Blaga P. Lectures on the differential geometry of - tiera.ru

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88 Chapter 2. Plane curves Figure 2.2: The evolute of an astroid Figure 2.3: The evolute of an cycloid Remark. Generally speaking, s is not a natural parameter along ρ. If (I, r = r(t)) is an arbitrary parameterized curve, the we can replace the parameter �t t by the arc length s = �r ′ (τ)�dτ and define the involute of r as being the involute of 0 the naturally parameterized curve equivalent to it, the natural parameter being the arc length. Is is easy to see that the following proposition holds true: Proposition 2.4.2. Let (I, r = r(t)) be a parameterized curve. The the involute of r with
2.4. The curvature center. The evolute and the involute of a plane curve 89 the origin at c ∈ I is given by where s = s(t) is the arc length of r. ρ(t) = r(t) + (c − s(t)) r′ (t) �r ′ (t)� , Example. Let r(t) = (a cos t, a sin t) be a circle. Then ⎧ r ⎪⎨ ⎪⎩ ′ (t) = {−a sin t, a cos t} x ′2 + y ′2 = a2 �t s(t) = adt = at, hence the equation of the involute is or, in projection, ρ(t) = (a cos t, a sin t) + 0 (c − at) {−a sin t, a cos t} a ⎧ ⎪⎨ X(t) = a cos t − (c − at) sin t ⎪⎩ Y(t) = a sin t + (c − at) cos t We represented in the figure 2.4 an involute of a circle of radius 1.5, with the origin at the point of parameter 0. The following lemma establishes the connection between the signed curvature of a parameterized curve and that of one of its involutes and will serve as a tool to establish the relation between evolute and involute. Lemma. Let (I, r = r(s)) be a naturally parameterized curve and ρ the involute of r with the origin at c ∈ I. Then the signed curvature of ρ is given by Proof. We have k±[ρ](s) = sgn(k±[r](s)) . |c − s| ρ ′ (s) = r ′ (s) + (c − s)r ′′ (s) − r ′ (s) = (c − s)r ′′ (s) = (c − s)k±[r](s) · Jr ′ (s) ρ ′′ (s) = −k±[r](s·Jr ′ (s) + (c − s)(k±[r](s)) ′ · Jr ′ (s) + (c − s)k±[r](s) · Jr ′′ (s) = [−k±[r](s) + (c − s)(k±[r](s)) ′ ] · Jr ′ (s) − (c − s)(k±[r](s)) 2 · r ′ (s), .
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Lectures on the Di
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Contents Foreword 9 I Curves 11 1 S
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Contents 7 4.5 The tangent vector s
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Foreword This book is based on the
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Part I Curves
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14 Chapter 1. Space curves from the
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16 Chapter 1. Space curves Definiti
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18 Chapter 1. Space curves Figure 1
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20 Chapter 1. Space curves (I, r) i
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22 Chapter 1. Space curves Remark.
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24 Chapter 1. Space curves and ρ
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26 Chapter 1. Space curves given by
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28 Chapter 1. Space curves • a sm
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30 Chapter 1. Space curves Implicit
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32 Chapter 1. Space curves Figure 1
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34 Chapter 1. Space curves Theorem
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36 Chapter 1. Space curves Explicit
Page 38 and 39: 38 Chapter 1. Space curves From thi
Page 40 and 41: 40 Chapter 1. Space curves Theorem
Page 42 and 43: 42 Chapter 1. Space curves 1.7 The
Page 44 and 45: 44 Chapter 1. Space curves Remark.
Page 46 and 47: 46 Chapter 1. Space curves Figure 1
Page 48 and 49: 48 Chapter 1. Space curves 1.9 Orie
Page 50 and 51: 50 Chapter 1. Space curves 1.10 The
Page 52 and 53: 52 Chapter 1. Space curves therefor
Page 54 and 55: 54 Chapter 1. Space curves Proposit
Page 56 and 57: 56 Chapter 1. Space curves or 1 + r
Page 58 and 59: 58 Chapter 1. Space curves or c0 ·
Page 60 and 61: 60 Chapter 1. Space curves Let ω b
Page 62 and 63: 62 Chapter 1. Space curves Corollar
Page 64 and 65: 64 Chapter 1. Space curves vector r
Page 66 and 67: 66 Chapter 1. Space curves a) If a
Page 68 and 69: 68 Chapter 1. Space curves d) We sh
Page 70 and 71: 70 Chapter 1. Space curves Then: τ
Page 72 and 73: 72 Chapter 1. Space curves 1.14.3 T
Page 74 and 75: 74 Chapter 1. Space curves
Page 76 and 77: 76 Chapter 2. Plane curves Proof. I
Page 78 and 79: 78 Chapter 2. Plane curves By subst
Page 80 and 81: 80 Chapter 2. Plane curves The foll
Page 82 and 83: 82 Chapter 2. Plane curves c) J(Jv)
Page 84 and 85: 84 Chapter 2. Plane curves therefor
Page 86 and 87: 86 Chapter 2. Plane curves whence,
Page 90 and 91: 90 Chapter 2. Plane curves whence w
Page 92 and 93: 92 Chapter 2. Plane curves where A(
Page 94 and 95: 94 Chapter 2. Plane curves
Page 96 and 97: 96 Chapter 3. The integration of th
Page 98 and 99: 98 Chapter 3. The integration of th
Page 100 and 101: 100 Chapter 3. The integration of t
Page 104 and 105: 104 Problems 6. A straight line seg
Page 106 and 107: 106 Problems 19. Show that the Bern
Page 108 and 109: 108 Problems 35. Find the envelopes
Page 110 and 111: 110 Problems d) x 2 y 2 = (a 2 −
Page 115 and 116: 4.1 Parameterized surfaces (patches
Page 117 and 118: 4.2. Surfaces 117 Explicit represen
Page 119 and 120: 4.3. The equivalence of local param
Page 121 and 122: 4.3. The equivalence of local param
Page 123 and 124: 4.5. The tangent vector space, the
Page 125 and 126: 4.5. The tangent vector space, the
Page 127 and 128: 4.6. The orientation of surfaces 12
Page 129 and 130: and 4.6. The orientation of surface
Page 131 and 132: 4.7. Differentiable maps on a surfa
Page 133 and 134: 4.7. Differentiable maps on a surfa
Page 135 and 136: where 4.8. The differential of a sm
Page 137 and 138: 4.9. The spherical map and the shap
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4.10. The first fundamental form of
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4.11. The matrix of the shape opera
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4.12. The second fundamental form o
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4.13. The normal curvature. The Meu
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4.14. Asymptotic directions and asy
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4.15. The classification of points
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4.15. The classification of points
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4.16. Principal directions and curv
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4.17. The fundamental equations of
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4.18. The Gauss’ egregium theorem
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4.19 Geodesics 4.19.1 Introduction
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4.19. Geodesics 177 will denote it
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4.19. Geodesics 179 where, as we kn
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Examples of geodesics 4.19. Geodesi
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4.19. Geodesics 183 Here, for the f
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5.1 Ruled surfaces CHAPTER 5 Specia
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5.1. Ruled surfaces 187 Figure 5.2:
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5.1. Ruled surfaces 189 Proof. Sinc
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5.1. Ruled surfaces 191 surfaces, g
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5.1. Ruled surfaces 193 The three c
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5.1. Ruled surfaces 195 We can rega
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5.1. Ruled surfaces 197 5.1.5 Devel
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5.1. Ruled surfaces 199 • The str
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5.2. Minimal surfaces 201 Given a c
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5.2. Minimal surfaces 203 The follo
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5.2. Minimal surfaces 205 Proof. Le
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5.2.3 Ruled minimal surfaces 5.2. M
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5.2. Minimal surfaces 209 asymptoti
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5.2. Minimal surfaces 211 as γ is
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5.3. Surfaces of constant curvature
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5.3. Surfaces of constant curvature
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1. We consider, in the coordinate p
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Problems 219 12. Show that the norm
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Problems 221 33. Find the envelope,
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Problems 223 45. Find the equation
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Problems 225 58. Find the second fu
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Problems 227 67. Through a point M
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[1] Bär, C. - Elementare Different
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Bibliography 231 [29] Jellet, J.H.
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affine normal, 109 arc length, 19,
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182, 185, 187-189, 202, 205, 213, 2
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torus, 118, 119, 154, 191, 217, 219
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Blaga P. Lectures on the differential geometry of - tiera.ru

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