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

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40 Chapter 1. Space curves Theorem 1.6.1. The osculating planes to two equivalent parameterized curves at corresponding biregular points coincides. Proof. Let (I, r = r(t)) and (J, ρ = ρ(s)) be two equivalent parameterized curves and λ : I → J – the parameter change. Then r(t) = ρ(λ(t)), r ′ (t) = ρ ′ (λ(t)) · λ ′ (t), r ′′ (t) = ρ ′′ (λ(t)) · (λ ′ (t)) 2 + ρ ′ (λ(t)) · λ ′′ (t). Since λ ′ (t) � 0, from these relations it follows that the system of vectors {r ′ (t), r ′′ (t)} and {ρ ′ (λ(t)), ρ ′′ (λ(t))} are equivalent, i.e. they generate the same vector subspace R 3 , hence the osculating planes to the two parameterized curves at the corresponding points t0 and λ(t0) have the same directing subspace, therefore, they are parallel. As they have a common point (since r(t0) = ρ(λ(t0)), they have to coincide. � Remark. From the previous theorem it follows that the notion of an osculating plane makes sense also for regular curves. As in the case of the tangent line, there is a more geometric way of defining the osculating plane which is, at the same time, more general, since it can be applied also for the case of the points which are not biregular. Let r(t0) ¸si r(t0 + ∆t) two neighboring points on a parameterized curve, with r(t0) biregular. We consider a plane α, of normal versor e, passing through r(t0). and denote d(∆t) = d(r(t0 + ∆t), α). Theorem 1.6.2. α is the osculating plane to the parameterized curve r = r(t) at the biregular point r(t0) iff d(∆t) lim = 0, ∆t→0 |∆t| 2 i.e. the curve and the plane have a second order contact. Proof. From the Taylor formula we have r(t0 + ∆t) = r(t0) + ∆t · r ′ (t0) + 1 2 (∆t)2 · r ′′ (t0) + (∆t) 2 · ɛ, with lim ɛ = 0. ∆t→0 On the other hand, d(∆t) = |e · (r(t0 + ∆t) − r(t0))| = = |(e · r ′ (t0)) · ∆t + 1 2 (e · r′′ (t0)) · (∆t) 2 + (e · ɛ) · (∆t) 2 |.
Thus, lim ∆t→0 d(∆t) = lim |∆t| 2 ∆t→0 1.6. The osculating plane 41 � � � � � e · r � � � ′ (t0) ∆t � � = lim �e · r � ∆t→0 � ′ (t0) ∆t + 1 2 · (e · r′′ � � � � (t0)) + �� e · ɛ � � � →0 � = + 1 2 · (e · r′′ � � (t0)) � � � . d(∆t) If lim ∆t→0 |∆t| 2 = 0, then e · r ′ (t0) = 0 and e · r ′′ (t0) = 0, i.e. e � r ′ (t0) × r ′′ (t0), meaning that α is the osculating plane. The converse is obvious. � Remarks. (i) The previous theorem justifies the name of the osculating plane. Actually, the name (coined by Johann Bernoulli), comes from the Latin verb osculare, which means to kiss end emphasizes the fact that, among all the planes that are passing through a given point of a curve, the osculating plane has the higher order (“the closer”) contact. (ii) Defining the osculating plane through the contact, one can define the notion of an osculating plane also for the points which are not biregular, but in that case any plane passing through the tangent is an osculating plane, in the sense that it has a second order contact with the curve. Saying that the osculating plane at a biregular point is the only plane which has at that point a second order contact with the curve is the same with saying that the osculating plane is the limit position of a plane determined by the considered point and two neighbouring points when this ones are approaching indefinitely to the given one. Also, on can define the osculating plane at biregular point of a parameterized curve as being the limit position of a plane passing through the tangent at the given point and a neighboring point from the curve, when this point is approaching indefinitely the given one. A natural question that one may ask is what happens with the osculating plane in the particular case of a plane parameterized curve. The answer is given by the following proposition, whose prove is left to the reader: Proposition 1.6.1. If a biregular parameterized curve is plane, i.e. its support is contained into a plane π, then the osculating plane to this curve at each point coincides to the plane of the curve. Conversely, if a given biregular parameterized curve has the same osculating plane at each point, then the curve is plane and its support is contained into the osculating plane.
Page 1: Lectures on the Di
Page 5 and 6: Contents Foreword 9 I Curves 11 1 S
Page 7 and 8: Contents 7 4.5 The tangent vector s
Page 9 and 10: Foreword This book is based on the
Page 11: Part I Curves
Page 14 and 15: 14 Chapter 1. Space curves from the
Page 16 and 17: 16 Chapter 1. Space curves Definiti
Page 18 and 19: 18 Chapter 1. Space curves Figure 1
Page 20 and 21: 20 Chapter 1. Space curves (I, r) i
Page 22 and 23: 22 Chapter 1. Space curves Remark.
Page 24 and 25: 24 Chapter 1. Space curves and ρ
Page 26 and 27: 26 Chapter 1. Space curves given by
Page 28 and 29: 28 Chapter 1. Space curves • a sm
Page 30 and 31: 30 Chapter 1. Space curves Implicit
Page 34 and 35: 34 Chapter 1. Space curves Theorem
Page 36 and 37: 36 Chapter 1. Space curves Explicit
Page 38 and 39: 38 Chapter 1. Space curves From thi
Page 42 and 43: 42 Chapter 1. Space curves 1.7 The
Page 44 and 45: 44 Chapter 1. Space curves Remark.
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
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Page 72 and 73: 72 Chapter 1. Space curves 1.14.3 T
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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
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90 Chapter 2. Plane curves whence w
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92 Chapter 2. Plane curves where A(
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94 Chapter 2. Plane curves
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96 Chapter 3. The integration of th
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98 Chapter 3. The integration of th
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100 Chapter 3. The integration of t
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104 Problems 6. A straight line seg
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106 Problems 19. Show that the Bern
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108 Problems 35. Find the envelopes
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110 Problems d) x 2 y 2 = (a 2 −
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4.1 Parameterized surfaces (patches
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4.2. Surfaces 117 Explicit represen
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4.3. The equivalence of local param
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4.3. The equivalence of local param
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4.5. The tangent vector space, the
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4.5. The tangent vector space, the
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4.6. The orientation of surfaces 12
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and 4.6. The orientation of surface
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4.7. Differentiable maps on a surfa
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4.7. Differentiable maps on a surfa
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where 4.8. The differential of a sm
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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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