Chapter 4: Geometry

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FIGURE 4.22 The folium of Descartes in two positions, and the strophoid. È ¼ È Ç ¿ FIGURE 4.23 Cassini’s ovals for ¼, ¼, , ½½ and ½ (from the inside to the outside). The foci (dots) are at Ü and Ü . The black curve, , is also called Bernoulli’s lemniscate. the shape: the curve has one smooth segment and one with a self-intersection, or two segments depending on whether is greater than, equal to, or smaller than (Figure 4.23). The case is also known as the lemniscate (of Jakob Bernoulli); the equation reduces to ´Ü ¾ · Ý ¾ µ ¾ ¾´Ü ¾ Ý ¾ µ, and upon a Æ rotation to ´Ü ¾ · Ý ¾ µ ¾ ¾ ¾ ÜÝ. Each Cassini oval is the section of a torus of revolution by a plane parallel to the axis of revolution. 2. A conchoid of Nichomedes is the set of points such that the signed distance È in Figure 4.24, left, equals a x ed real number (the line Ä and the origin Ç being x ed). If Ä is the line Ü , the conchoid’s polar equation is Ö × · . Once more, is a scaling parameter, and the value of controls the shape: when the curve is smooth, when there is a cusp, and when there is a self-intersection. The curves for and can also be considered two leaves of the same conchoid, with Cartesian equation ´Ü µ ¾´Ü ¾ · Ý ¾ µ ¾ Ü ¾ 3. A limaçon of Pascal is the set of points such that the distance È in Figure 4.25, left, equals a x ed positive number measured on either side (the © 2003 by CRC Press LLC
FIGURE 4.24 De ning property of the conchoid of Nichomedes (left), and curves for ¦¼, ¦, and ¦½ (right). Ä Ç È Ç Ç Ç FIGURE 4.25 De ning property of the limaçon of Pascal (left), and curves for ½, , and ¼ (right). The middle curve is the cardioid; the one on the right a trisectrix. È Ç diameter Ç Ç circle and the origin Ç being x ed). If has diameter and center at ´¼ ½ µ, the limaçon’s polar equation is Ö Ó× · , and its Cartesian ¾ equation is ´Ü ¾ · Ý ¾ Üµ ¾ ¾´Ü ¾ · Ý ¾ µ (4.7.1) The value of controls the shape, and there are two particularly interesting cases. For , we get a cardioid (see also page 341). For ½ , we get ¾ a curve that can be used to trisect an arbitrary angle «. If we draw a line Ä through the center of the circle making an angle « with the positive Ü-axis, and if we call È the intersection of Ä with the limaçon ½ , the line from ¾ Ç to È makes an angle with Ä equal to ½ «. ¿ Hypocycloids and epicycloids with rational ratios (see next section) are also algebraic curves, generally of higher degree. © 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: FIGURE 4.21 The semi-cubic parabola
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 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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