Plane Geometry - Bruce E. Shapiro

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304 SECTION 51. THE POINCARE DISK MODELisoceles and β = β ′ . Hence ∠CRP = π − 2β (measured in radians). Thus∠P RO = 2β. Henceθ = π 2 − 2βtan θ 2 = tan ( π4 − β )= 1 − tan β1 + tan βSince tan β = OP = (e d − 1)/(e d + 1) by theorem 51.5,tan θ 2 = 1 − ed − 1e d + 11 + ed − 1e d + 1= e −d« CC BY-NC-ND 3.0. Revised: 18 Nov 2012
Section 52Arc LengthIn this section we begin our consideration of geometry on the surface of asphere. We first need to figure out what we mean by a “line” on the surfaceof a sphere. Since we would like out geometry to satisfy as m much ofneutral geometry as possible (we will find that it is not possible to satsifyall the postulates of neutral geometry in spherical geometry), we will beginwith the result that the distance between two points is the length of a linecontaining those two points. By the triangle inequality, we know that wecan not construct any shorter path between those two points by any pairof line segments. Applying this concept repeatedly, any path composed ofline segments will have an overall length at least as long as the length of asingle segment.Figure 52.1: The segment AB is the shortest path between A and B inneutral geometry.305
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Foundations of GeometryLecture Note
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ivCONTENTS30 Triangles in Neutral G
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2 SECTION 1. PREFACEare enrolled in
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4 SECTION 1. PREFACEre-learning it
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6 SECTION 2. NCTM3. To guide in the
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8 SECTION 2. NCTMTechnology. Techno
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10 SECTION 2. NCTM« CC BY-NC-ND 3.
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12 SECTION 3. CA STANDARDIn 2005 Ca
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14 SECTION 3. CA STANDARDThe Califo
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16 SECTION 3. CA STANDARDCalifornia
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18 SECTION 4. COMMON COREAt all lev
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20 SECTION 5. LOGIC AND PROOFour th
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22 SECTION 5. LOGIC AND PROOFRene D
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24 SECTION 5. LOGIC AND PROOF“I
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26 SECTION 6. REAL NUMBERSThe inter
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28 SECTION 6. REAL NUMBERS“The Su
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30 SECTION 7. EUCLID’S ELEMENTSEu
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32 SECTION 7. EUCLID’S ELEMENTSFi
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34 SECTION 8. HILBERT’S AXIOMSiom
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36 SECTION 8. HILBERT’S AXIOMS«
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38 SECTION 9. BIRKHOFF/MACLANE AXIO
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40 SECTION 9. BIRKHOFF/MACLANE AXIO
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42 SECTION 10. THE SMSG AXIOMSP is
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44 SECTION 10. THE SMSG AXIOMS« CC
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46 SECTION 11. THE UCSMP AXIOMSin s
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48 SECTION 11. THE UCSMP AXIOMS« C
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50 SECTION 12. VENEMA’S AXIOMS3.
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52 SECTION 12. VENEMA’S AXIOMS«
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54 SECTION 13. INCIDENCE GEOMETRYFi
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56 SECTION 13. INCIDENCE GEOMETRYEx
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58 SECTION 13. INCIDENCE GEOMETRYFi
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60 SECTION 13. INCIDENCE GEOMETRY«
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62 SECTION 14. BETWEENNESSFigure 14
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64 SECTION 14. BETWEENNESSTheorem 1
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66 SECTION 14. BETWEENNESS(a) To sh
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68 SECTION 14. BETWEENNESSThe follo
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70 SECTION 14. BETWEENNESSExample 1
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72 SECTION 14. BETWEENNESSorf(A) >
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74 SECTION 14. BETWEENNESS« CC BY-
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76 SECTION 15. THE PLANE SEPARATION
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78 SECTION 15. THE PLANE SEPARATION
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80 SECTION 16. ANGLESFigure 16.1: T
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82 SECTION 16. ANGLESCorollary 16.6
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84 SECTION 16. ANGLESFigure 16.6: I
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86 SECTION 16. ANGLESD cannot be in
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88 SECTION 17. THE CROSSBAR THEOREM
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90 SECTION 17. THE CROSSBAR THEOREM
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92 SECTION 18. LINEAR PAIRSFigure 1
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94 SECTION 18. LINEAR PAIRSγ + ∠
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96 SECTION 19. ANGLE BISECTORSwith
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98 SECTION 19. ANGLE BISECTORSAngle
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100 SECTION 20. THE CONTINUITY AXIO
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102 SECTION 20. THE CONTINUITY AXIO
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104 SECTION 21. SIDE-ANGLE-SIDEBirk
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106 SECTION 21. SIDE-ANGLE-SIDEFigu
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108 SECTION 22. NEUTRAL GEOMETRYEll
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110 SECTION 22. NEUTRAL GEOMETRY«
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112 SECTION 23. ANGLE-SIDE-ANGLEFig
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114 SECTION 23. ANGLE-SIDE-ANGLE«
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116 SECTION 24. EXTERIOR ANGLESTheo
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118 SECTION 24. EXTERIOR ANGLES« C
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120 SECTION 25. ANGLE-ANGLE-SIDEBy
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122 SECTION 26. SIDE-SIDE-SIDECase
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124 SECTION 26. SIDE-SIDE-SIDEBy si
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126 SECTION 27. SCALENE AND TRIANGL
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128 SECTION 27. SCALENE AND TRIANGL
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130 SECTION 28. CHARACTERIZATION OF
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132 SECTION 28. CHARACTERIZATION OF
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134 SECTION 29. TRANSVERSALSFigure
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136 SECTION 29. TRANSVERSALSCorolla
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138 SECTION 30. TRIANGLES IN NEUTRA
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146 SECTION 31. QUADRILATERALS IN N
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156 SECTION 32. THE EUCLIDEAN PARAL
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168 SECTION 33. RECTANGLESLemma 33.
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170 SECTION 33. RECTANGLESFigure 33
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172 SECTION 33. RECTANGLES« CC BY-
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174 SECTION 34. THE PARALLEL PROJEC
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176 SECTION 34. THE PARALLEL PROJEC
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178 SECTION 35. SIMILAR TRIANGLESBy
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180 SECTION 35. SIMILAR TRIANGLESCo
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182 SECTION 36. TRIANGLE CENTERSFig
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184 SECTION 36. TRIANGLE CENTERSThe
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186 SECTION 36. TRIANGLE CENTERSCen
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188 SECTION 37. AREADefinition 37.5
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190 SECTION 37. AREAToday’s Lesso
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192 SECTION 38. THE PYTHAGOREAN THE
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202 SECTION 39. CIRCLESFigure 39.1:
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208 SECTION 39. CIRCLESTheorem 39.1
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210 SECTION 39. CIRCLESLet D be the
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212 SECTION 39. CIRCLES« CC BY-NC-
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214 SECTION 40. CIRCLES AND TRIANGL
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220 SECTION 41. EUCLIDEAN CIRCLESFi
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228 SECTION 41. EUCLIDEAN CIRCLESBy
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230 SECTION 42. AREA AND CIRCUMFERE
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238 SECTION 43. INDIANA BILL 246the
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240 SECTION 44. ESTIMATING πFigure
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242 SECTION 44. ESTIMATING πso tha
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244 SECTION 44. ESTIMATING πn π n
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246 SECTION 44. ESTIMATING πTable
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248 SECTION 44. ESTIMATING πnums =
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250 SECTION 44. ESTIMATING π« CC
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252 SECTION 45. EUCLIDEAN CONSTRUCT
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Page 268 and 269: 264 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 288 and 289: 284 SECTION 49. TRIANGLES IN HYPERB
Page 294 and 295: 290 SECTION 50. AREA IN HYPERBOLIC
Page 302 and 303: 298 SECTION 51. THE POINCARE DISK M
Page 310 and 311: 306 SECTION 52. ARC LENGTHAs we see
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Page 324 and 325: 320 APPENDIX A. SYMBOLS USEDAppendi
Page 326 and 327: 322 APPENDIX A. SYMBOLS USEDTechnol
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Plane Geometry - Bruce E. Shapiro

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