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Unformatted text preview: One way to compute the curvature of an ellipse Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ ∼ squash/ 11 February, 2009 (at 12:50 ) [ NB: We use angle-brackets, h v , a i , to mean the inner-product ( dot-product ) of vectors v and a . ] We want to compute the curvature function of the ellipse, E , whose axes-of-symmetry are the coordinate axes and whose semi-axis lengths are A and B ; so A,B > 0. In cartesian coordinates, E is the solution set of the equation x 2 A 2 + y 2 B 2 = 1 2 . 1 : If A = B then this ellipse is circle. If A > B then the major-axis direction is horizontal, etc. The first issue in computing the curvature function of E is to find a representation of E for which we have a corresponding curvature formula. We don’t yet have a formula which applies to (1); so far, our only curvature formulae apply to either the graph of a 1-variable function or to the image of a parameterized curve....
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