Reparametrization and Subdivision of Interval Bezier Curves

Abstract — Interval Bezier curve are new representation forms of parametric curves. Using this new representation, the problem of lack of robustness in all state-of-the art CAD systems can be largely overcome. In this paper this concept has been discussed to form a new curve over rectangular domain such that its parameter varies in an arbitrary range, - instead of standard parameter, -. Where and are real and, we also want that curve gets generated within the given error tolerance limit. The four fixed Kharitonov's polynomials (four fixed Bezier curves) associated with the original interval Bezier curve are obtained. A new parameterization is applied to the four fixed Kharitonov's polynomials (four fixed Bezier curves). Finally, the required interval control points are obtained from the fixed control points of the four fixed Kharitonov's polynomials. Subdividing a parametric interval Bezier curve into two interval segments is also presented. The two interval segments have the same shape as the original interval Bezier curve, but they are defined by more entities (interval control points or interval vectors) thereby making it possible to fine-tune the interval Bezier curve. Using matrix representation, it has been shown how to determine the control polygon that covers an arbitrary interval [a, b] of the original interval Bezier curve. Numerical examples are included in order to demonstrate the effectiveness of the proposed method