Aspect Ratio- and Size-Controlled Patterned Triangulations of Parametric Surfaces

O.E. Ruiz (Columbia), S. Peña (Germany), and J. Duque (Columbia)


2-Manifold Triangulation, Grid Triangulation, Constrained Delaunay, Parametric Space Triangulation Glossary S(u, v) a parametric surface function. S : R2 → R3 . S(u, v) = [x(u, v), y(u, v), z(u, v)] F a FACE in a boundary representation. F−1 pre-image of F via S(u, v): S : F−1 → F. LF set of loops of F. LF = {L0, L1, ..., Lm} . L0 external loop of F. Li internal loop of F (i = 1, 2, ...). e an edge of a loop of F. pi point of e (pi ∈ F). (u, v) parametric point in F−1 . Γe pre-


A method to produce patterned, controlled size triangula tion of Boundary Representations is presented. Although the produced patterned triangulations are not immediately suited for fast visualization, they were used in Fixed Grid Finite Element Analysis, and do provide a control on the aspect ratio or shape factor of the triangles produced. The method presented first calculates a triangulation in the pa rameter space of the faces in which the B-Rep is partitioned and then maps it to 3D space. Special emphasis is set in en suring that the triangulations of neighboring faces meet in a seamless manner, therefore ensuring that a borderless C2 2-manifold would have as triangulation a C0 borderless 2 manifold. The method works properly under the conditions (i) the parametric form of the face is a 1-1 function, (ii) the parametric pre-image of a parametric face is a connected region, and (iii) the user-requested sampling frequency ( samples per length unit ) is higher than twice the spatial frequency of the features in the B-Rep ( thus respecting the Nyquist principle ). As the conditions (i) and (ii) are pos sible under face reparameterization or sub-division and the condition (iii) is the minimum that a triangulation should comply with, the method is deemed as generally applica ble.

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