An efficient scheme is introduced to construct interpolatory curves and surfaces passing through a set of given scattered data points. The scheme is based on an interpolatory basis derived from the sinc function with a Guassian multiplier previously applied in other fields for signal or function reconstruction. In connection with its application addressed in this article for spatial curve and surface construction, the interpolatory basis possesses various nice properties, such as partition of unity, linear precision, and local support, etc., under a small tolerance. By using these basis functions, free-form curves and surfaces can be conveniently constructed. A designer can adjust the shape of the constructed curve and surface by moving some interpolating points or by inserting new interpolating points. The resulting interpolatory curves and surfaces are C ∞ continuous. Smooth connection between curves or surfaces can easily be achieved. Closed curves and surfaces can also be expressed using the proposed interpolatory basis functions.

This paper presents a novel approach for constructing a piecewise triangular cubic polynomial surface with C/sup 1/ continuity around a common corner vertex. A C/sup 1/ continuity condition between two cubic triangular patches is first derived using mixed directional derivatives. An approach for constructing a surface with C/sup 1/ continuity around a corner is then developed. Our approach is easy and fast with the virtue of cubic reproduction, local shape controllability, C/sup 2/ continuous at the corner vertex. Some experimental results are presented to show the applicability and flexibility of the approach.

Subdivision, intersection and triangulation of a surface are the basic and common operations in computer aided design and computer graphics. A complicated surface can be approximated by some triangular patches. The key technique of this problem is the estimation of approximation error. This work has been done in some papers by estimating second derivatives of the surface, thus very complicated computation must be executed and hence time is overly consumed. Avoiding this estimation and exerting the influence of the control points of the surfaces, this paper gives a formula for computing the exact distance between a degree 2 Bernstein-Bezier triangular surface and its base triangle. Also, this paper gives estimates of maximal distance between a degree 2 or 3 rational Bernstein-Bezier triangular surface and its base triangle with a high precision. These results are easy for programming, simple for computing, and effective for improving the subdivision/triangulation algorithms of common rational Bezier surfaces in a design system.