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
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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.

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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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