Celebratio Mathematica

X. Song, T. W. Sederberg, J. Zheng, R. T. Farouki, and J. Hass: “Linear perturbation methods for topologically consistent representations of free-form surface intersections,” Comput. Aided Geom. Design 21 : 3 (March 2004), pp. 303–319. A corrigendum to this article was published in Comput. Aided Geom. Design 21:3 (2004). MR 2042022 Zbl 1069.65567 article

By applying displacement maps to slightly perturb two free-form surfaces, one can ensure exact agreement between the images in \( \mathbb{R}^3 \) of parameter-domain approximations to their curve of intersection. Thus, at the expense of slightly altering the surfaces in the vicinity of their intersection, a perfect matching of the surface trimming curves is guaranteed. This exact agreement of contiguous trimmed surfaces is essential to achieving topologically consistent solid model constructions through Boolean operations, and has a profound impact on the efficiency and reliability of applications such as meshing, rendering, and computing volumetric properties. Moreover, the control point perturbations require only the solution of a linear system for their determination. The basic principles of this approach to topologically consistent surface trimming curves are described, and example results from the implementation of a simple instance of the method are presented.

Thomas W. Sederberg	Related
Xiaowen Song	Related
Jian Min Zheng	Related
Rida T. Farouki	Related

X. Song, T. W. Sederberg, J. Zheng, R. T. Farouki, and J. Hass: “Corrigendum to ‘Linear perturbation methods for topologically consistent representations of free-form surface intersections’,” Comput. Aided Geom. Design 21 : 3 (2004), pp. 321. corrigendum to an article published in Comput. Aided Geom. Design 21:3 (2004). MR 2042023 Zbl 1069.65568 article

Thomas W. Sederberg	Related
Xiaowen Song	Related
Jian Min Zheng	Related
Rida T. Farouki	Related

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

R. T. Farouki and J. Hass: “Evaluating the boundary and covering degree of planar Minkowski sums and other geometrical convolutions,” J. Comput. Appl. Math. 209 : 2 (2007), pp. 246–266. MR 2387129 Zbl 1140.65020 article

Algorithms are developed, based on topological principles, to evaluate the boundary and “internal structure” of the Minkowski sum of two planar curves. A graph isotopic to the envelope curve is constructed by computing its characteristic points. The edges of this graph are in one-to-one correspondence with a set of monotone envelope segments. A simple formula allows a degree to be assigned to each face defined by the graph, indicating the number of times its points are covered by the Minkowski sum. The boundary can then be identified with the set of edges that separate faces of zero and non-zero degree, and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy. For applications that require only the Minkowski sum boundary, the algorithm minimizes geometrical computations on the “internal” envelope edges, that do not contribute to the final boundary. In other applications, this internal structure is of interest, and the algorithm provides comprehensive information on the covering degree for different regions within the Minkowski sum. Extensions of the algorithm to the computation of Minkowski sums in \( R^3 \), and other forms of geometrical convolution, are briefly discussed.

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