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

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102 Chapter 3. The integration of the natural equations of a curve Then therefore and � y = α = � x = � sin α ds = Figure 3.3: The tractrix � � 1 ds = R � cos α ds = 1 (−a tan t) dt = −t, a tan t cos t(−a tan t) dt = a cos t � � 1 + sin t sin t · a · tan t dt = a ln − sin t . cos t
Problems 1. Write the equations of the geometric locus of the points for which the product of their distances to two fixed points F1 and F2, with d(F1, F2) = 2b, is a constant quantity, equal to a 2 (the curve is called a Cassini oval). 2. We consider a circle of diameter OA = 2a and its tangent at the point A. Through the point O passes a half-line OC, where C is the intersection point between the half-line and the circle. On this half-line one takes a segment OM, of length BC, where B is the intersection point between the half-line and the tangent to the circle at A. If we let the half-line OC rotate around the point O, then the point M describe a curve, called the Diocles’ cissoid. Find the equations of this curve. 3. An arbitrary half-line OE, starting from the origin O, intersects at the points D and E the circle x 2 � + y − a �2 = 2 a2 4 and its tangent at C, the opposite of O on the circle. We draw, through the points D and E, two straight line, parallel to the axes Ox and Oy, respectively. The two lines intersect at M. Find the equations of the geometrical locus of M ( the witch of Maria Agnesi). 4. A point M is moving uniformly on a straight line ON, which is rotating with constant angular velocity around the point O. Find the equations of the trajectory of M (Archimede’s spiral). 5. A straight line OL rotates around the point O with a constant angular velocity ω. A point M moves along the line OL with a speed proportional to the distance OM. Find the equation of the curve described by the point M (the logarithmical spiral).
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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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34 Chapter 1. Space curves Theorem
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36 Chapter 1. Space curves Explicit
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38 Chapter 1. Space curves From thi
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40 Chapter 1. Space curves Theorem
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42 Chapter 1. Space curves 1.7 The
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44 Chapter 1. Space curves Remark.
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48 Chapter 1. Space curves 1.9 Orie
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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 88 and 89: 88 Chapter 2. Plane curves Figure 2
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 112 and 113: 112 Chapter 3. The integration of t
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
Page 139 and 140: 4.10. The first fundamental form of
Page 145 and 146: 4.11. The matrix of the shape opera
Page 147 and 148: 4.12. The second fundamental form o
Page 149 and 150: 4.13. The normal curvature. The Meu
Page 151 and 152: 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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