Abstract:  In many applications of geometric modeling, curves of desirable shape should eliminate the unnecessary singularities and inflection points. Therefore, to avoid potential risk in shape design, it is essential to predict and analyze the shape features of parametric curves. In order to quickly determine the shape features of parametric curves, the shape conditions of the parametric curve are simplified due to the homogeneous property of cones, and the cusp conditional cone and two boundary loop conditional cones are obtained for a quadratic trigonometric polynomial Bézier curve characterized with two shape parameters. These three characteristic cones and their tangent planes divide the characteristic space into different characteristic regions. The curve's shape features are completely determined by the distribution region which the characteristic point locates in the characteristic space. It is shown that the shape diagrams obtained by the method based on the theory of envelopes and topological mappings can be derived from characteristic space by virtue of planar slices, which are vertical to one of the axes. Furthermore, the influences of shape parameters on the associated characteristic regions are also discussed. The obtained results enable the user to place control points or choose shape parameters so that the resulting curve is globally or locally convex, possessing wanted singularities or inflection points, or enjoying the desired shape features.

Key words:  trigonometric Bézier curves, shape features, cusp cones, loop cones