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# Optimal Boundary Triangulations of an Interpolating Ruled Surface

[+] Author and Article Information
Charlie C. Wang

Department of Automation and Computer-Aided Engineering,  Chinese University of Hong Kong, Shatin, N.T., Hong Kong, People’s Republic of Chinacwang@acae.cuhk.edu.hk

Kai Tang1

Department of Mechanical Engineering,  Hong Kong University of Science and Technology, Clear Water Bay, KLN, ong Kong, People’s Republic of Chinamektang@ust.hk

1

Corresponding author.

J. Comput. Inf. Sci. Eng 5(4), 291-301 (Feb 22, 2005) (11 pages) doi:10.1115/1.2052850 History: Received November 25, 2004; Revised February 22, 2005

## Abstract

We investigate how to define a triangulated ruled surface interpolating two polygonal directrices that will meet a variety of optimization objectives which originate from many CAD/CAM and geometric modeling applications. This optimal triangulation problem is formulated as a combinatorial search problem whose search space however has the size tightly factorial to the numbers of points on the two directrices. To tackle this bound, we introduce a novel computational tool called multilayer directed graph and establish an equivalence between the optimal triangulation and the single-source shortest path problem on the graph. Well known graph search algorithms such as the Dijkstra’s are then employed to solve the single-source shortest path problem, which effectively solves the optimal triangulation problem in $O(mn)$ time, where $n$ and $m$ are the numbers of vertices on the two directrices respectively. Numerous experimental examples are provided to demonstrate the usefulness of the proposed optimal triangulation problem in a variety of engineering applications.

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## Figures

Figure 10

Example I: comparison of paths on the validity map: (a) path of minimal twist BBT, and (b) path of maximal convexity BBT

Figure 1

Different parameterizations on the same two rails lead to different ruled surfaces

Figure 2

Different BBTs on the same directrices P and Q in (a)

Figure 9

Example I: strip triangulation results of different objectives: (a) the directrices, (b) minimal area, (c) minimal twist, (d) maximal convexity, (e) minimal bending, and (f) minimal mean curvature variation

Figure 11

Example II: strip triangulation with coupled objectives: (a) the directrices, (b) minimal area, (c) maximal convexity, (d) maximal convexity+minimal bending energy, and (e) maximal convexity+minimal mean curvature variation

Figure 12

Example III: strip triangulation for ribbon design: (a) the directrices, (b) minimal area triangulation, and (c) minimal bending triangulation

Figure 13

Contour-based surface reconstruction in human body modeling: (a) the point cloud, (b) contours generated, (c) surface by “sewing” the contours, and (d) shaded result

Figure 14

Example IV: surface wrinkle design: (a) the skirt and the directrices to specify surface wrinkles, (b) wrinkle strip generated with the maximal convexity objective, and (c) wrinkle strip generated with the coupled objective of maximal convexity+minimal bending

Figure 15

Example V: strip blending in shoe design: (a) shoe last A, (b) shoe last B, (c) the rear part of A+ the front part of B, (d) mesh representation of (c), (d) the blending strip with minimal bending energy, and (e) the blending strip with minimal normal variation

Figure 16

Example VI: strip triangulation for design of a flange: (a) the sheet metal part to add a flange, (b) the directrices for optimal triangulation, (c) the flange as a BBT with minimal bending, and (d) top view

Figure 3

Bending energy calculation on a bridge edge

Figure 4

Failure of local optimum approach in finding a global optimum

Figure 5

An example single layer graph constructed from P and Q with m and n points, respectively

Figure 6

An example single layer graph for developable triangulation

Figure 7

Building the dual layer graph for global minimum bending triangulation: (a) four configurations of triangles neighboring a bridge edge, and (b) dual layer graph

Figure 8

Building the quadruple layer graph for the BBT with minimal mean curvature variation

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