Interpolation with Curvature Constraints
LETHBRIDGE UNIV (ALBERTA) DEPT OF MATHEMATICS AND COMPUTER SCIENCE
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We address the problem of controlling the curvature of a Bezier curve interpolating a given set of data. More precisely, given two points M and N, two directions uright arrow and upsilonright arrow and a constant kappa, we would like to find two quadratic Bezier curves Gamma 1 and Gamma 2 joined with continuity Gsup 1, and interpolating the two points M and N, such that the tangent vectors at M and N have directions uright arrow and upsilonright arrow respectively, the curvature is everywhere upper bounded by kappa, and some evaluating function, the length of the resulting curve for example, is minimized. In order to solve this problem, we first need to determine the maximum curvature of quadratic Bezier curves. This problem was solved by Sapidis and Frey in 1992. Here we present a simpler formula that has an elegant geometric interpretation in terms of distances and areas determined by the control points. We then use this formula to solve the variant of the curvature control problem in which Gamma 1 and Gamma 2 are joined with continuity Csup 1, where the length alpha between the first two control points of Gamma 1 is equal to the length between the last two control points of Gamma 2, and where alpha is the evaluating function to be minimized.
- Numerical Mathematics
- Theoretical Mathematics