Topologically consistent trimmed surface approximations based on triangular patches

Topologically consistent algorithms for the intersection and trimming of free-form parametric surfaces are of fundamental importance in computer-aided design, analysis, and manufacturing. Since the intersection of (for example) two bicubic tensor-product surface patches is not a rational curve, it is usually described by approximations in the parameter domain of each surface. If these approximations are employed as “trim curves”, their images in R do not agree precisely, and the resulting trimmed surfaces may exhibit “gaps” and “overlaps” along their common edge, an artifact that often incurs failure of downstream applications. We present a direct and simple approach to the problem, wherein the intersection curve is described explicitly by the sides of a sequence of triangular Bézier patches. Instead of representing trimmed surfaces by trim curves in the surface parameter domain, together with appropriate control point perturbations to guarantee consistency, we use triangular patches to directly approximate the intersection curve and the trimmed surfaces it defines. The triangular patches are constructed so as to maintain smooth (i.e., tangent-plane continuous) connections to untrimmed patches of the original surface. We assume that the original intersecting surfaces are subject to a subdivision process, such that the intersection segment (if any) on each sub-patch is a smooth arc between diametrically opposite corners. This guarantees that all intersection segments, and the trimmed surfaces they delineate, are “simple” enough to admit accurate approximation using triangular Bézier patches. Ensuring position and tangent plane agreement of degree-n triangular trimmed patch approximations p̂(r, s, t) and q̂(u, v, w) with given degree-(m,m) tensor-product patches p(r, s) and q(u, v) along the boundaries r = 0, s = 0 and u = 0, v = 0, and that the curve p̂(ξ, 1− ξ, 0) ≡ q̂(ξ, 1 − ξ, 0) matches the end points and tangents of the exact intersection, entails solving a linear system of 8m + 4n− 14 equations in 12n− 28 scalar variables. Although much of the ensuing discussion will be cast in a general context, our primary emphasis is the case of greatest practical interest—namely, the approximation of trimmed bicubic patches by quintic triangular patches. In this case, the tangent-plane matching conditions on the common patch sides entail solution of a linear system of 30 equations in 32 unknowns. Subsequent to solving this system, the degrees of freedom that remain to improve the approximation accuracy are the end-derivative magnitudes of the approximate intersection curve and is its two middle control points, and one “interior” control point for each triangular patch. Additional degrees of freedom may be introduced, to further improve the approximation accuracy, by elevating the degree of the quintic triangular patches. The elegant simplicity of this method, and the well-conditioned nature of the linear system that expresses the boundary conditions, makes it eminently suited to practical implementation. Many of the basic principles hold in contexts other than interesting bicubic patches with trimmed surfaces approximated by quintic triangular patches, although we expect this combination will be of greatest practical interest. The final trimmed surfaces resulting from the procedure described herein are hybrid collections of tensor-product (rectangular) patches and triangular patches. If homogeneity of the patch types in the final surface representation is an important consideration, known algorithms for converting a triangular patch into a set of tree rectangular patches or for splitting a rectangular patch into two triangular patches may be invoked.