The conditions on the coefficient vectors of parametric polynomial minimal surface of arbitrary degree with isothermal paprameter is investigated. From the conditions, polynomial minimal surface of arbitrary degree with shape parameters and elegant properties can be constructed for shape design.

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. This paper gives an upper bound of the Bézout number, the maximum number of intersections between two linear piecewise algebraic curves whose intersections are finite, over arbitrary triangulations.

Abstract This paper presents a new interpolatory Loop scheme and an unified and mixed interpolatory and approximation subdivision scheme for triangular meshes. The former which is C1 continuous as same as the modified Butterfly scheme has better effect in some complex models. The latter can be used to solve the “popping effect” problem when switching between meshes at different levels of resolution. The scheme generates surfaces coincident with the Loop subdivision scheme in the limit condition having the coefficient k equal 0. When k equal 1, it will be changed into a new interpolatory subdivision scheme. Eigen‐structure analysis demonstrates that subdivision surfaces generated using the new scheme are C1 continuous. All these are achieved only by changing the value of a parameter k. The method is a completely simple one without constructing and solving equations. It can achieve local interpolation and solve the “popping effect” problem which are the method's advantages over the modified Butterfly scheme.

In this paper, we study the bivariate cubic spline space S/sub 3//sup 2,1/ (/spl Delta//sub mn//sup (2)/) and construct a kind of bivariate cubic NURBS surfaces by using the bivariate B-spline bases in S/sub 3//sup 2,1/ (/spl Delta//sub mn//sup (2)/). Compared with the bicubic NURBS surfaces, this kind of bivariate NURBS surfaces are non-tensor product surfaces on the type-2 triangulation with many good properties.