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Unformatted text preview: Math 4441 Sep 20, 2007 1 Differential Geometry Fall 2007, Georgia Tech Lecture Notes 5 1.13 Osculating Circle and Radius of Curvature Recall that in a previous section we defined the osculating circle of a planar curve : I R 2 at a point a of nonvanishing curvature t I as the circle with radius r ( t ) and center at ( t ) + r ( t ) N ( t ) where r ( t ) := 1 ( t ) is called the radius of curvature of . If we had a way to define the osculating circle independently of curvature, then we could define curvature simply as the reciprocal of the radius of the osculating circle, and thus obtain a more geometric definition for curvature. Exercise 1. Let r ( s, t ) be the radius of the circle which is tangent to at ( t ) and is also passing through ( s ). Show that ( t ) = lim s t r ( s, t ) . To do the above exercise first recall that, as we showed in the previous lecture, curvature is invaraint under rigid motions. Thus, after a rigid motion, we may assume that ( t ) = (0 , 0) and ( t ) is parallel to the x-axis. Then, we may assume that ( t ) = ( t, f ( t )), for some function f : R R with f (0) = 0 and f (0) = 0. Further, recall that ( t ) = | f 00 ( t ) | ( p 1 + f ( t ) 2 ) 3 . 1 Last revised: October 2, 2007 1 Thus (0) = | f 00 (0) | . Next note that the center of the circle which is tangent to at (0 , 0) must lie on the y-axis at some point (0 , r ), and for this circle to also pass through the point ( s, f ( s )) we must have: r 2 = s 2 + ( r- f ( s )) 2 . Solving the above equation for r and taking the limit as s 0, via the LHopitals rule, we have lim s 2 | f ( s ) | f 2 ( s ) + s 2 = | f 00 (0) | = (0) , which is the desired result. Note 2. The above limit can be used to define a notion of curvature for curves that are not twice differentiable. In this case, we may define the upper curvature and lower curvature respectively as the upper and lower limit of 2 | f ( s ) | f 2 ( s ) + s 2 . as s 0. We may even distinguish between right handed and left handed upper or lower curvature, by taking the right handed or left handed limits respectively. Exercise* 3. Let : I R 2 be a planar curve and t , t 1 , t 2 I with t 1 t t 2 . Show that ( t ) is the reciprocal of the limit of the radius of the circles which pass through ( t ), ( t 1 ) and ( t 2 ) as t 1 , t 2 t ....
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