Geometric shapes of C-Bezier curves

摘要

In this paper, we focus on the geometric shapes of the C-Bezier curves for the space span{1, t, ..., t~n, sin t, cos t}. First, any C-Bezier curve is divided into a Bezier curve and a trigonometric part. So any C-Bezier curve describes the trajectory of a point orbiting around a center in an elliptical orbit while the orbital plane is moving as the ellipse center translating along a Bezier curve. Second, the geometric characters of the C-Bezier curve (the control points of the center Bezier curve, the trajectories of the vertices and the foci of the ellipse, etc.), can all be explicitly presented by the control points of the C-Bezier curve. Third, considering some special cases, we give the sufficient and necessary conditions of C-Bezier basis forming Bezier curve, ellipse, circle, common helix, and so on. Lastly, we show how to build some geometrically intuitive curves through the C-Bezier basis without rational forms.