. Yet they are raised to prevent nonhuman animals from possessing legal rights against those humans who actually harm them. No one asks whether human fetuses should have any rights – they certainly have many rights now - because the law would then have to grapple with whether they should testify in court or have the right to vote or the right to contract or participate in political deliberation or have the right to worship. No one asks whether children should have any rights and for the same reasons. Humans have no legal rights to be protected from being attacked by nonhuman animals or hit by lightning or hurricanes or having trees fall on them. The rights of nonhuman animals, like the rights of humans, are rights against other humans and no one claims differently. Moreover the claim that, as judges would have to grapple with new legal claims, an entity should have no rights at all, is unprecedented. And every time lawyers advocate for a new human right those who will be unable to exploit other humans try to block it. Who believes that humans should not take steps toward preventing climate change or environmental damages just because later steps might have more serious economic impacts on those who are causing climate change or ruining the environment or which involve more complex questions? Finally, the only job judges have is to decide. Their compass, as always, is the fundamental values that have long guided judges in deciding cases involving humans. Richard Epstein 4. Many animals benefit from being treated as property and might actually fare less well if treated as “persons” entitled to their freedom. Another way of framing this argument is to ask: “Who’s to say animals want to be liberated?” Organizations like the Nonhuman Rights Project might say that they speak for the animals, but obviously there are some difficulties when it comes to determining the precise wishes of an animal. It is easy to imagine that in some cases freeing an animal from what appears to be a sub-optimal situation might lead to an even worse situation. Richard Epstein puts the argument this way: “Animals that are left to their own devices may have no masters; nor do they have any peace. Life in the wild leaves them exposed to the elements; to attacks by other animals; to the inability to find food or shelter; to accidental injury; and to disease. The expected life of animals in the wild need not be solitary, poor, nasty, brutish and short. But it is often rugged, and rarely placid and untroubled. . . . Because they use and value animals, owners will spend resources for their protection. Veterinary medicine may not be at the level of human medicine, but it is only a generation or so behind. When it comes to medical care, it’s better to be a sick cat in a middle-class United States household than a sick peasant in a third-world country. Private ownership of many pets gives them access to food and shelter (and sometimes clothing) which creates long lives of ease and comfort.” (RE, 10) STEVEN WISE'S RESPONSE:Humans often need to persuade themselves and others that the entities we enslave for our selfish purposes actually benefit from their enslavement. In one 18 th century debate in the English House of Lords, one Lord claimed that slaves in the Caribbean lived in a state of humble happiness. Another said slaves appeared so happy to that he often wished to be a slave himself. In the 19 th century United States Vice-President John C. Calhoun said “(n)ever before has the black races of Central Africa, from the dawn of civilization to the present day attained a condition so civilized and so improved, not only physically, but morally and intellectually.” After hearing evidence in a 2012 case alleging that elephants in the Los Angeles Zoo were being treated badly, a Los Angeles judge noted that the senior elephant keeper engaged in an “anthropomorphic fantasy that the elephants are happy to see her and live their lives in captivity” and observed that “(c)aptivity is a terrible existence for any intelligent self-aware species, which the undisputed evidence shows elephants are. To believe otherwise, as some high-ranking employees appear to believe, is delusional.” 5. If we don’t limit personhood to human beings, but grant personhood to other species based on their cognitive capacities, the net result might be that the law will give less respect to human rights and human dignity. Richard Posner makes this argument: “It is that if we fail to maintain a bright line between animals and human beings, we may end up by treating human beings as badly as we treat animals, rather than treating animals as well as we treat (or aspire to treat) human beings. Equation is a reflexive relation. If chimpanzees equal human infants, human infants equal chimpanzees.” (RP, 535) Richard Cupp makes this argument a central part of his amicus brief against extending rights to animals: “The pervasive view that all humans have distinctive and intrinsic human dignity regardless of their capabilities may have cultural, religious, or even instinctual foundations. Morally autonomous humans have unique natural bonds with other humans who have cognitive impairments, and thus denying rights to them also harms the interests of society—we are all in a community together. Infants’ primary identities are as humans, and adults with severe cognitive impairments’ primary identities are as humans who are other humans’ parents, siblings, children or spouse. However, good intentions sometimes create disastrous results. There should be deep concern that over a long horizon, allowing animal legal personhood based on cognitive abilities could unintentionally lead to gradual erosion of protections for these especially vulnerable humans. The sky would not immediately fall if courts started treating chimpanzees as persons. . . .But, over time, both the courts and society might be tempted not only to view the most intelligent animals more like we now view humans but also to view the least intelligent humans more like we now view animals. (Cupp brief, 14-18). Cupp concludes: “Deciding chimpanzees are legal persons based on the cognitive abilities we have seen in them may open a door that swings in both directions regarding rights for humans as well as for animals, and later generations may well wish we had kept it closed." (Cupp brief, 21) STEVEN WISE'S RESPONSE:Enormous concern is expressed that granting any legal rights to any nonhuman animal “may”, “could,” or “might” cause us humans to treat other humans as badly as we treat nonhuman animals, though no reasons are given for why that could happen, other than another “may” and a “sometimes,” with some pop psychology thrown in, and no claim is made that such a thing has ever happened. That is because it never has happened. When the question arose as to whether women should have the vote no one argued that if they got the vote some men might end up losing their right to vote vote. When the abolition of black chattel slavery arose no one claimed that some white people could end up being enslaved. And nowhere in this thicket of great unlikelihoods can one locate any concern about changing the admittedly bad way we humans actually treat nonhuman animals. And that’s a certainty. Richard Cupp 6. When push comes to shove, we have to favor our own species and cannot give anything like equal rights to another species. Our first moral obligation is to our own species. Richard Epstein raises the issue thusly: “Suppose that we manufacture limited supplies of a new pill that is a cure for some disease that is ravaging both human and chimpanzee populations. There is not enough to go around for both man and beast. Is there some kind of affirmative duty to assist chimps to the same extent that we assist other human beings? I should be stunned if any real world scenario would ever produce any result other than humans first, chimps second.” Epstein concludes: “The blunt point is that we have, and will continue to have, different moral obligations to our own conspecifics than we do to chimps or members of any other species.” (RE, 23) Gary Francione asks whether “we cannot prefer human over animals in situations of true emergencies or conflicts.” In his book, Your Child or the Dog? Francione asks how an individual should choose between saving the life of his or her child or his or her dog if both are trapped inside a burning house. Epstein argues the right answer is obvious: “The child, darn it, even if the child is unrelated and the dog is one’s own.” (RE, 23) Epstein agrees that line-drawing is uncomfortable, even though necessary: “The root of our discontent is that in the end we have to separate ourselves from (the rest of) nature from which we evolved. Unhappily but insistently, the ‘collective’ we is prepared to do just that. Such is our lot, and perhaps our desire, as human beings.” (RE, 27) Richard Posner believes that the special status humans accord themselves is not based on empirical evidence or logic. Rather, he says, it is based on “a moral intuition deeper than any reason that could be given for it and impervious to any reason that you or anyone could give against it.” (Posner debate with Peter Singer, 2001). STEVEN WISE'S RESPONSE:This sort of line-drawing by which powerful line-drawers accord themselves a special status that carries with it the entitlement to enslave others has been made many times throughout history, Egyptians of Hebrews, Greeks of barbarians, Romans of Greeks, Christians of indigenous peoples and Muslims, Muslims of Christians, and whites of blacks, the last exemplified perhaps most famously, and execrably, in Chief Justice Taney’s opinion in the Dred Scott case. The slave-holding Taney, referring both to the legal status of black people in the 1780s and in the present, wrote that blacks “had for more than a century before been regarded as beings of an inferior order, and altogether unfit to associate with the white race, either in social or political relations; and so far inferior, that they had no rights which the white man was bound to respect; and that the negro might justly and lawfully be reduced to slavery for his benefit … This opinion was at that time fixed and universal in the civilized portion of the white race. It was regarded as an axiom in morals as well as in politics, which no one thought of disputing, or supposed to be open to dispute; and men in every grade and position in society daily and habitually acted upon it in their private pursuits, as well as in matters of public concern, without doubting for a moment the correctness of this opinion … a perpetual and impassable barrier was intended to be erected between the white race and the one which they had reduced to slavery, and governed as subjects with absolute and despotic power, and which they then looked upon as so far below them in the scale of created beings …” Don E. Fehrenbacher noted in his Pulitzer Prize winning book, The Dred Scott Case,“Taney’s statement was to place Negroes … on the same level, legally, as domestic animals.” (p. 349). It is time that nonhuman animals themselves are no longer placed on the same level as nonhuman animals. 7. The concept of rights is essentially human and can have force and applicability only in the human moral world. The philosopher Carl Cohen puts the argument this way: “Animals cannot be the bearers of rights because the concept of right is essentially human; it is rooted in the human moral world and has force and applicability only within that world.” (Carl Cohen & Tom Regan, The Animal Rights Deba te. 2001, p. 30). Richard Cupp argues that because chimps should not be legally accountable, they also should not be given legal rights: “The pertinent question is not whether chimpanzees possess anything that could be characterized as a sense of responsibility, but rather whether they possess a sufficient level of moral agency to be justly held legally accountable as well as to possess legal rights under our human legal system. When, in 2012, an adult chimpanzee at the Los Angeles Zoo beat a three-month-old baby chimpanzee in the head until the baby died, doubtless no authorities seriously contemplated charging the perpetrator in criminal court." (Cupp brief, 5--6) STEVEN WISE'S RESPONSE:In 2018 New York Court of Appeals Judge Eugene Fahey, the only high court judge in the United States who has opined to date on whether nonhuman animals should have any legal rights, wrote in a habeas corpus case brought on behalf of a chimpanzee, that the lower appellate court had “reasoned that chimpanzees are not persons because they lack ‘the capacity or ability ... to bear legal duties, or to be held legally accountable for their actions. (The Nonhuman Rights Project and) law professors Laurence H. Tribe, Justin Marceau, and Samuel Wiseman question this assumption. Even if it is correct, however, that nonhuman animals cannot bear duties, the same is true of human infants or comatose human adults, yet no one would suppose that it is improper to seek a writ of habeas corpus on behalf of one’s infant child or a parent suffering from dementia. In short, being a ‘moral agent’ who can freely choose to act as morality requires is not a necessary condition of being a ‘moral patient’ who can be wronged and may have the right to redress wrongs. The (lower appellate court’s) conclusion that a chimpanzee cannot be considered a ‘person’ and is not entitled to habeas relief is in fact based on nothing more than the premise that a chimpanzee is not a member of the human species … The better approach … is to ask not whether a chimpanzee fits the definition of a person or whether a chimpanzee has the same rights and duties as a human being, but instead whether he or she has the right to liberty … Moreover, the answer to that question will depend on our assessment of the intrinsic nature of chimpanzees as a species. The record before us … contains unrebutted evidence, in the form of affidavits from eminent primatologists, that chimpanzees have advanced cognitive abilities and demonstrate autonomy by self–Initiating intentional, adequately informed actions, free of controlling influences.” 8. Most all of what animal rights proponents seek to accomplish can be accomplished better by seeking additional legislative protection against animal cruelty. Posner argues that seeking legislative protection is “more conservative, methodologically as well as politically, but possibly more efficacious, than rights-mongering.” He contends we should “extend, and move vigorously to enforce, laws designed to prevent gratuitous cruelty to animals. We should be able to agree without help from philosophers and constitutional theorists that gratuitous cruelty is bad.” (RP, 539) Posner writes: “Wise is another deer frozen in the headlights of Brown v. Board of Education.” Wise overlooks better approaches to securing animal welfare through legislative action or redefining property rights. Posner suggests we should simply “forbid treating chimpanzees, or any other animals with whom we sympathize, cruelly.” Animal rights, as Posner sees it, is “just an impediment to clear thought as well as a provocation in some legal and philosophical quarters.” (RP, 540) Richard Cupp agrees that the focus should be on legislative change to protect animal welfare. He approves of the language of New York’s Third Department, which refused to extend personhood to Tommy in the Lavery appeal: “Our rejection of a rights paradigm for animals does not, however, leave them defenseless,” noting that advocates for animals remained "free to importune the Legislature to extend further legal protections to chimpanzees.” (Cupp brief, 25). STEVEN WISE'S RESPONSE:Centuries of legal history have demonstrated that the only way in which a human’s most fundamental interests can be protected is through fundamental legal rights. That is why such international human rights documents as the Universal Declaration of Human Rights and the International Convention on Civil and Political Rights require that every individual be deemed a “person.” I invite any humans who claim that it would better for nonhuman animals to forgo all legal rights and be protected solely by welfare statutes to give up all their legal rights. That way even your most fundamental interests will be protected solely by human welfare statutes. As with animal welfare statutes, you too will be unable to enforce the statutes, their enforcement will be left to the complete discretion of a public prosecutor, and any prosecution will lead at best to the violator being convicted of a crime, with no remedy coming to you.. If you are unhappy you can always “importune the legislature for further legal protections” that don’t amount to legal rights. Not interested? I thought not. HomeTrial Account Other Resources Animal Rights on Trial: An Account Animals and Humans: A Brief History Animal Rights (Personhood) Trials: A Chronology The Case FOR Animal Personhood The Case AGAINST Animal Personhood Comparing Species: Autonomy, Intelligence, and Morality First Petition for Habeas Corpus (Tommy, a chimpanzee vs Patrick Lavery) What Chimps Can Do: The Cognitive Capacities of Chimpanzees Transcript: Hearing on Personhood for Herecules and Leo, Two Chimps NhRP Petition for for Writ of Habeas Corpus for Three Elephants (2017) What Elephants Can Do: The Cognitive Capacities of Elephants Transcript: Hearing on Personhood for Happy, an Elephant The James Sommersett Case (1772) Animal Rights Debate: Peter Singer vs Richard Posner Philosopher Tom Regan on Animal Rights Are All Species Created Equal? Animal Rights on Trial: Bibliography and Links Donate to Famous-Trials.com:With your help, Famous-Trials.com can expand and update its library of landmark cases and, at the same time, support the next generation of legal minds from UMKC School of Law. Donate Now Powered by UMKC School of Law Copyright © 1995 - 2025 Professor Douglas O. Linder To Top
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11	https://math.stackexchange.com/questions/4824606/circles-formed-by-points-of-common-tangency-of-two-circles	geometry - Circles formed by points of common tangency of two circles - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Circles formed by points of common tangency of two circles Ask Question Asked 1 year, 9 months ago Modified1 year, 9 months ago Viewed 457 times This question shows research effort; it is useful and clear 2 Save this question. Show activity on this post. I was playing around in Geogebra with circles and their common tangents. For any two random circles (yellow), I noticed that the four points of tangency of direct common tangents form a circle, and so do the four points of tangency of the transverse common tangents. The two circles (blue) are concentric, and their common centre M also happens to be the midpoint of the line segment joining the centres of the original two circles. My question: how do I find the radii of the concentric circles, just using the radii of the original circles and the distance between them? (Also, are there any more interesting properties of these circles, which have been documented somewhere?) geometry euclidean-geometry circles tangent-line Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications edited Dec 12, 2023 at 8:41 algorhythmalgorhythm asked Dec 11, 2023 at 17:12 algorhythmalgorhythm 77 7 7 bronze badges Add a comment\| 3 Answers 3 Sorted by: Reset to default This answer is useful 2 Save this answer. Show activity on this post. The well-known geometric construction of direct common tangent, which is shown in the diagram below, can be used to easily determine the dimeter of the mentioned circle that passes through the four points of tangency. To construct the direct common tangents, we need to draw two auxiliary circles. One of those circles has the radius R−r R−r and the center C 1 C 1. The other is drawn with the radius d 2 d 2 and has the mod point of C 1 C 2 C 1 C 2 as its center. One of their points of intersection is Q Q. Draw the segment C 1 Q C 1 Q and then extend it so that it cuts the circle Ω 1 Ω 1 at P 1 P 1 as shown in the diagram. We also draw a line segment joining Q Q and C 2 C 2. To get the direct common tangent, draw a line parallel to Q C 2 Q C 2 through P 1 P 1 to through the circle Ω 2 Ω 2 at P 2 P 2. As per the construction, we can state that, C 1 Q=R−r,(0)(0)C 1 Q=R−r, P 1 P 2=Q C 2 and(1)(1)P 1 P 2=Q C 2 and ∡C 2 Q C 1=90 o.(2)(2)∡C 2 Q C 1=90 o. Now, mark N N as the midpoint of P 1 P 2 P 1 P 2. Since M M is the midpoint of C 1 C 2 C 1 C 2, M N M N is parallel to both C 1 P 1 C 1 P 1 and C 2 P 2 C 2 P 2. Therefore, we have, M N=1 2(C 1 P 1+C 2 P 2)=1 2(R+r).(3)(3)M N=1 2(C 1 P 1+C 2 P 2)=1 2(R+r). Because ∡C 2 Q C 1=90 o∡C 2 Q C 1=90 o, the triangle C 1 C 2 Q C 1 C 2 Q is right-angled. Using (0), we apply Pythagoras theorem to this triangle to obtain, Q C 2 2=C 1 C 2 2−C 1 Q 2=d 2−(R−r)2.(4)(4)Q C 2 2=C 1 C 2 2−C 1 Q 2=d 2−(R−r)2. Since Q C 2 P 2 P 1 Q C 2 P 2 P 1 is a rectangle, using (4), we shall write, P 1 P 2=Q C 2=d 2−(R−r)2−−−−−−−−−−−√.P 1 P 2=Q C 2=d 2−(R−r)2. N N is the midpoint of P 1 P 2 P 1 P 2. Therefore, P 1 N 2=1 4 P 1 P 2 2=1 4(d 2−(R−r))2.(5)(5)P 1 N 2=1 4 P 1 P 2 2=1 4(d 2−(R−r))2. Finally, using (1) and (5), apply Pythagoras theorem to the right-angled triangle M N P 1 M N P 1 to express M P 1 M P 1 in terms of the diameters of the two circles Ω 1 Ω 1 and Ω 2 Ω 2, and the distance d d between their centers. M P 2 1=M N 2+P 1 N 2==1 4(R+r)2+1 4(d 2−(R−r))2 M P 1 2=M N 2+P 1 N 2==1 4(R+r)2+1 4(d 2−(R−r))2 a=M P 1=1 2 d 2+4 R r−−−−−−−−√a=M P 1=1 2 d 2+4 R r Added at OP's request - Transverse Common Tangent: Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Dec 12, 2023 at 18:50 answered Dec 12, 2023 at 13:17 YNKYNK 4,650 1 1 gold badge 12 12 silver badges 18 18 bronze badges 1 (+1) Thanks! Is it possible to also determine the radius of the circle formed by the transverse common tangency points?algorhythm –algorhythm 2023-12-12 13:49:53 +00:00 Commented Dec 12, 2023 at 13:49 Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. Let C 1=(x 1,y 1)C 1=(x 1,y 1) and C 2=(x 2,y 2)C 2=(x 2,y 2). Call the points of tangency on one of the direct tangents A 1 A 1 and A 2 A 2. Then because each of the segments A 1 C 1 A 1 C 1 (of length r 1 r 1) and A 2 C 2 A 2 C 2 (of length r 2 r 2) is perpendicular to that tangent, they are parallel. So if O O is the intersection of the two direct tangents, then △O A 1 C 1△O A 1 C 1 and △O A 2 C 2△O A 2 C 2 are similar right triangles. The former has one leg r 1 r 1 and hypotenuse h h, And the latter has one leg r 2 r 2 and hypotenuse h+(x 1−x 2)2+(y 1−y 2)2−−−−−−−−−−−−−−−−−−√.h+(x 1−x 2)2+(y 1−y 2)2. Solving for h h will allow you to find ∠C 1 C 2 A 2∠C 1 C 2 A 2, and thus the coordinates of A 2 A 2, which, of course, allows you to calculate the length of M A 2 M A 2. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications answered Dec 11, 2023 at 17:54 Paul TanenbaumPaul Tanenbaum 2,621 9 9 silver badges 19 19 bronze badges 1 (+1). Is it possible to find the radius using a more Euclidean approach? Like without finding the coordinates.algorhythm –algorhythm 2023-12-11 18:09:39 +00:00 Commented Dec 11, 2023 at 18:09 Add a comment\| This answer is useful 1 Save this answer. Show activity on this post. We use this property that the distances of points of intersection of the circle , passing the tangency points, with the line connecting the centers of two circles EN (in figure points P and O) from the centers of two circles are equal( in figure AP=CO=0.41). In particular case where the projection of point of tangency H on EN is coincided with reflect of point P about point A (the center of biger circle) we have AC, A H=r A A H=r A and C F=r C C F=r C as known of problem and we can write: ∠Q C A=∠P′H A∠Q C A=∠P′H A ⇒A P=P′A A H=r A=r A−r C A C⇒A P=P′A A H=r A=r A−r C A C in this way A P=C O A P=C O can be foud and the diameter of the circle passing the tangency points is: D=A C+2 A P D=A C+2 A P Update: to calculate the radius we can also use the power of E to the circle it's radius is required: E F×E H=E S 2 E F×E H=E S 2 t a n H E A=t a n Q C A=r A−r C C Q t a n H E A=t a n Q C A=r A−r C C Q s i n Q C A=r A−r C A C s i n Q C A=r A−r C A C These relations give measure of Q C=F H Q C=F H ∠F E C=∠Q C A∠F E C=∠Q C A E H=r c t a n F E C E H=r c t a n F E C E H=E F+F H E H=E F+F H E S 2=E F×E H E S 2=E F×E H E A=r A t a n H E A E A=r A t a n H E A E C=r c t a n H E A E C=r c t a n H E A E R=E A+E C 2 E R=E A+E C 2 Finally: r=R S=E R 2−E S 2−−−−−−−−−−√r=R S=E R 2−E S 2 Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Dec 12, 2023 at 14:40 answered Dec 12, 2023 at 7:52 siroussirous 13.1k 3 3 gold badges 16 16 silver badges 25 25 bronze badges Add a comment\| You must log in to answer this question. 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