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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 α : I → R 2 at a point a of nonvanishing curvature t ∈ I as the circle with radius r ( t ) and center at α ( t ) + r ( t ) N ( t ) where r ( t ) := 1 κ ( t ) is called the radius of curvature of α . 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 α at α ( t ) and is also passing through α ( s ). Show that κ ( 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 α ( t ) = (0 , 0) and α ( t ) is parallel to the x-axis. Then, we may assume that α ( t ) = ( t, f ( t )), for some function f : R → R with f (0) = 0 and f (0) = 0. Further, recall that κ ( t ) = | f 00 ( t ) | ( p 1 + f ( t ) 2 ) 3 . 1 Last revised: October 2, 2007 1 Thus κ (0) = | f 00 (0) | . Next note that the center of the circle which is tangent to α 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 L’Hopital’s rule, we have lim s → 2 | f ( s ) | f 2 ( s ) + s 2 = | f 00 (0) | = κ (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 α : I → R 2 be a planar curve and t , t 1 , t 2 ∈ I with t 1 ≤ t ≤ t 2 . Show that κ ( t ) is the reciprocal of the limit of the radius of the circles which pass through α ( t ), α ( t 1 ) and α ( t 2 ) as t 1 , t 2 → t ....
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- Spring '08
- Geometry, Osculating circle, Radius of curvature, total curvature