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Unformatted text preview: Math 4441 Sep 14, 2007 1 Differential Geometry Fall 2007, Georgia Tech Lecture Notes 6 1.15 The four vertex theorem for convex curves A vertex of a planar curve : I R 2 is a point where the signed curvature of has a local max or min. Exercise 1. Show that an ellipse has exactly 4 vertices, unless it is a circle. We say that a planar curve is convex if through each point in the image of it there passes a line with respect to which the curve lies on side. The main aim of this section is to show that: Theorem 2. Any convex C 3 planar curve has (at least) four vertices. In fact any simple closed curve has 4 vertices, and it is not necessary to assume that is C 1 , but the proof is hader. On the other hand if the curve is not simple, then the 4 vertex property may no longer be true: Exercise 3. Sketch the limacon : [0 , 2 ] R 2 given by ( t ) := (2 cos t + 1)(cos t, sin t ) and show that it has only two vertices. ( Hint : It looks like a loop with a smaller loop inside) The proof of the above theorem is by contradiction. Suppose that has fewer than 4 vertices, then it must have exactly 2. Exercise 4. Verify the last sentence. Suppose that these two vertices occur at t and t 1 . Then ( t ) will have one sign on ( t 1 , t 2 ) and the opposite sign on I- [ t 1 , t 2 ]. Let ` be the line passing through ( t 1 ) and ( t 2 ). Then, since is convex, restricted to ( t 1 , t 2 ) lies entirely in one of the closed half planes determined by ` and restricted to I- [ t 1 , t 2 ] lies in the other closed half plane. 1 Last revised: September 20, 2007 1 Exercise 5. Verify the last sentence, i.e., show that if : I R 2 is a simple closed convex planar curve, and ` is any line in the plane which intersects ( I ), then either ` intersects in exactly two points, or ( I ) lies on one side ` .( Hint : Show that if intersects ` at 3 points, then it lies on one side of ` .) Let p be a point of ` and v be a vector orthogonal to ` , then f : I R , given by f ( t ) := h ( t )- p, v i has one sign on ( t 1 , t 2 ) and has the opposite sign on I- [ t 1 , t 2 ]. Consequently, ( t ) f ( t ) is always nonnegative. So < Z I ( t ) h ( t )- p, v i dt. On the other hand Z I ( t ) h ( t )- p, v i dt = ( t ) h ( t )- p, v i| b a- Z I ( t ) h T ( t ) , v i dt = 0- Z I h- N ( t ) , v i dt = h N ( t ) , v i| b a = 0 . So we have a contradiction, as desired. Exercise 6. Justify each of the lines in the above computation. 1.16 Shurs Arm Lemma The following result describes how the distance between the end points of a planar curve is effected by its curvature: Theorem 7 (Shurs Arm Lemma) . Let 1 , 2 : [0 , L ] R 3 be unit speed C 1 curves such that the union of each i with the line segment from i (0) to i ( L ) is a convex curve. Suppose that for almost all t [0 , L ] , i ( t ) is well defined, e.g., i is piecewise C 2 , and 1 ( t ) 2 ( t ) for almost all...
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