Chapter 4: Geometry

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FIGURE 4.39 The ve non-degenerate real quadrics. Top left: ellipsoid. Top right: hyperboloid of two sheets (one facing up and one facing down). Bottom left: elliptic paraboloid. Bottom middle: hyperboloid of one sheet. Bottom right: hyperbolic paraboloid. Conversely, an equation of the form Ü ¾ · Ý ¾ · Þ ¾ ·¾Ü ·¾Ý ·¾Þ · ¼ (4.18.7) defines a sphere if ¾ · ¾ · ¾ ; the center is ´ µ Ô and the radius is ¾ · ¾ · ¾ . 1. Four points not in the same plane determine a unique sphere. If the points have coordinates ´Ü ½ Ý ½ Þ ½ µ, ´Ü ¾ Ý ¾ Þ ¾ µ, ´Ü ¿ Ý ¿ Þ ¿ µ, and ´Ü Ü Þ µ, the © 2003 by CRC Press LLC
FIGURE 4.40 Left: a spherical cap. Middle: a spherical zone (of two bases). Right: a spherical segment. Ô equation of the sphere is ¬ Ü ¾ · Ý ¾ · Þ ¾ Ü Ý Þ ½ Ü ¾ ½ · Ý¾ ½ · Þ¾ ½ Ü ½ Ý ½ Þ ½ ½ Ü ¾ ¾ · Ý¾ ¾ · Þ¾ ¾ Ü ¾ Ý ¾ Þ ¾ ½ Ü ¾ ¿ · Ý¾ ¿ · Þ¾ ¿ Ü ¿ Ý ¿ Þ ¿ ½ Ü ¾ · Ý¾ · Þ¾ Ü Ý Þ ½ ¬ ¼ (4.18.8) 2. Given two points È ½ ´Ü ½ Ý ½ Þ ½ µ and È ¾ ´Ü ¾ Ý ¾ Þ ¾ µ, there is a unique sphere whose diameter is È ½ È ¾ ; its equation is ´Ü Ü ½ µ´Ü Ü ¾ µ·´Ý Ý ½ µ´Ý Ý ¾ µ·´Þ Þ ½ µ´Þ Þ ¾ µ¼ (4.18.9) 3. The area of a sphere of radius Ö is Ö ¾ , and the volume is ¿ Ö¿ . 4. The area of a spherical polygon (that is, of a polygon on the sphere whose sides are arcs of great circles) is Ë ´Ò ¾µ Ö ¾ (4.18.10) Ò ½ where Ö is the radius of the sphere, Ò is the number of vertices, and are the internal angles of the polygons in radians. In particular, the sum of the angles of a spherical triangle is always greater than ½¼ Æ , and the excess is proportional to the area. 4.18.1.1 Spherical cap Let the radius be Ö (Figure 4.40, left). The area of the curved region is ¾Ö Ô ¾ . The volume of the cap is ½ ¾´¿Ö µ ½ ¿ ´¿¾ · ¾ µ. 4.18.1.2 Spherical zone (of two bases) Let the radius be Ö (Figure 4.40, middle). The area of the curved region (called a spherical zone)is¾Ö. The volume of the zone is ½ ´¿¾ ·¿ ¾ · ¾ µ. © 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
Page 19 and 20: (In an oblique coordinate system, e
Page 21 and 22: 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: Given a general quadratic equation
Page 73 and 74: FIGURE 4.41 Right spherical triangl
Page 75 and 76: ØÒ ØÒ ØÒ ¾ ¾ ¾
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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