Chapter 4: Geometry

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¼ Æ · · ¿¼ Æ · · ½¼ Æ ½¼ Æ · · ¼ Æ ¡Ê ¾ ½¼ ØÒ ÖØÒ ½ × ¾ ØÒ ½ ´× ¾ µ ØÒ ½ ´× ¾ µØÒ ½ ´× µ ¾ ½¼ Æ ¼ ½¼ Æ ¼ ½¼ Æ ¼ ½¼ Æ ¼ ½¼ Æ ¼ ½¼ Æ ¼ 4.19.2.1 Spherical law of sines ×Ò ×Ò ×Ò ×Ò ×Ò ×Ò 4.19.2.2 Spherical law of cosines for sides Ó× Ó× Ó× ·×Ò ×Ò Ó× Ó× Ó× Ó× · ×Ò ×Ò Ó× Ó× Ó× Ó× · ×Ò ×Ò Ó× 4.19.2.3 Spherical law of cosines for angles Ó× Ó× Ó× · ×Ò ×Ò Ó× Ó× Ó× Ó× ·×Ò ×Ò Ó× Ó× Ó× Ó× ·×Ò ×Ò Ó× 4.19.2.4 Spherical law of tangents ØÒ ½ ´ µ ¾ ØÒ ½ ´ ¾ · µ ØÒ ½ ´ µ ¾ ØÒ ½ ´ · µ ¾ ØÒ ½ ´ µ ¾ ØÒ ½ ´ ¾ · µ ØÒ ½ ´ µ ¾ ØÒ ½ ´ · µ ¾ ØÒ ½ ´ µ ¾ ØÒ ½ ´ ¾ · µ ØÒ ½ ´ µ ¾ ØÒ ½ ´ · µ ¾ 4.19.2.5 Spherical half angle formulae Define ¾ ´ØÒÖµ ¾ ×Ò´× µ×Ò´× µ×Ò´× µ . Then ×Ò × © 2003 by CRC Press LLC
ØÒ ØÒ ØÒ ¾ ¾ ¾ ×Ò´× µ ×Ò´× (4.19.1) µ ×Ò´× µ 4.19.2.6 Spherical half side formulae Define Ã ¾ ´ØÒ Êµ ¾ Ó× Ë Ó×´Ë µ Ó×´Ë µ Ó×´Ë . Then µ ØÒ´¾µ Ã Ó×´Ë µ ØÒ´¾µ Ã Ó×´Ë µ ØÒ´¾µ Ã Ó×´Ë µ (4.19.2) 4.19.2.7 Gauss’s formulae ×Ò ½ ´ µ ¾ ×Ò ½ ×Ò ½ ´ µ ¾ Ó× ½ ¾ ¾ ×Ò ½ ´ · µ ¾ ×Ò ½ Ó× ½ ´ µ ¾ ×Ò ½ ¾ ¾ Ó× ½ ´ µ ¾ Ó× ½ ×Ò ½ ´ · µ ¾ Ó× ½ ¾ ¾ Ó× ½ ´ · µ ¾ Ó× ½ Ó× ½ ´ · µ ¾ ×Ò ½ ¾ ¾ 4.19.2.8 Napier’s analogs ×Ò ½ ´ µ ¾ ×Ò ½ ´ ¾ · µ ØÒ ½ ´ µ ¾ ØÒ ½ ¾ Ó× ½ ´ µ ¾ Ó× ½ ´ ¾ · µ ØÒ ½ ´ · µ ¾ ØÒ ½ ¾ ×Ò ½ ´ µ ¾ ×Ò ½ ´ ¾ · µ ØÒ ½ ´ µ ¾ ÓØ ½ ¾ Ó× ½ ´ µ ¾ Ó× ½ ´ ¾ · µ ØÒ ½ ´ · µ ¾ ÓØ ½ ¾ 4.19.2.9 Rules for determining quadrant 1. If , then . 2. A side (angle) which differs by more than ¼ Æ from another side (angle) is in the same quadrant as its opposite angle (side). 3. Half the sum of any two sides and half the sum of the opposite angles are in the same quadrant. © 2003 by CRC Press LLC
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Chapter Geometry 4.1 COORDINATE SY
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4.19.2 Oblique spherical triangles
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FIGURE 4.1 Change of coordinates by
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4.1.4.1 Relations between Cartesian
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This formula also covers passing fr
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EXAMPLE To nd the matrix for a rota
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Æ p1 ¢¢ pg ££ pm ¢£ cm ¾¾
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Æ p1 ¿¿¿ p3 £ ¿¿¿ p3m1 ¿
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(also known as a homothety) with an
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(In an oblique coordinate system, e
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4.4.4 CONCURRENCE AND COLLINEARITY
Page 23 and 24: FIGURE 4.10 Notations for an arbitr
Page 25 and 26: FIGURE 4.12 Left: Ceva’s theorem.
Page 27 and 28: 2. The rhombus or diamond Ð , wher
Page 29 and 30: 4.6 CONICS A conic (or conic sectio
Page 31 and 32: FIGURE 4.17 Hyperbola with transver
Page 33 and 34: (c) If ¼, the equation again repr
Page 35 and 36: (a) The area of the ellipse is ¾
Page 37 and 38: 8. Let Ä be any line in the plane.
Page 39 and 40: FIGURE 4.18 The arc of a circle sub
Page 41 and 42: FIGURE 4.21 The semi-cubic parabola
Page 43 and 44: FIGURE 4.24 De ning property of the
Page 45 and 46: FIGURE 4.27 Left: initial con gurat
Page 47 and 48: FIGURE 4.29 The Bernoulli or logari
Page 49 and 50: 4.7.7 CLASSICAL CONSTRUCTIONS The a
Page 51 and 52: and the positive polar axis, and th
Page 53 and 54: mations of the following types: 1.
Page 55 and 56: 7. Re ec tion in a plane with equat
Page 57 and 58: 4.10.3 PROJECTIVE TRANSFORMATIONS A
Page 59 and 60: 6. The angle between two planes ¼
Page 61 and 62: Angle between lines with direction
Page 63 and 64: FIGURE 4.35 The Platonic solids. To
Page 65 and 66: FIGURE 4.36 Left: an oblique cylind
Page 67 and 68: For a frustum of a right circular c
Page 69 and 70: Given a general quadratic equation
Page 71 and 72: FIGURE 4.40 Left: a spherical cap.
Page 73: FIGURE 4.41 Right spherical triangl
Page 77 and 78: The angle between these vectors, «
Page 79 and 80: esults hold at point f´Øµ of :
Page 81 and 82: Asymptotic direction Asymptotic lin
Page 83 and 84: 8. The principal directions Ù Ú
Page 85 and 86: 4.21 ANGLE CONVERSION Degrees Radia
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Chapter 4: Geometry

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