This paper shows that under a necessary and sufficient condition, there exists a cubic rational Bezier curve with convex hull property and convexity preserving property which passes two given points inside the control polygon.

By comparison with the approximate degree 0(h2+12) of the formula presented in [1], the accuracy is raised so highly that in the condition of not inereasing the nodes and the amount of computation, the approximate degree can be achieved to 0(h2+12), and the convexity can be guaranteed.

In this paper,we give the definition of convexity-preserving inter polation of a given function f(x).

Finally,we prove that theW-surface on the parallelogram have the convexity-preserving.

The contents includes the blending-based curve modeling methods, conic blending scheme, several generalized blending-based interpolation schemes, the fairness and convexity-preserving analysis, and blending-based surface research, etc.

G~1 convexity preserving quadratic polynomial parametric interpolating curve

We present a new dynamic convexity preserving subdivision scheme based on the bi-quadratic uniform trigonometric polynomial B-spline.

Based on the analysis of the convexity for planar discrete data points, a type of local, interpolatory and convexity preserving discrete subdivision scheme for the generation of smooth curves is presented, and the schemes proposed in [1] and [2] are the special cases.

At the same time this method realizes a convexity-preserving.

This paper is concerned with the convexity conditions of implicit algebraic surfaces whose formulas are represented through the relations of the coefficients of the Bernstein-Bezier-representation.

Necessary and sufficient conditions for convexity preserving of developable surface patches are presented.