Lecture Notes for 120 - UCLA Department of Mathematics

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2.6. CONVEX CURVES 52 Proof. Any curve with nonnegative curvature always locally lies on the left of its tangent lines. So if it comes back to intersect a tangent l after having travelled to the left of l, then there will be a point of locally maximal distance to the left of l. Atthislocalmaximumthetangentlinel ⇤ must be parallel but not equal to l. If they are oriented in the same direction, then the curve will locally be on the right of l ⇤ . As that does not happen they have opposite direction. This shows that the total curvature is ⇡. However, the curve will have strictly larger total curvature as it still has to make its way back so that it can intersect l. ⇤ Theorem 2.6.4. If a closed curve has nonnegative signed curvature and total curvature apple 2⇡, then it is convex. Proof. The argument is similar to the one above. Assume that we have a tangent line l such that the curve lies on both sides of this line. As the curve is closed there’ll be points one both sides of this tangent at maximal distance from the tangent. The tangent lines at the points l ⇤ and l ⇤⇤ are then parallel to l. Thus we have three parallel tangent lines that are not equal. Two of these must correspond to unit tangents that point in the same direction. As the curvature does not change sign this implies that the total curvature of part of the curve is 2⇡. The total curvature must then be > 2⇡ as these two tangent lines are different and the curve still has to return to both of the points of contact. ⇤ Example 2.6.5. Euler’s construction: A piecewise smooth simple closed planar curve with n>2 cusps and the property that tangents at different points are never parallel has the property that its involutes are curves of constant width. The cusps are the points where the curve is not smooth and we assume that the unit tangents are opposite at those points, i.e., the interior angles are zero at the non-smooth points. Exercises. (1) Let q (✓) be a simple closed planar curve with apple>0 parametrized by ✓, where ✓ is defined as the arclength parameter of the unit tangent field T. Show that dq d✓ = 1 apple T dT d✓ = N dN d✓ = T T (✓ + ⇡) = T (✓) (2) Let q (✓) be a simple closed planar curve with apple>0 parametrized by ✓, where ✓ is defined as the arclength parameter of the unit tangent field T. Define v (✓) as the distance from the origin to the tangent line through q (✓). (a) Show that v (✓) = q (✓) · N (✓) (b) Show that the width (distance) between the parallel tangent lines through q (✓) and q (✓ + ⇡) is w (✓) =v (✓)+v (✓ + ⇡) =N (✓) · (q (✓ + ⇡) q (✓))
2.6. CONVEX CURVES 53 (c) Show that: L (q) = ˆ 2⇡ 0 v (✓) d✓ (d) Show that 1 apple = v + d2 v d✓ 2 (e) Let A denote the area enclosed by the curve. Establish the following formulas for A A = 1 ˆ L vds = 1 ˆ 2⇡ ✓ ◆ v 2 + v d2 v 2 0 2 0 d✓ 2 d✓ = 1 ˆ 2⇡ ✓ ◆ ! 2 dv v 2 d✓ 2 0 d✓ (3) Let q (✓) be a simple closed planar curve with apple>0 parametrized by ✓, where ✓ is defined as the arclength parameter of the unit tangent field T. Show that the width from the previous problem satisfies: d 2 w d✓ 2 + w = 1 apple (✓) + 1 apple (✓ + ⇡) . (4) Let q (✓) be a simple closed planar curve with apple>0 parametrized by ✓, where ✓ is defined as the arclength parameter of the unit tangent field T. With the width defined as in the previous exercises show that: ˆ 2⇡ 0 wd✓ =2L (q) (5) Let q (✓) be a simple closed planar curve of constant width with apple>0 parametrized by ✓, where ✓ is defined as the arclength parameter of the unit tangent field T. (a) Show that if ✓ corresponds to a local maximum for apple, thentheopposite point ✓ + ⇡ corresponds to a local minimum. (b) Assume for the remainder of the exercise that apple has a finite number of critical points and that they are all local maxima or minima. Show that the number of vertices is even 2n, n =3, 4, 5... (c) Show that each point on the evolute corresponds to two points on the curve. (d) Show that the evolute consists of n convex curves that are joined at n cusps that correspond to pairs of vertices on the curve. (e) Show that the evolute has no double tangents. (6) (Euler) Reverse the construction in the previous exercise to create curves of constant width by taking involutes of suitable curves. (7) Let q be a closed convex curve and l aline. (a) Show that l can only intersect q in one point, two points, or a line segment. (b) Show that if l is also a tangent line then it cannot intersect q in only two points. (c) Show that the interior of q is convex, i.e., the segment between any two points in the interior also lies in the interior. (8) Let q be a planar curve with nonnegative signed curvature. Show that if q has a double tangent, then its total curvature is 2⇡. Note that it is possible for the double tangent to have opposite directions at the points of tangency.
Page 1: Classical Differential Geometry Pet
Page 4 and 5: CONTENTS 4 Chapter 7. Other Topics
Page 6 and 7: 1.1. CURVES 6 measurements are need
Page 8 and 9: 1.1. CURVES 8 as we then get q (✓
Page 10 and 11: 1.1. CURVES 10 these derivatives ar
Page 12 and 13: 1.1. CURVES 12 (a) Show that if r (
Page 14 and 15: 1.2. ARCLENGTH AND LINEAR MOTION 14
Page 20 and 21: 1.3. CURVATURE 20 relates the side
Page 22 and 23: 1.3. CURVATURE 22 of a that is norm
Page 24 and 25: 1.3. CURVATURE 24 As the piece of t
Page 26 and 27: 1.4. INTEGRAL CURVES 26 (14) Let q
Page 28 and 29: 1.4. INTEGRAL CURVES 28 So at q 0 =
Page 30 and 31: CHAPTER 2 Planar Curves 2.1. Genera
Page 32 and 33: 2.2. THE FUNDAMENTAL EQUATIONS 32 M
Page 34 and 35: 2.2. THE FUNDAMENTAL EQUATIONS 34 (
Page 36 and 37: 2.3. LENGTH AND AREA 36 (22) (Newto
Page 38 and 39: As ( 2.3. LENGTH AND AREA 38 sin
Page 40 and 41: 2.3. LENGTH AND AREA 40 The choice
Page 42 and 43: 2.4. THE ROTATION INDEX 42 the hypo
Page 44 and 45: 2.4. THE ROTATION INDEX 44 where an
Page 46 and 47: 2.4. THE ROTATION INDEX 46 Exercise
Page 48 and 49: 2.5. TWO SURPRISING RESULTS 48 unit
Page 50 and 51: 2.6. CONVEX CURVES 50 Exercises. Po
Page 54 and 55: 2.6. CONVEX CURVES 54 (9) Give an e
Page 56 and 57: 3.1. THE FUNDAMENTAL EQUATIONS 56 T
Page 58 and 59: 3.1. THE FUNDAMENTAL EQUATIONS 58 B
Page 60 and 61: 3.1. THE FUNDAMENTAL EQUATIONS 60 (
Page 62 and 63: 3.2. CHARACTERIZATIONS OF SPACE CUR
Page 64 and 65: 3.2. CHARACTERIZATIONS OF SPACE CUR
Page 66 and 67: 3.3. CLOSED SPACE CURVES 66 (12) Pr
Page 68 and 69: 3.3. CLOSED SPACE CURVES 68 Thus th
Page 70 and 71: 3.3. CLOSED SPACE CURVES 70 (b) Sho
Page 72 and 73: CHAPTER 4 Basic Surface Theory 4.1.
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
Page 94 and 95: 4.5. SPECIAL MAPS AND PARAMETRIZATI
Page 96 and 97: 4.5. SPECIAL MAPS AND PARAMETRIZATI
Page 98 and 99: 4.6. THE GAUSS FORMULAS 98 and then
Page 100 and 101: CHAPTER 5 The Second Fundamental Fo
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5.1. CURVES ON SURFACES 102 This sh
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5.1. CURVES ON SURFACES 104 We also
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5.1. CURVES ON SURFACES 106 Exercis
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5.2. THE GAUSS AND WEINGARTEN MAPS
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5.2. THE GAUSS AND WEINGARTEN MAPS
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5.3. THE GAUSS AND MEAN CURVATURES
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5.4. PRINCIPAL CURVATURES 118 The f
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5.4. PRINCIPAL CURVATURES 120 By le
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5.5. RULED SURFACES 122 (16) Let q
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5.5. RULED SURFACES 124 Definition
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i.e., d dv and X are parallel. In p
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5.5. RULED SURFACES 128 This shows
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5.5. RULED SURFACES 130 (b) Show th
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5.5. RULED SURFACES 132 (e) Show th
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6.1. CALCULATING CHRISTOFFEL SYMBOL
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6.1. CALCULATING CHRISTOFFEL SYMBOL
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6.2. GENERALIZED AND ABSTRACT SURFA
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6.3. ACCELERATION 140 Exercises. (1
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6.3. ACCELERATION 142 Alternately t
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6.4. THE GAUSS AND CODAZZI EQUATION
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6.5. LOCAL GAUSS-BONNET 150 (b) Use
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6.5. LOCAL GAUSS-BONNET 152 We furt
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6.5. LOCAL GAUSS-BONNET 154 clockwi
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6.5. LOCAL GAUSS-BONNET 156 where t
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7.1. GEODESICS 158 we must solve a
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7.2. UNPARAMETRIZED GEODESICS 160 T
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7.5. ISOMETRIES 162 since the v-cur
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7.6. THE UPPER HALF PLANE 164 Proof
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7.6. THE UPPER HALF PLANE 166 and t
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7.6. THE UPPER HALF PLANE 168 so so
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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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