Chapter 4: Geometry

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The choice of sign is arbitrary and is any number in Á . 6. A natural representation of class of the regular curve defined by the regular parametric representation f is defined by g´×µ f´« ½´×µµ, for all × ¾ ¼Ä℄. 7. A property of a regular curve is any property of a regular parametric representation representing which is invariant under any allowable change of parameter. 8. Let g be a natural representation of a regular curve . The following quantities may be defined at each point x g´×µ of : Binormal line y b´×µ ·x Curvature ´×µ n´×µ ¡ k´×µ Curvature vector k´×µ t´×µ Moving trihedron t´×µ n´×µ b´×µ Normal plane ý xµ ¡ t´×µ ¼ Osculating plane ý xµ ¡ b´×µ ¼ Osculating sphere ý cµ ¡ ý cµ Ö ¾ where c x · ´×µn´×µ ´´×µ´ ¾´×µ ´×µµµb´×µ and Ö ¾ ¾´×µ ·¾´×µ´´×µ ¾´×µµ Principal normal line y n´×µ ·x Principal normal unit vector n´×µ ¦k´×µk´×µ, for k´×µ ¼defined to be continuous along Radius of curvature ´×µ ½ ´×µ, when ´×µ ¼ Rectifying plane ý xµ ¡ n´×µ ¼ Tangent line y t´×µ ·x Torsion Unit binormal vector Unit tangent vector ´×µ n´×µ ¡ b´×µ b´×µ t´×µ ¢ n´×µ t´×µ g´×µ with g´×µ g × 4.20.1.2 Results The arc length Ä and the arc length parameter × of any regular parametric representation f are invariant under any allowable change of parameter. Thus, Ä is a property of the regular curve defined by f. ¬ The arc length parameter satisfies × «¼´Øµ ¦ ¬ f ¼´Øµ ¬, which implies that ¬ ¬ Ø f ¼´×µ ½, if and only if Ø is an arc length parameter. Thus, arc length parameters are uniquely determined up to the transformation × × ¦× · × ¼ , where × ¼ is any constant. The curvature, torsion, tangent line, normal plane, principal normal line, rectifying plane, binormal line, and osculating plane are properties of the regular curve defined by any regular parametric representation f. If x f´Øµ is any regular representation of a regular curve , the following © 2003 by CRC Press LLC
esults hold at point f´Øµ of : x¼¼ ¢ x ¼ x ¼ ¿ x¼ ¡ ´x ¼¼ ¢ x ¼¼¼ µ x ¼ ¢ x ¼¼ (4.20.3) ¾ The vectors of the moving trihedron satisfy the Serret–Frenet equations t n n t · b b n (4.20.4) For any plane curve represented parametrically by x f´Øµ ´Ø ´Øµ ¼µ, ¬ ¬ ¾ Ü ¬¬ Ø ¾ ¡ ¾ ¿¾ (4.20.5) ½· Ü Ø Expressions for the curvature vector and curvature of a plane curve corresponding to different representations are given in the following table: Representation Curvature vector k Curvature, ½ Ü ´Øµ Ý ´Øµ Ý ´Üµ Ö ´µ ´ÜĐÝ ÝĐÜµ ´Ü ¾ · Ý ¾ µ ¾ ´ Ý ÜĐÝ ÝĐÜ Üµ ´Ü ¾ · Ý ¾ µ ¿¾ Ý ¼¼ Ý ¼¼ ´½ · Ý ¼¾ µ ´ Ý¼ ½µ ¾ ´½ · Ý ¼¾ µ ¿¾ ´Ö ¾ ·¾Ö ¼ ¾ ÖÖ ¼¼ µ ´ Ö ×Ò Ö Ó× Ö ¾ ·¾Ö ¼ ¾ ÖÖ ¼¼ ´Ö ¾ · Ö ¼ ¾ µ ¾ Ö Ó× Ö ×Ò µ ´Ö ¾ · Ö ¼ ¾ µ ¿¾ The equation of the osculating circle of a plane curve is given by where c x · ¾ k is the center of curvature. ý cµ ¡ ý cµ ¾ (4.20.6) THEOREM 4.20.1 (Fundamental existence and uniqueness theorem) Let ´×µ and ´×µ be any continuous functions de ne d for all × ¾ ℄. Then there exists, up to a congruence, a unique space curve for which is the curvature function, is the torsion function, and × an arc length parameter along . 4.20.1.3 Example A regular parametric representation of the circular helix is given by x f´Øµ ´ Ó× Ø ×Ò Ø Øµ, for all Ø ¾ Ê, where ¼ and ¼are constant. By successive differentiation, x ¼ ´ ×Ò Ø Ó× Ø µ x ¼¼ ´ Ó× Ø ×Ò Ø ¼µ x ¼¼¼ ´ so that × Ø x¼ Ô ¾ · ¾ . Hence, ×Ò Ø Ó× Ø ¼µ 1. Arc length parameter: × «´Øµ Ø´ ¾ · ¾ µ ¾ ½ 2. Curvature vector: k t × Ø t × Ø ´¾ · ¾ µ ½´ Ó× Ø ×Ò Ø ¼µ (4.20.7) © 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
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Æ p1 ¢¢ pg ££ pm ¢£ cm ¾¾
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Æ p1 ¿¿¿ p3 £ ¿¿¿ p3m1 ¿
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(also known as a homothety) with an
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(In an oblique coordinate system, e
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4.4.4 CONCURRENCE AND COLLINEARITY
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FIGURE 4.10 Notations for an arbitr
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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 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: The angle between these vectors, «
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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