Lecture Notes for 120 - UCLA Department of Mathematics

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5.5. RULED SURFACES 122 (16) Let q (u, v) be a parametrized surface and q ✏ = q+✏N the parallel surface at distance ✏ from q. (a) Show that @q ✏ @w = @q @w + ✏@N @w =(I + ✏L) ✓ @q @w where I is the identity map I (v) =v. n (b) Show that q ✏ is a parametrized surface if |✏| < min that N is also the natural normal to q ✏ . (c) Show that L ✏ = L (I + ✏L) 1 by using that L ✓ ◆ @q = @N ✓ ◆ @q ✏ @w @w = L✏ @w ◆ 1 |apple , 1 1| |apple 2| o and (d) Show that these surfaces all have the same principal directions and that the principal curvatures satisfy 1 = apple i 1+✏apple i apple ✏ i (17) Let q (u, v) be a parametrized surface and q ✏ = q+✏N the parallel surface at distance ✏ from q. (a) Show that I ✏ =I 2✏II + ✏ 2 III (b) Show that II ✏ =II ✏III (c) Show that III ✏ =III (d) How do you reconcile this with the formula L ✏ = L (I + ✏L) 1 from the previous exercise? (e) Show that K ✏ K = 1+2✏H + ✏ 2 K and H ✏ H ✏K = 1+2✏H + ✏ 2 K 5.5. Ruled Surfaces Aruledsurfaceisparameterizedbyselectingacurve↵ (v) and then considering the surface one gets by adding a line through each of the points on the curve. If the directions of those lines are given by X (v), thenthesurfacecanbeparametrized by q (u, v) =↵ (v) +uX (v). We can without loss of generality assume that X is a unit field. The condition for obtaining a parametrized surface is that @q @u = X and @q @v = d↵ dv + u dX dv are linearly independent. Even though we don’t always obtain asurfaceforallparametervaluesitisimportanttoconsidertheextendedlinesin the rulings for all values of v. For example, generalized cones are best recognized by their cone point which is not part of the surface. Similarly tangent developables
5.5. RULED SURFACES 123 are best recognized by the generating curve which is also not part of the surface. One of our goals is to understand when these two situations occur as that might not be so obvious from a general parametrization. An example of a cone that is not rotationally symmetric is the elliptic cone The elliptic hyperboloid x 2 x 2 a 2 + y2 b 2 = z2 a 2 + y2 b 2 = z2 +1 is an example of a surface that is ruled in two different ways, but which does not have zero Gauss curvature. We can let ↵ (t) =(a cos (t) ,bsin (t) , 0) be the ellipse where z =0. The fields generating the lines are given by X = d↵ +(0, 0, ±1) dt and it is not difficult to check that ✓ ◆ d↵ q (s, t) =↵ (t)+s dt +(0, 0, ±1) are both rulings of the elliptic hyperboloid. Proposition 5.5.1. Ruled surfaces have non-positive Gauss curvature and the Gauss curvature vanishes if and only if ✓ X ⇥ d↵ ◆ · dX dv dv =0. In particular, generalized cylinders, generalized cones, and tangent developables have vanishing Gauss curvature. Proof. Since @2 q @u =0it follows that L 2 uu =0. Thus K = g uu g vv Moreover, K vanishes precisely when L 2 uv guv 2 apple 0 @ 2 q @u@v = dX dv is perpendicular to the normal. Since the normal is given by this translates to 0 = = N = X ⇥ X ⇥ which is what we wanted to prove. d↵ dv + u dX dv d↵ dv + u dX dv ✓ ✓ ◆◆ d↵ X ⇥ dv + udX · dX dv dv ✓ X ⇥ d↵ ◆ · dX dv dv ⇤
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Classical Differential Geometry Pet
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CONTENTS 4 Chapter 7. Other Topics
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1.1. CURVES 6 measurements are need
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1.1. CURVES 8 as we then get q (✓
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1.1. CURVES 10 these derivatives ar
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1.1. CURVES 12 (a) Show that if r (
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1.2. ARCLENGTH AND LINEAR MOTION 14
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1.3. CURVATURE 20 relates the side
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1.3. CURVATURE 22 of a that is norm
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1.3. CURVATURE 24 As the piece of t
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1.4. INTEGRAL CURVES 26 (14) Let q
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1.4. INTEGRAL CURVES 28 So at q 0 =
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CHAPTER 2 Planar Curves 2.1. Genera
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2.2. THE FUNDAMENTAL EQUATIONS 32 M
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2.2. THE FUNDAMENTAL EQUATIONS 34 (
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2.3. LENGTH AND AREA 36 (22) (Newto
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As ( 2.3. LENGTH AND AREA 38 sin
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2.3. LENGTH AND AREA 40 The choice
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2.4. THE ROTATION INDEX 42 the hypo
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2.4. THE ROTATION INDEX 44 where an
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2.4. THE ROTATION INDEX 46 Exercise
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2.5. TWO SURPRISING RESULTS 48 unit
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2.6. CONVEX CURVES 50 Exercises. Po
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2.6. CONVEX CURVES 52 Proof. Any cu
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2.6. CONVEX CURVES 54 (9) Give an e
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3.1. THE FUNDAMENTAL EQUATIONS 56 T
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3.1. THE FUNDAMENTAL EQUATIONS 58 B
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3.1. THE FUNDAMENTAL EQUATIONS 60 (
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3.2. CHARACTERIZATIONS OF SPACE CUR
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3.2. CHARACTERIZATIONS OF SPACE CUR
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3.3. CLOSED SPACE CURVES 66 (12) Pr
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3.3. CLOSED SPACE CURVES 68 Thus th
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3.3. CLOSED SPACE CURVES 70 (b) Sho
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Page 74 and 75: 4.1. SURFACES 74 Definition 4.1.6.
Page 76 and 77: 4.1. SURFACES 76 (6) Many classical
Page 78 and 79: 4.2. TANGENT SPACES AND MAPS 78 Thi
Page 80 and 81: 4.2. TANGENT SPACES AND MAPS 80 We
Page 82 and 83: 4.2. TANGENT SPACES AND MAPS 82 Def
Page 84 and 85: 4.3. THE ABSTRACT FRAMEWORK 84 Befo
Page 86 and 87: 4.3. THE ABSTRACT FRAMEWORK 86 in o
Page 88 and 89: 4.4. THE FIRST FUNDAMENTAL FORM 88
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Page 98 and 99: 4.6. THE GAUSS FORMULAS 98 and then
Page 100 and 101: CHAPTER 5 The Second Fundamental Fo
Page 102 and 103: 5.1. CURVES ON SURFACES 102 This sh
Page 104 and 105: 5.1. CURVES ON SURFACES 104 We also
Page 106 and 107: 5.1. CURVES ON SURFACES 106 Exercis
Page 108 and 109: 5.2. THE GAUSS AND WEINGARTEN MAPS
Page 110 and 111: 5.2. THE GAUSS AND WEINGARTEN MAPS
Page 112 and 113: 5.3. THE GAUSS AND MEAN CURVATURES
Page 118 and 119: 5.4. PRINCIPAL CURVATURES 118 The f
Page 120 and 121: 5.4. PRINCIPAL CURVATURES 120 By le
Page 124 and 125: 5.5. RULED SURFACES 124 Definition
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Page 128 and 129: 5.5. RULED SURFACES 128 This shows
Page 130 and 131: 5.5. RULED SURFACES 130 (b) Show th
Page 132 and 133: 5.5. RULED SURFACES 132 (e) Show th
Page 134 and 135: 6.1. CALCULATING CHRISTOFFEL SYMBOL
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Page 140 and 141: 6.3. ACCELERATION 140 Exercises. (1
Page 142 and 143: 6.3. ACCELERATION 142 Alternately t
Page 144 and 145: 6.4. THE GAUSS AND CODAZZI EQUATION
Page 150 and 151: 6.5. LOCAL GAUSS-BONNET 150 (b) Use
Page 152 and 153: 6.5. LOCAL GAUSS-BONNET 152 We furt
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Page 156 and 157: 6.5. LOCAL GAUSS-BONNET 156 where t
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Page 160 and 161: 7.2. UNPARAMETRIZED GEODESICS 160 T
Page 162 and 163: 7.5. ISOMETRIES 162 since the v-cur
Page 164 and 165: 7.6. THE UPPER HALF PLANE 164 Proof
Page 166 and 167: 7.6. THE UPPER HALF PLANE 166 and t
Page 168 and 169: 7.6. THE UPPER HALF PLANE 168 so so
Page 170 and 171: 8.2. RIEMANNIAN GEOMETRY 170 The ke
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APPENDIX A Vector Calculus A.1. Vec
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A.2. GEOMETRY 174 the characteristi
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A.4. DIFFERENTIAL EQUATIONS 176 The
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A.4. DIFFERENTIAL EQUATIONS 178 Nex
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APPENDIX B Special Coordinate Repre
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This leaves us with finding L s s.
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B.3. MONGE PATCHES 184 So we immedi
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B.4. SURFACES GIVEN BY AN EQUATION
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B.6. CHEBYSHEV NETS 188 (1) Show th
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B.7. ISOTHERMAL COORDINATES 190 K =
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Lecture Notes for 120 - UCLA Department of Mathematics

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