Chapter 4: Geometry

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´Ü ¾ Ý ¾ Þ ¾ µ, and ´Ü ¿ Ý ¿ Þ ¿ µ is given by volume ½ ¬ Ü ½ Ý ½ Þ ½¬¬¬¬¬ Ü ¾ Ý ¾ Þ ¾ (4.14.3) ¬ Ü ¿ Ý ¿ Þ ¿ In a tetrahedron with vertices È ¼ , È ½ , È ¾ , È ¿ , let be the distance (edge length) from È to È . Form the determinants ¡ ¬ ¼ ½ ½ ½ ½ ½ ¼ ¾ ¼½ ¾ ¼¾ ¾ ¼¿ ½ ¾ ¼½ ¼ ¾ ½¾ ¾ ½¿ ½ ¾ ¼¾ ¾ ½¾ ¼ ¾ ¾¿ ½ ¾ ¼¿ ¾ ½¿ ¾ ¾¿ ¼ ¬ ¬ ¼ ¾ ¼½ ¾ ¼¾ ¾ ¼¿ and ¾ ¼½ ¼ ¾ ½¾ ¾ ½¿ ¾ ¼¾ ¾ ½¾ ¼ ¾ (4.14.4) ¾¿ ¾ ¼¿ ¾ ½¿ ¾ ¾¿ ¼ Then the volume of the tetrahedron is Ô ¡¾, and the radius of the circumscribed sphere is ½ ¾Ô ¾¡. Expanding the determinant we find that the volume Î satisfies the formula: ½Î ¾ ¾ ¼½ ¾ ½¾ ¾ ¼¾ ¾ ¼½ ¾ ½¿ ¾ ¼¿ ¾ ½¾ ¾ ½¿ ¾ ¾¿ ¾ ¼¾ ¾ ¼¿ ¾ ¾¿ · ¾ ¼½ ¾ ¼¾ ¾ ½¿ · ¾ ½¾ ¾ ¼¾ ¾ ½¿ · ¾ ¼½ ¾ ½¾ ¾ ¼¿ · ¾ ½¾ ¾ ¼¾ ¾ ¼¿ ¬ · ¾ ½¾ ¾ ½¿ ¾ ¼¿ · ¾ ¼¾ ¾ ½¿ ¾ ¼¿ · ¾ ¼½ ¾ ½¾ ¾ ¾¿ · ¾ ¼½ ¾ ¼¾ ¾ ¾¿ · ¾ ¼½ ¾ ½¿ ¾ ¾¿ · ¾ ¼¾ ¾ ½¿ ¾ ¾¿ · ¾ ¼½ ¾ ¼¿ ¾ ¾¿ · ¾ ½¾ ¾ ¼¿ ¾ ¾¿ (4.14.5) ¾ ¼¾ ¾ ¼¾ ¾ ½¿ ¾ ¼¾ ¾ ½¿ ¾ ½¿ ¾ ½¾ ¾ ½¾ ¾ ¼¿ ¾ ½¾ ¾ ¼¿ ¾ ¼¿ ¾ ¼½ ¾ ¼½ ¾ ¾¿ ¾ ¼½ ¾ ¾¿ ¾ ¾¿ (Mnemonic: Each of the first four negative terms corresponds to a closed path around a face; each positive term to an open path along three consecutive edges; each remaining negative term to a pair of opposite edges with weights 2 and 1. All such edge combinations are represented.) For an arbitrary tetrahedron, let È be a vertex and let , , be the lengths of the edges converging on È .If, , are the angles between the same three edges, the volume of the tetrahedron is Î ½ Ô ½ Ó× ¾ Ó× ¾ Ó× ¾ · ¾ Ó× Ó× Ó× (4.14.6) 4.14.1 CONVEX REGULAR POLYHEDRA Figure 4.35 shows the five regular polyhedra, or Platonic solids. In the following tables and formulae, is the length of an edge, the dihedral angle at each edge, Ê the radius of the circumscribed sphere, Ö the radius of the inscribed sphere, Î the volume, Ë the total surface area, Ú the total number of vertices, the total number of edges, the total number of faces, Ô the number of edges in a face (3 for equilateral triangles, 4 for squares, 5 for regular pentagons), and Õ the number of edges meeting at a vertex. © 2003 by CRC Press LLC
FIGURE 4.35 The Platonic solids. Top: the tetrahedron (self-dual). Middle: the cube and the octahedron (dual to one another). Bottom: the dodecahedron and the icosahedron (dual to one another). ¾×Ò ½ Ó×´½¼Æ Õµ ×Ò´½¼ Æ Ê ½¼Æ ½¼Æ Ôµ ØÒ ØÒ Ö Ô Õ ½ Ê ¾ ×Ò´½¼Æ Õµ ×Ò´½¼ Æ Ôµ Ó× ½ Ë Ô ¾ ¾ ½¼Æ ÓØ Ô and (4.14.7) Ö ½ ½¼Æ ÓØ ¾ Ô ØÒ Î ½ ÖË ¿ ¾ Name Ú Ô Õ ×Ò Regular tetrahedron ¿ ¿ ¾ Ô ¾¿ ¼ Æ ¿½ ¼ ¼¼ Cube ½¾ ¿ ½ ¼ Æ Regular octahedron ½¾ ¿ ¾ Ô ¾¿ ½¼ Æ ¾ ¼ ½ ¼¼ Regular dodecahedron ¾¼ ¿¼ ½¾ ¿ ¾ Ô ½½ Æ ¿¿ ¼ ¼¼ Regular icosahedron ½¾ ¿¼ ¾¼ ¿ ¾¿ ½¿ Æ ½½ ¼ ¾¿ ¼¼ © 2003 by CRC Press LLC
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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 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: Angle between lines with direction
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
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Chapter 4: Geometry

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