Geometric shapes of C-Bézier curves

In this paper, we focus on the geometric shapes of the C-Bézier curves for the space span{1,t,…,tn,sint,cost}. First, any C-Bézier curve is divided into a Bézier curve and a trigonometric part. So any C-Bézier 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 Bézier curve. Second, the geometric characters of the C-Bézier curve (the control points of the center Bézier 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-Bézier curve. Third, considering some special cases, we give the sufficient and necessary conditions of C-Bézier basis forming Bézier curve, ellipse, circle, common helix, and so on. Lastly, we show how to build some geometrically intuitive curves through the C-Bézier basis without rational forms.