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Abstract: Coons’ construction generates a surface patch that interpolates four groups of specified boundary curves and the corresponding cross-boundary derivative curves. This constructive method is simple and widely used in computer aided design. However, at the corner points, it requires compatibility of the boundary conditions, which is usually difficult to satisfy in practice. In order to handle the incompatible case where the normal directions respectively indicated by two adjacent boundaries do not agree with each other at the common corner point, we utilize the property of degenerate parametric surfaces that the normal directions can converge to multiple values at degenerate points, and therefore the local degenerate geometry can satisfy conflicting conditions simultaneously. Following this idea, we use a single patch of -degree polynomial spline surface with four degenerate corners to interpolate incompatible boundary conditions, which are represented by -degree polynomial spline curves with continuity. This method is based on symbolic operations and polynomial reparameterizations for polynomial splines, and without introducing any theoretical errors, it achieves continuity on the boundary except for the four corner points. [Copyright &y& Elsevier]

Curves on surfaces play an important role in computer-aided geometric design. Because of the considerably high degree of exact curves on surfaces, approximation algorithms are preferred in CAD systems. To approximate the exact curve with a reasonably low degree curve which also lies completely on the -spline surface, an algorithm is presented in this paper. The Hausdorff distance between the approximate curve and the exact curve is controlled under the user-specified distance tolerance. The approximate curve is continuous, where is the user-specified angle tolerance. Examples are given to show the performance of our algorithm. [Copyright &y& Elsevier]

Abstract: -sided hole filling plays an important role in vertex blending. To deal with the case that the corner is surrounded by rational surfaces (i.e. NURBS surfaces), an algorithm to fill -sided holes with - G1 continuous NURBS patches that interpolate the given boundary curves and approximate the given cross-boundary derivatives is presented based on Piegl’s method. The NURBS surfaces joining along inner or boundary curves have normal vectors that do not deviate more than the user-specified angular tolerance . The boundaries as well as cross-boundary derivatives can all be NURBS curves. No restrictions are imposed on the number of boundary curves, and the cross-boundary derivatives can be specified independently. [Copyright &y& Elsevier]

Abstract: By Bezout''s theorem, three quadric surfaces have at most eight isolated intersections although they may have infinitely many intersections. In this paper, we present an efficient and robust algorithm, to obtain the isolated and the connected components of, or to determine the number of isolated real intersections of, three quadric surfaces by reducing the problem to computing the real intersections of two planar curves obtained by Levin''s method. [Copyright &y& Elsevier]