Lecture Notes for 120 - UCLA Department of Mathematics

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2.2. THE FUNDAMENTAL EQUATIONS 32 Moreover, given an initial position q (0) and unit direction T (0) the curve q (t) is uniquely determined by its speed and signed curvature. Proof. The three equations are simple to check as T, N ± form an orthonormal basis. For fixed speed and signed curvature functions these equations form a differential equation which has a unique solution given the initial values q (0), T (0) and N ± (0). The normal vector is determined by the unit tangent so we have all of that data. ⇤ We offer a combined characterization of lines and circles as the only curves with constant curvature. Theorem 2.2.3. A planar curve is part of a line if and only if its signed curvature vanishes. A planar curve is part of a circle if and only if its signed curvature is non-zero and constant. Proof. If the curvature vanishes then we already know that it has to be a straight line. If the curve is a circle of radius R with center c, then Differentiating this yields |q (s) c| 2 = R 2 T · (q (s) c) =0 Thus the unit tangent is perpendicular to the radius vector q (s) again yields apple ± N ± · (q (s) c)+1=0 c. Differentiating However the normal and radius vectors must be parallel so their inner product is ±R. This shows that the curvature is constant. We also obtain the equation q = c 1 apple ± N ± This indicates that, if we take a curve with constant curvature, then we should attempt to show that c = q + 1 apple ± N ± is constant. Since apple ± is constant the derivative of this curve is so it is constant and dc ds = T + 1 ( apple ± apple ± T)=0 |q (s) c| 2 = 1 apple ± N ± 2 = 1 apple 2 ± showing that q is a circle of radius 1 apple ± centered at c. ⇤ Proposition 2.2.4. The evolute of a unit speed planar curve q (s) of non-zero curvature is given by q ⇤ = q + 1 apple ± N ± = q + 1 apple N
2.2. THE FUNDAMENTAL EQUATIONS 33 Proof. This follows from dq ⇤ ds = T + 1 ( apple ± T)+ d apple ± ds ✓ ◆ 1 N ± = d ✓ ◆ 1 N ± apple ± ds apple ± ⇤ Exercises. (1) Show that a planar curve is part of a line if all its tangent lines pass through a fixed point. (2) Compute the signed curvature of q (t) = t, t 3 and show that it has a critical point at t =0,isnegativefort0. (3) Let q (s) =(x (s) ,y(s)) : [0,L] ! R 2 be a unit speed planar curve with signed curvature apple ± (s) and q ⇤ (s) =x (s) e 1 + y (s) e 2 + x another planar curve where e 1 ,e 2 is a positively oriented orthonormal basis and x apoint. (a) Show that q ⇤ is a unit speed curve with curvature apple ⇤ ± (s) =apple ± (s). (b) Show that a planar unit speed curve with the same curvature as q is of the form q ⇤ . (4) Compute the signed curvature of the logarithmic spiral ae bt (cos t, sin t) (5) Compute the signed curvature of the spiral of Archimedes: (a + bt) (cos t, sin t) (6) Show that if a planar unit speed curve q (s) satisfies: apple ± (s) = 1 es + f for constants e, f > 0, thenitisalogarithmicspiral. (7) Show that a planar curve is part of a circle if all its normal lines pass through a fixed point. ds (8) Show that apple ± dt =det⇥ ⇤ T dT dt . (9) Let q (t) =r (t) (cos t, sin t). Showthatthespeedisgivenby ✓ ◆ 2 ✓ ◆ 2 ds dr = + r 2 dt dt and the curvature apple ± = 2 dr 2 dt + r 2 r d2 r dt ⇣ ⌘ 2 3 2 2 + r 2 dr dt Parametrize the curve 1 x 2 x 2 = y 2 in this way and compute its curvature. Note that such a parametrization won’t be valid for all t. (10) Assume a planar curve is given as a level set F (x, y) =c where rF 6= 0 everywhere along the curve. We orient and parametrize the curve so that ⇣ v = @F @y , @F @x ⌘ . (a) Show that the signed normal is given by N ± = rF |rF |
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 =
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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 52 and 53: 2.6. CONVEX CURVES 52 Proof. Any cu
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
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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
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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
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4.2. TANGENT SPACES AND MAPS 82 Def
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4.3. THE ABSTRACT FRAMEWORK 84 Befo
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4.3. THE ABSTRACT FRAMEWORK 86 in o
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4.4. THE FIRST FUNDAMENTAL FORM 88
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4.5. SPECIAL MAPS AND PARAMETRIZATI
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4.5. SPECIAL MAPS AND PARAMETRIZATI
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4.6. THE GAUSS FORMULAS 98 and then
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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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