Celebratio Mathematica

R. T. Farouki, C. Y. Han, J. Hass, and T. W. Sederberg: “Topologically consistent trimmed surface approximations based on triangular patches,” Comput. Aided Geom. Design 21 : 5 (2004), pp. 459–478. MR 2058392 Zbl 1069.65552 article

A scheme to approximate the trimmed surfaces defined by two tensor-product surface patches, intersecting in a smooth curve segment that extends between diametrically opposite patch corners, is formulated. The trimmed surface approximations are specified by triangular Bézier patches, whose tangent planes agree precisely with those of the tensor-product surfaces along the two sides where they coincide. Topological consistency of the two trimmed surfaces is achieved by requiring the “hypotenuse” sides of the triangular patches to be coincident. In the case of bicubic tensor-product patches and quintic triangular trimmed surface approximations, enforcing these conditions entails the solution of a linear system of 30 equations in 32 unknowns. The two remaining scalar freedoms, together with four additional free control points, are employed to enhance the accuracy and/or smoothness properties of the intersection curve and trimmed surface approximations. By means of an appropriate subdivision preprocessing, the trimmed surface scheme may be used on models described by arbitrary bicubic surface patches.

Thomas W. Sederberg	Related
Chang Yong Han	Related
Rida T. Farouki	Related

R. T. Farouki, C. Y. Han, and J. Hass: “Boundary evaluation algorithms for Minkowski combinations of complex sets using topological analysis of implicit curves,” Numer. Algorithms 40 : 3 (2005), pp. 251–283. MR 2189407 Zbl 1087.65011 article

Minkowski geometric algebra is concerned with sets in the complex plane that are generated by algebraic combinations of complex values varying independently over given sets in \( \mathbb{C} \). This algebra provides an extension of real interval arithmetic to sets of complex numbers, and has applications in computer graphics and image analysis, geometrical optics, and dynamical stability analysis. Algorithms to compute the boundaries of Minkowski sets usually invoke redundant segmentations of the operand-set boundaries, guided by a “matching” criterion. This generates a superset of the true Minkowski set boundary, which must be extracted by the laborious process of identifying and culling interior edges, and properly organizing the remaining edges. We propose a new approach, whereby the matching condition is regarded as an implicit curve in the space \( \mathbb{R}^n \) whose coordinates are boundary parameters for the \( n \) given sets. Analysis of the topological configuration of this curve facilitates the identification of sets of segments on the operand boundaries that generate boundary segments of the Minkowski set, and rejection of certain sets that satisfy the matching criterion but yield only interior edges. Geometrical relations between the operand set boundaries and the implicit curve in \( \mathbb{R}^n \) are derived, and the use of the method in the context of Minkowski sums, products, planar swept volumes, and Horner terms is described.

Chang Yong Han	Related
Rida T. Farouki	Related

J. Hass, R. T. Farouki, C. Y. Han, X. Song, and T. W. Sederberg: “Guaranteed consistency of surface intersections and trimmed surfaces using a coupled topology resolution and domain decomposition scheme,” Adv. Comput. Math. 27 : 1 (2007), pp. 1–26. MR 2317919 Zbl 1118.65011 article

We describe a method that serves to simultaneously determine the topological configuration of the intersection curve of two parametric surfaces and generate compatible decompositions of their parameter domains, that are amenable to the application of existing perturbation schemes ensuring exact topological consistency of the trimmed surface representations. To illustrate this method, we begin with the simpler problem of topology resolution for a planar algebraic curve \( F(x,y) = 0 \) in a given domain, and then extend concepts developed in this context to address the intersection of two tensor-product parametric surfaces \( \mathbf{p}(s,t) \) and \( \mathbf{q}(u,v) \) defined on \( (s,t)\in [0,1]^2 \) and \( (u,v)\in [0,1]^2 \). The algorithms assume the ability to compute, to any specified precision, the real solutions of systems of polynomial equations in at most four variables within rectangular domains, and proofs for the correctness of the algorithms under this assumption are given.

Thomas W. Sederberg	Related
Xiaowen Song	Related
Rida T. Farouki	Related
Chang Yong Han	Related

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