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


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

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.

Copyright © 2005 by American Society of Mechanical Engineers
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Figure 1

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

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Figure 2

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

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Figure 3

Bending energy calculation on a bridge edge

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Figure 4

Failure of local optimum approach in finding a global optimum

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Figure 5

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

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Figure 6

An example single layer graph for developable triangulation

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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

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Figure 8

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

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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

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Figure 10

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

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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

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Figure 12

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

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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

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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

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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

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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




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