Chapter 4: Geometry

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Name Ê Ö Ô Ô Tetrahedron ¼½¾¿¾ ½¾ ¼¾¼½¾ Ô ½ Cube ¿¾ ¼¼¾ ¼ Ô Ô ¾ Octahedron ¾¾ ¼¼½¼ ¼¼¾ Ô Ô Ô Ô ½ ½ Dodecahedron ½ · ¿µ ½¼½¾ ¾¼ · ½½¼ ½½½¿½ Icosahedron Ô ´ ½ ½¼ · ¾ Ô ¼½¼ ¾¼ ½ ½¾ Ô ¾ · ½ Ô ¼½ Name Ë ¾ Î ¿ Ô Ô Tetrahedron ¿ ½¿¾¼ ¾½¾ ¼½½½ Cube ½ ½ Octahedron Ô Ô ¾ ¿ ¿½¼ ¾¿ ¼½¼ Ô Dodecahedron Ô ½ ¿ ¾·½¼ ¾¼¿ ´½ · Ôµ ¿½½ Ôµ Icosahedron Ô ¿ ¼¾ ´¿ · ¾½½ ½¾ 4.14.2 POLYHEDRA NETS Nets for the five Platonic solids are shown: a) tetrahedron, b) octahedron, c) icosahedron, d) cube, and e) dodecahedron. Paper models can be made by making an enlarged photocopy of each, cutting them out along the exterior lines, folding on the interior lines, and using tape to join the edges. (a) (b) (c) (d) (e) © 2003 by CRC Press LLC
FIGURE 4.36 Left: an oblique cylinder with generator Ä and directrix . Right: a right circular cylinder. Ä È Ö 4.15 CYLINDERS Given a line Ä and a curve in a plane È , the cylinder with generator Ä and directrix is the surface obtained by moving Ä parallel to itself, so that a point of Ä is always on . IfÄ is parallel to the Þ-axis, the surface’s implicit equation does not involve the variable Þ. Conversely, any implicit equation that does not involve one of the variables (or that can be brought to that form by a change of coordinates) represents a cylinder. If is a simple closed curve, we also apply the word cylinder to the solid enclosed by the surface generated in this way (Figure 4.36, left). The volume contained between È and a plane È ¼ parallel to È is Î Ð ×Ò (4.15.1) where is the area in the plane È enclosed by , is the distance between È and È ¼ (measured perpendicularly), Ð is the length of the segment of Ä contained between È and È ¼ , and is the angle that Ä makes with È . When ¼ Æ we have a right cylinder, and Ð. For a right cylinder, the lateral area between È and È ¼ is ×, where × is the length (circumference) of . The most important particular case is the right circular cylinder (often simply called a cylinder). If Ö is the radius of the base and is the altitude (Figure 4.36, right), the lateral area is ¾Ö, the total area is ¾Ö´Ö · µ, and the volume is Ö ¾ . The implicit equation of this surface can be written Ü ¾ · Ý ¾ Ö ¾ ; see also page 364. 4.16 CONES Given a curve in a plane È and a point Ç not in È , the cone with vertex Ç and directrix is the surface obtained as the union of all rays that join Ç with points of . IfÇ is the origin and the surface is given implicitly by an algebraic equation, that equation is homogeneous (all terms have the same total degree in the variables). Conversely, any homogeneous implicit equation (or one that can be made homogeneous by a change of coordinates) represents a cone. © 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
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 and 62: Angle between lines with direction
Page 63: FIGURE 4.35 The Platonic solids. To
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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