Chapter 4: Geometry

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4.8.4 RELATIONS BETWEEN CARTESIAN, CYLINDRICAL, AND SPHERICAL COORDINATES Consider a Cartesian, a cylindrical, and a spherical coordinate system, related as shown in Figure 4.33. The Cartesian coordinates ´Ü Ý Þµ, the cylindrical coordinates ´ÖÞµ, and the spherical coordinates ´ µ of a point are related as follows (where the ØÒ cart ° cyl cyl ° sph cart ° sph ½ function must be interpreted correctly in all quadrants): Ô Ý Ü Ö Ó× Ö Ü¾ · Ý ¾ ×Ò Ô Ü¾ Ý Ö ×Ò Ý · Ý ¾ ØÒ ½ Ü Ü Ó× Ô Ü¾ · Ý ¾ Þ Þ Ö ×Ò Þ Ó× Ü Ó× ×Ò Ý ×Ò ×Ò Þ Ó× Þ Þ Ô Ö¾ · Þ ¾ ØÒ ½ Ö Þ ×Ò Ó× Ô Ü¾ · Ý ¾ · Þ ¾ ØÒ ½ Ý Ü Þ Þ Ô ØÒ ½ Ü¾ · Ý ¾ Þ Þ Ó× ½ Ô Ü¾ · Ý ¾ · Þ ¾ Ö Ô Ö¾ · Þ ¾ Þ Ô Ö¾ · Þ ¾ 4.8.5 HOMOGENEOUS COORDINATES IN SPACE A quadruple of real numbers ´Ü Ý Þ Øµ, with Ø ¼, is a set of homogeneous coordinates for the point È with Cartesian coordinates ´ÜØ ÝØ ÞØµ. Thus the same point has many sets of homogeneous coordinates: ´Ü Ý Þ Øµ and ´Ü ¼ Ý ¼ Þ ¼ Ø ¼ µ represent the same point if, and only if, there is some real number « such that Ü ¼ «Ü, Ý ¼ «Ý, Þ ¼ «Þ, Ø ¼ «Ø. IfÈ has Cartesian coordinates ´Ü ¼ Ý ¼ Þ ¼ µ, one set of homogeneous coordinates for È is ´Ü ¼ Ý ¼ Þ ¼ ½µ. Section 4.1.5 has more information on the relationship between Cartesian and homogeneous coordinates. Section 4.9.2 has formulae for space transformations in homogeneous coordinates. 4.9 SPACE SYMMETRIES OR ISOMETRIES A transformation of space (invertible map of space to itself) that preserves distances is called an isometry of space. Every isometry of space is a composition of transfor- © 2003 by CRC Press LLC
mations of the following types: 1. The identity (which leaves every point fixed) 2. A translation by a vector v 3. A rotation through an angle « around a line Ä 4. A screw motion through an angle « around a line Ä, with displacement 5. A re ection in a plane È 6. A glide-re e ction in a plane È with displacement vector v 7. A rotation-re ection (rotation through an angle « around a line Ä composed with reflection in a plane perpendicular to Ä). The identity is a particular case of a translation and of a rotation; rotations are particular cases of screw motions; reflections are particular cases of glide-reflections. However, as in the plane case, it is more intuitive to consider each case separately. 4.9.1 FORMULAE FOR SYMMETRIES: CARTESIAN COORDINATES In the formulae below, multiplication between a matrix and a triple of coordinates should be carried out regarding the triple as a column vector (or a matrix with three rows and one column). 1. Translation by ´Ü ¼ Ý ¼ Þ ¼ µ: ´Ü Ý Þµ ´Ü · Ü ¼ Ý· Ý ¼ Þ· Þ ¼ µ (4.9.1) 2. Rotation through « (counterclockwise) around the line through the origin with direction cosines (see page 353): ´Ü Ý Þµ Å ´Ü Ý Þµ where Å is ¾the matrix ¿ ¾´½ Ó× «µ ·Ó×« ´½ Ó× «µ ×Ò « ´½ Ó× «µ · ×Ò « ´½ Ó× «µ · ×Ò « ¾´½ Ó× «µ · Ó× « ´½ Ó× «µ ×Ò « ´½ Ó× «µ ×Ò « ´½ Ó× «µ · ×Ò « ¾´½ Ó× «µ · Ó× « (4.9.2) 3. Rotation through « (counterclockwise) around the line with direction cosines through an arbitrary point ´Ü ¼ Ý ¼ Þ ¼ µ: ´Ü Ý Þµ ´Ü ¼ Ý ¼ Þ ¼ µ·Å ´Ü Ü ¼ Ý Ý ¼ Þ Þ ¼ µ (4.9.3) where Å is given by Equation (4.9.2). 4. Arbitrary rotations and Euler angles: Any rotation of space fixing the origin can be decomposed as a rotation by about the Þ-axis, followed by a rotation by about the Ý-axis, followed by a rotation by about the Þ-axis. The numbers , , and are called the Euler angles of the composite rotation, which acts as: ´Ü Ý Þµ Å ´Ü Ý Þµ where Å is the matrix given by ¾ Ó× Ó× Ó× ×Ò ×Ò ×Ò Ó× Ó× Ó× ×Ò ×Ò Ó× Ó× Ó× ×Ò ·×Ò Ó× ×Ò Ó× ×Ò · Ó× Ó× ×Ò ×Ò Ó× ×Ò ×Ò ×Ò Ó× (4.9.4) ¿ © 2003 by CRC Press LLC
Page 1 and 2: Chapter Geometry 4.1 COORDINATE SY
Page 3 and 4: 4.19.2 Oblique spherical triangles
Page 5 and 6: FIGURE 4.1 Change of coordinates by
Page 7 and 8: 4.1.4.1 Relations between Cartesian
Page 9 and 10: This formula also covers passing fr
Page 11 and 12: EXAMPLE To nd the matrix for a rota
Page 13 and 14: Æ p1 ¢¢ pg ££ pm ¢£ cm ¾¾
Page 15 and 16: Æ p1 ¿¿¿ p3 £ ¿¿¿ p3m1 ¿
Page 17 and 18: (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 the positive polar axis, and th
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 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

Chapter 4: Geometry

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