Types of parametric curves geometric properties

Smoothing curve modeling computer-aided geometric design, do not want the curve with a double junction cusp and excess inflection point, geometric properties (including parametric curves the odd inflection point distribution and convexity) control its curved shape of the key. The cubic algebraic polynomial parametric curves problem that has been solved, However lapsed the used geometric invariant method for non-algebraic curve. In this paper, a method based on the theory and continuous mapping of the envelope, the method that cusp conditions line inflection point region of the envelope curve, has been cleverly re-node area and through the continuous mapping for cubic algebraic curve with fourth-order non-algebraic curves are applicable. Analysis by this method some important geometric properties of the curve, that contain sharp points on the curve segments, necessary and sufficient conditions for re-node and the inflection point, and does not contain these points with the relative position of control vertices. The main work is as follows: 1 for the first time the function l, t, φ (t), ψ (t) as the base plane parametric curves odd inflection point distribution and the convexity of the necessary and sufficient conditions; construct the Bézier type curve and B-spline curves, and discuss their properties. Study the general plane of the C-curve, C-Bézier curve CB-spline curves, rational C-Bézier curves and rational CB spline curves odd inflection point distribution and convexity properties obtained cusp curve contains heavy node and the inflection point, and do not contain any of these points the necessary and sufficient conditions on the relative position of control vertices used in [15] analysis method geometric intuitive, simple division of the shape of the distribution area than and easy to judge the impact of change on the the odd inflection point region, also discussed the shape parameter (or weight factor). 3. Gives the entire round, The cubic uniform of a cycle on the cycloid and sinusoidal CB the spline said, with increasing spline node control polygon approximation increasingly generate curves than text [16] and [18 the degree of approximation is superior. Systematic analysis of the space quartic polynomial curve and the nature of the Fourth the Bézier curve singularity Pan inflection point, the space has been a necessary and sufficient to contain a sharp point, re-node and Pan inflection point on the curve, and does not contain these points conditions to solve the space quartic polynomial curve geometric properties of the problem.