Chapter 4: Geometry

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2. Line through ´Ü ¼ Ý ¼ Þ ¼ µ and ´Ü ½ Ý ½ Þ ½ µ: Ü Ü Ý Ý ¼ ¼ Þ Þ ¼ (4.13.3) Ü ½ Ü ¼ Ý ½ Ý ¼ Þ ½ Þ ¼ This line is parallel to the vector ´Ü ½ Ü ¼ Ý ½ Ý ¼ Þ ½ Þ ¼ µ. 4.13.1 DISTANCES 1. The distance between two points in space is the length of the line segment joining them. The distance between the points ´Ü ¼ Ý ¼ Þ ¼ µ and ´Ü ½ Ý ½ Þ ½ µ is Ô ´Ü ½ Ü ¼ µ ¾ ·´Ý ½ Ý ¼ µ ¾ ·´Þ ½ Þ ¼ µ ¾ (4.13.4) 2. The point ± of the way from È ¼ ´Ü ¼ Ý ¼ Þ ¼ µ to È ½ ´Ü ½ Ý ½ Þ ½ µ is Ü½ ·´½¼¼ µÜ ¼ Ý ½ ·´½¼¼ µÝ ¼ Þ ½ · ´½¼¼ µÞ ¼ (4.13.5) ½¼¼ ½¼¼ ½¼¼ (The same formula also applies in oblique coordinates.) This point divides the segment È ¼ È ½ in the ratio ´½¼¼ µ. As a particular case, the midpoint of È ¼ È ½ is given by Ü½ · Ü ¼ (4.13.6) ¾ Ý ½ · Ý ¼ ¾ Þ ½ · Þ ¼ ¾ 3. The distance between the point ´Ü ¼ Ý ¼ Þ ¼ µ and the line through ´Ü ½ Ý ½ Þ ½ µ in direction ´ µ: ÚÙ Ù Ø ¬ ¬ Ý ¼ Ý ½ Þ ¼ Þ ½ ¬ ¾ ¬ ¬ ¾ · ¬ Þ ¼ Þ ½ Ü ¼ Ü ½ ¬ · ¬ Ü ¼ Ü ½ Ý ¼ Ý ½ ¬ ¾ · ¾ · ¾ (4.13.7) ¬ ¬ ¾ 4. The distance between the line through ´Ü ¼ Ý ¼ Þ ¼ µ in direction ´ ¼ ¼ ¼ µ and the line through ´Ü ½ Ý ½ Þ ½ µ in direction ´ ½ ½ ½ µ: ¬ Ü ½ Ü ¼ Ý ½ Ý ¼ Þ ½ Þ ¼¬¬¬¬¬ ¼ ¼ ¼ ¬ ½ ½ ½ × ¬¬¬¬ ¬ ¼ ¼¬¬¬ ¾ ¬ · ¬ ½ ½ ¬ ¼ ¼¬¬¬ ¾ ¬ · ½ ½ ¬ ¼ ¼¬¬¬ ¾ ½ ½ ¬ (4.13.8) 4.13.2 ANGLES Angle between lines with directions ´Ü ¼ Ý ¼ Þ ¼ µ and ´Ü ½ Ý ½ Þ ½ µ: Ó× ½ ¼ ½ · ¼ ½ · ¼ Ô Ô ½ ¾ ¼ · ¾ ¼ · ¾ ¼ ¾ ½ · ¾ ½ · ¾ ½ (4.13.9) In particular, the two lines are parallel when ¼ ¼ ¼ ½ ½ ½ , and perpendicular when ¼ ½ · ¼ ½ · ¼ ½ ¼. © 2003 by CRC Press LLC
Angle between lines with direction angles « ¼ ¬ ¼ ¼ and « ½ ¬ ½ ½ : Ó× ½´Ó× « ¼ Ó× « ½ · Ó× ¬ ¼ Ó× ¬ ½ ·Ó× ¼ Ó× ½ µ (4.13.10) 4.13.3 CONCURRENCE, COPLANARITY, PARALLELISM Two lines, each specified by point and direction, are coplanar if, and only if, the determinant in the numerator of Equation (4.13.8) is zero. In this case they are concurrent (if the denominator is non-zero) or parallel (if the denominator is zero). Three lines with directions ´ ¼ ¼ ¼ µ, ´ ½ ½ ½ µ, and ´ ¾ ¾ ¾ µ are parallel to a common plane if and only if ¬ ¬¬¬¬¬ ¬ ¼ ¼ ¼¬¬¬¬¬ ½ ½ ½ ¼ (4.13.11) ¾ ¾ ¾ 4.14 POLYHEDRA For any polyhedron topologically equivalent to a sphere—in particular, for any convex polyhedron—the Euler formula holds: Ú · ¾ (4.14.1) where Ú is the number of vertices, is the number of edges, and is the number of faces. Many common polyhedra are particular cases of cylinders (Section 4.15) or cones (Section 4.16). A cylinder with a polygonal base (the base is also called a directrix) is called a prism. A cone with a polygonal base is called a pyramid. A frustum of a cone with a polygonal base is called a truncated pyramid. Formulae (4.15.1), (4.16.1), and (4.16.2) give the volumes of a general prism, pyramid, and truncated pyramid. A prism whose base is a parallelogram is a parallelepiped. The volume of a parallelepiped with one vertex at the origin and adjacent vertices at ´Ü ½ Ý ½ Þ ½ µ, ´Ü ¾ Ý ¾ Þ ¾ µ, and ´Ü ¿ Ý ¿ Þ ¿ µ is given by volume ¬ Ü ½ Ý ½ Þ ½¬¬¬¬¬ Ü ¾ Ý ¾ Þ ¾ (4.14.2) ¬ Ü ¿ Ý ¿ Þ ¿ The rectangular parallelepiped is a particular case: all of its faces are rectangles. If the side lengths are , the volume is , the total surface area is ¾´··µ, and each diagonal has length Ô ¾ · ¾ · ¾ . When we get the cube. See Section 4.14.1. A pyramid whose base is a triangle is a tetrahedron. The volume of a tetrahedon with one vertex at the origin and the other vertices at ´Ü ½ Ý ½ Þ ½ 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
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: 6. The angle between two planes ¼
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
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Chapter 4: Geometry

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