Chapter 4: Geometry

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4.19.2.10 Summary of solution of oblique spherical triangles Given Solution Check Three sides Half-angle formulae Law of sines Three angles Half-side formulae Law of sines Two sides and included angle Two angles and included side Two sides and an opposite angle Two angles and an opposite side Napier’s analogies (to find sum and difference of unknown angles); then law of sines (to find remaining side). Napier’s analogies (to find sum and difference of unknown sides); then law of sines (to find remaining angle). Law of sines (to find an angle); then Napier’s analogies (to find remaining angle and side). Note the number of solutions. Law of sines (to find a side); then Napier’s analogies (to find remaining side and angle). Note the number of solutions. 4.19.2.11 Haversine formulae Ú ½ Ó× ×Ò ¾ ¾ ¾ Ú´ µ · ×Ò ×Ò Ú ×Ò´× µ ×Ò´× µ Ú ×Ò ×Ò Ú Ú´ µ ×Ò ×Ò Ú½¼ Æ ´ · µ℄· ×Ò ×Ò Ú Gauss’s formulae Gauss’s formulae Gauss’s formulae Gauss’s formulae 4.19.2.12 Finding the distance between two points on the earth To find the distance between two points on the surface of a spherical earth, let point È ½ have a (latitude, longitude) of ´ ½ ½ µ and point È ¾ have a (latitude, longitude) of ´ ¾ ¾ µ. Two different computational methods are as follows: 1. Let be the North pole and let and be the points È ½ and È ¾ . Then the spherical law of cosines for sides gives the central angle, , subtended by the desired distance: Ó×´µ Ó×´µÓ×´µ · ×Ò´µ×Ò´µ Ó×´µ where the angle is the difference in longitudes, and and are the angles of the points from the pole (i.e., ¼ Æ latitude). Scale by Ê¨ (the radius of the earth) to get the desired distance. 2. In ´Ü Ý Þµ space (with ·Þ being the North pole) points È ½ and È ¾ are represented as vectors from the center of the earth in spherical coordinates: ¢ £ v ½ ¢ Ê¨ Ó×´ ½ µÓ×´ ½ µ Ê¨ Ó×´ ½ µ×Ò´ ½ µ Ê¨ ×Ò´ ½ µ £ v ¾ Ê¨ Ó×´ ¾ µÓ×´ ¾ µ Ê¨ Ó×´ ¾ µ×Ò´ ¾ µ Ê¨ ×Ò´ ¾ µ (4.19.3) © 2003 by CRC Press LLC
The angle between these vectors, «, is given by Ó× « v ½ ¡ v v ¾ v ½ v ¾ ½ ¡ v ¾ Ê ¾¨ Ó×´ ½ µÓ×´ ¾ µ Ó×´ ½ ¾ µ · ×Ò´ ½ µ ×Ò´ ¾ µ where ×Ò ¾ ½ ¾ ¾ ¾ ØÒ ½ Ö ½ ·Ó×´ ½ µÓ×´ ¾ µ ×Ò ¾ ½ ¾ ¾ ¡ . (The formula using is more accurate numerically when È ½ and È ¾ are close.) The great circle distance between È ½ and È ¾ is then Ê¨«. EXAMPLE The angle between È New York with ´ ½ ¼ Æ ½ ¿ Æ µ and È Beijing with ´ ¾ ¿¿ Æ ¾ ¾¿ Æ µ is « Æ . Using Ê¨ ¿the great circle distance between New York and Beijing is about 11,000 km. 4.20 DIFFERENTIAL GEOMETRY 4.20.1 CURVES 4.20.1.1 De nitions 1. A regular parametric representation of class , ½, is a vector valued function f Á Ê ¿ , where Á Ê is an interval that satisfies (i) f is of class (i.e., has continuous th order derivatives), and (ii) f ¼´Øµ ¼, for all Ø ¾ Á. In terms of a standard basis of Ê ¿ , we write x f´Øµ ´ ½´Øµ, ¾´Øµ ¿´Øµµ, where the real valued functions , ½ ¾ ¿ are the component functions of f. 2. An allowable change of parameter of class is any function Â Á, where Â is an interval and ´Âµ Á, that satisfies ¼´ µ ¼, for all ¾ Â. 3. A regular parametric representation f is equivalent to a regular parametric representation g if and only if an allowable change of parameter exists so that ´Á µÁ , and g´ µf´´ µµ, for all ¾ Á . 4. A regular curve of class is an equivalence class of regular parametric representation under the equivalence relation on the set of regular parametric representations defined above. 5. The arc length of any regular curve defined by the regular parametric representation f, with Á ℄, is defined by Ä ¬ ¬ f ¼´Ùµ Ù (4.20.1) An arc length parameter along is defined by Ø × «´Øµ ¦ ¬ ¬ f ¼´Ùµ Ù (4.20.2) © 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
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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 and 74: FIGURE 4.41 Right spherical triangl
Page 75: ØÒ ØÒ ØÒ ¾ ¾ ¾
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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