Finding the best conic approximation to the convolution curve of two compatible conics based on Hausdorff distance

Authors:
Seok Hur;Tae-wan Kim
Affiliations:
Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul 151-744, South Korea;Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul 151-744, South Korea and Research Institute of Marine Systems Engineering, Seoul National University, Seoul ...
Venue:
Computer-Aided Design
Year:
2009

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Abstract

We consider the convolution of two compatible conic segments. First, we find an exact parametric expression for the convolution curve, which is not rational in general, and then we find the conic approximation to the convolution curve with the minimum error. The error is expressed as a Hausdorff distance which measures the square of the maximal collinear normal distance between the approximation and the exact convolution curve. For this purpose, we identify the necessary and sufficient conditions for the conic approximation to have the minimum Haudorff distance from the convolution curve. Then we use an iterative process to generate a sequence of weights for the rational quadratic Bezier curves which we use to represent conic approximations. This sequence converges to the weight of the rational quadratic Bezier curve with the minimum Hausdorff distance, within a given tolerance. We verify our method with several examples.