Geometric shape analysis for convolution curve of two compatible quadratic Bézier curves

摘要

In this paper we consider the local and global geometric properties of the convolution curve of two compatible quadratic Bézier curves. We characterize all shapes of convolution curves using the ratios of lengths of the corresponding control polygon of two Bézier curves. Especially we show that there are only three cases in the classification of local shapes with respect to the tangent direction and sign of curvature at each endpoint of the convolution curve. This special property can be extended to the convolution curve of two compatible Bézier curves of any degree n. We also classify all cases of global shapes of the convolution curves using the local shapes. The geometric properties of convolution curves are also presented when the ratio is critical point. Some examples are given to illustrate our characterization.