Geometric meanings of the parameters on rational conic segments

Using algebraic and geometric methods, functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic segment is presented by a rational quadratic or cubic Bézier curve. That is, the inverse mappings of the mappings represented by the expressions of rational conic segments are given. These formulae relate some triangular areas or some angles, determined by the selected point on the curve and the control points of the curve, as well as by the weights of the rational Bézier curve. Also, the relationship can be expressed by the corresponding parametric angles of the selected point and two endpoints on the conic segment, as well as by the weights of the rational Bézier curve. These results are greatly useful for optimal parametrization, reparametrization, etc., of rational Bézier curves and surfaces.