Chapter 4: Geometry

More documents

Recommendations

Info

3. Curvature: k ´ ¾ · ¾ µ ½ 4. Principal normal unit vector: n k k ´ Ó× Ø ×Ò Ø ¼µ 5. Unit tangent vector: t x¼ x ¼ ´¾ · ¾ µ ½ ¾ ´ ×Ò Ø Ó× Ø µ 6. Unit binormal vector: b t ¢ n ´ ¾ · ¾ µ ½ ¾ ´ ×Ò Ø Ó× Ø µ b Ø b × Ø ´¾ · ¾ µ ½´Ó× Ø ×Ò Ø ¼µ 7. Torsion: n ¡ b ´ ¾ · ¾ µ ½ The values of and can be verified using the formulae in (4.20.3). The sign of (the invariant) determines whether the helix is right handed, ¼, or left handed, ¼. 4.20.2 SURFACES 4.20.2.1 Definitions 1. A coordinate patch of class , ½ on a surface Ë Ê ¿ is a vector valued function Í Ë, where Í Ê ¾ is an open set, that satisfies (i) f is class on Í, (ii) Ù´Ù f f Úµ ¢ Ú´Ù Úµ ¼, for all ´Ù Úµ ¾ Í, and (iii) f is one-to-one and bi-continuous on Í. 2. In terms of a standard basis of Ê ¿ we write x f´Ù Úµ ´ ½´Ù Úµ, ¾´Ù Úµ, ¿´Ù Úµµ, where the real valued functions ½ ¾ ¿ are the component functions of f. The notation x ½ x Ù f Ù , x ¾ x Ú f Ú , Ù½ Ù Ù ¾ Ú, is frequently used. 3. A Monge patch is a coordinate patch where f has the form f´Ù Úµ ´Ù Ú ´Ù Úµµ, where is a real valued function of class . 4. The Ù-parameter curves Ú Ú ¼ on Ë are the images of the lines Ú Ú ¼ in Í. They are parametrically represented by x f´Ù Ú ¼ µ. The Ú-parameter curves Ù Ù ¼ are defined similarly. 5. An allowable parameter transformation of class is a one-to-one function Í Î , where Í Î Ê ¾ are open, that satisfies ½ Ø Ù ´Ù Úµ ½ Ú ´Ù Úµ ¾ Ù ´Ù Úµ ¾ Ú ´Ù Úµ ¼ (4.20.8) for all ´Ù Úµ ¾ Í, where the real valued functions, ½ and ¾ , defined by ´Ù Úµ ´ ½´Ù Úµ, ¾´Ù Úµµ are the component functions of . One may also write the parameter transformation as Ù ½ ½´Ù½ Ù ¾ µ, Ù ¾ ¾´Ù½ Ù ¾ µ. 6. A local property of surface Ë is any property of a coordinate patch that is invariant under any allowable parameter transformation. 7. Let f define a coordinate patch on a surface Ë. The following quantities may be defined at each point x f´Ù Úµ on the patch: © 2003 by CRC Press LLC
Asymptotic direction Asymptotic line A direction Ù Ú for which Ò ¼ A curve on Ë whose tangent line at each point coincides with an asymptotic direction Dupin’s indicatrix Ü ¾ ·¾Ü ½ ½Ü ¾ · Ü ¾ ¾ ¦½ Elliptic point ¾ ¼ First fundamental form Á x ¡ x «¬´Ù Úµ Ù « Ù ¬ First fundamental metric coef cients ´Ù Úµ Ù ¾ ·¾ ´Ù Úµ Ù Ú · ´Ù Úµ Ú ¾ ´Ù Úµ ½½´Ù Úµ x ½ ¡ x ½ ´Ù Úµ ½¾´Ù Úµ x ½ ¡ x ¾ ´Ù Úµ ¾¾´Ù Úµ x ¾ ¡ x ¾ Fundamental differential x x « Ù « x Ù Ù · x Ú Ú (a repeated upper and lower index signifies a summation over the range « ½ ¾µ Gaussian curvature Ã ½ ¾ ¾ ¾ Geodesic curvature vector of curve on Ë through Ü k k ´k ¡ nµn ĐÙ « · ¬ Ù¬ Ù ℄x « where « ¬ denote the Christoffel symbols of the second kind for the metric «¬ , defined in Section 5.10 Geodesic on Ë A curve on Ë which satisfies k ¼at each point Hyperbolic point ¾ ¼ Line of curvature A curve on Ë whose tangent line at each point coincides with a principal direction Mean curvature À ½ · ¾ · ¾ ¾ ¾´ ¾ µ Normal curvature in the Ò k ¡ n ÁÁ Á Ù Ú direction Normal curvature vector of curve on Ë through Ü Normal line Normal vector Parabolic point Planar point Principal curvatures Principal directions k Ò ´k ¡ nµn « y Æ · x Æ x Ù ¢ x Ú ¾ ¼not all of ¼ ¼ The extreme values ½ and ¾ of Ò The perpendicular directions Ù Ú in which Ò attains its extreme values Second fundamental form ÁÁ x ¡ n «¬´Ù Úµ Ù « Ù ¬ Second fundamental metric coef cients ´Ù Úµ Ù ¾ ·¾ ´Ù Úµ Ù Ú · ´Ù Úµ Ú ¾ ´Ù Úµ ½½´Ù Úµ x ½½ ¡ n ´Ù Úµ ½¾´Ù Úµ x ½¾ ¡ n ´Ù Úµ ¾¾´Ù Úµ x ¾¾ ¡ n © 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 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 and 78: The angle between these vectors, «
Page 79: esults hold at point f´Øµ of :
Page 83 and 84: 8. The principal directions Ù Ú
Page 85 and 86: 4.21 ANGLE CONVERSION Degrees Radia
show all

Chapter 4: Geometry

Create successful ePaper yourself

Delete template?

Save as template?