Interpolation with Curvature Constraints
Abstract
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 u(right arrow) and upsilon(right arrow) and a constant kappa, we would like to find two quadratic Bezier curves Gamma 1 and Gamma 2 joined with continuity G(sup 1), and interpolating the two points M and N, such that the tangent vectors at M and N have directions u(right arrow) and upsilon(right 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 C(sup 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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 2000
- Accession Number
- ADP011986
Entities
People
- Hafsa Deddi
- Hazel Everett
- Sylvain Lazard
Organizations
- University of Lethbridge