Title: Finding the point on Bezier curves with the normal vector passing an external point

Authors: Xia Hong, W.S. Harwin

Addresses: School of Systems Engineering, University of Reading, Reading RG6 6AY, UK. ' School of Systems Engineering, University of Reading, Reading RG6 6AY, UK

Abstract: Two algorithms for finding the point on non-rational/rational Bezier curves of which the normal vector passes through a given external point are presented. The algorithms are based on Bezier curves generation algorithms of de Casteljau|s algorithm for non-rational Bezier curve or Farin|s recursion for rational Bezier curve, respectively. Orthogonal projections from the external point are used to guide the directional search used in the proposed iterative algorithms. Using Lyapunov|s method, it is shown that each algorithm is able to converge to a local minimum for each case of non-rational/rational Bezier curves. It is also shown that on convergence the distance between the point on curves to the external point reaches a local minimum for both approaches. Illustrative examples are included to demonstrate the effectiveness of the proposed approaches.

Keywords: Bezier curves; de Casteljau algorithm; derivatives; haptics; Lyapunov method; normal vector; Farin recursion; haptic rendering.

DOI: 10.1504/IJMIC.2006.012620

International Journal of Modelling, Identification and Control, 2006 Vol.1 No.4, pp.316 - 324

Published online: 27 Feb 2007 *

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