Plane Geometry - Bruce E. Shapiro
Plane Geometry - Bruce E. Shapiro Plane Geometry - Bruce E. Shapiro
318 SECTION 53. SPHERICAL GEOMETRYProof. Define the x − axis to pass through vertex A and place C on theequator in the xy plane.Let α be the central angle determined by ˜BC. Hence a = αr.Let β be the central angle determined by ÃC. Hence b = βr.Let γ be the central angle determined by ÃB. Hence c = γr.Then the coordinates of the three vertices in R 3 are⎫A = (r, 0, 0)B = (r cos β sin(π/2 − α), ⎪⎬r sin β sin(π/2 − α),cos(π/2 − α) ⎪⎭C = (r cos β, r sin β, 0)(53.3)HenceSince |A| = |B| = r,cos γ = A · B|A||B|cos γ = cos β sin(π/2 − α) = cos β cos αSubstituting the equations a = αr, b = βr, and c = γr gives the desiredresult.For the law of cosines and law of sines we assume that in △ABC the sidesopposite vertices A, B and C have central angles a, b, and c, respectively(their arc lengths are ar, br, and cr).Theorem 53.12 (Spherical Law of Sines) 1sin asin A = sin bsin B = sin csin CTheorem 53.13 (Spherical Laws of Cosines) 2cos a = cos b cos c + sin b sin c cos Acos b = cos c cos a + sin c sin a cos Bcos c = cos a cos b + sin a sin b cos C1 Attributed to Abul Wafa Buzjana (940-998)2 Often attributed to Jamshid al-Kashi (1380-1429) who put it into its present form,although it was likely already know much earlier.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
319
- Page 272 and 273: 268 SECTION 46. HYPERBOLIC GEOMETRY
- Page 274 and 275: 270 SECTION 46. HYPERBOLIC GEOMETRY
- Page 276 and 277: 272SECTION 47.PERPENDICULAR LINES I
- Page 278 and 279: 274SECTION 47.PERPENDICULAR LINES I
- Page 280 and 281: 276 SECTION 48. PARALLEL LINES IN H
- Page 282 and 283: 278 SECTION 48. PARALLEL LINES IN H
- Page 284 and 285: 280 SECTION 48. PARALLEL LINES IN H
- Page 286 and 287: 282 SECTION 48. PARALLEL LINES IN H
- Page 288 and 289: 284 SECTION 49. TRIANGLES IN HYPERB
- Page 290 and 291: 286 SECTION 49. TRIANGLES IN HYPERB
- Page 292 and 293: 288 SECTION 49. TRIANGLES IN HYPERB
- Page 294 and 295: 290 SECTION 50. AREA IN HYPERBOLIC
- Page 296 and 297: 292 SECTION 50. AREA IN HYPERBOLIC
- Page 298 and 299: 294 SECTION 50. AREA IN HYPERBOLIC
- Page 300 and 301: 296 SECTION 50. AREA IN HYPERBOLIC
- Page 302 and 303: 298 SECTION 51. THE POINCARE DISK M
- Page 304 and 305: 300 SECTION 51. THE POINCARE DISK M
- Page 306 and 307: 302 SECTION 51. THE POINCARE DISK M
- Page 308 and 309: 304 SECTION 51. THE POINCARE DISK M
- Page 310 and 311: 306 SECTION 52. ARC LENGTHAs we see
- Page 312 and 313: 308 SECTION 52. ARC LENGTHProof. Su
- Page 314 and 315: 310 SECTION 52. ARC LENGTHThus the
- Page 316 and 317: 312 SECTION 52. ARC LENGTHMeasuring
- Page 318 and 319: 314 SECTION 53. SPHERICAL GEOMETRYW
- Page 320 and 321: 316 SECTION 53. SPHERICAL GEOMETRYF
- Page 324 and 325: 320 APPENDIX A. SYMBOLS USEDAppendi
- Page 326 and 327: 322 APPENDIX A. SYMBOLS USEDTechnol
319