Research Papers

Variational Discrete Developable Surface Interpolation

[+] Author and Article Information
Wen-Yong Gong

Institute of Mathematics,
Jilin University,
Changchun 130012, China
e-mail: gongwenyong@gmail.com

Yong-Jin Liu

Department of Computer
Science and Technology,
Tsinghua University,
Beijing 100084, China
e-mail: liuyongjin@tsinghua.edu.cn

Kai Tang

Department of Mechanical Engineering,
Hong Kong University of
Science and Technology,
Hong Kong 00852, China
e-mail: mektang@ust.hk

Tie-Ru Wu

Institute of Mathematics,
Jilin University,
Changchun 130012, China
e-mail: wutr@jlu.edu.cn

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF Computing AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received February 27, 2012; final manuscript received December 16, 2013; published online February 26, 2014. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 14(2), 021002 (Feb 26, 2014) (9 pages) Paper No: JCISE-12-1028; doi: 10.1115/1.4026291 History: Received February 27, 2012; Revised December 16, 2013

Modeling using developable surfaces plays an important role in computer graphics and computer aided design. In this paper, we investigate a new problem called variational developable surface interpolation (VDSI). For a polyline boundary P, different from previous work on interpolation or approximation of discrete developable surfaces from P, the VDSI interpolates a subset of the vertices of P and approximates the rest. Exactly speaking, the VDSI allows to modify a subset of vertices within a prescribed bound such that a better discrete developable surface interpolates the modified polyline boundary. Therefore, VDSI could be viewed as a hybrid of interpolation and approximation. Generally, obtaining discrete developable surfaces for given polyline boundaries are always a time-consuming task. In this paper, we introduce a dynamic programming method to quickly construct a developable surface for any boundary curves. Based on the complexity of VDSI, we also propose an efficient optimization scheme to solve the variational problem inherent in VDSI. Finally, we present an adding point condition, and construct a G1 continuous quasi-Coons surface to approximate a quadrilateral strip which is converted from a triangle strip of maximum developability. Diverse examples given in this paper demonstrate the efficiency and practicability of the proposed methods.

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Fig. 1

(a) Discrete developable surface. (b) Normal map

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Fig. 2

Boundary bridge triangulation (a) and boundary triangulation (b) generated by the same closed polyline

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Fig. 3

Two halves P(l+) and P(l-) by a rung 〈i,j〉. (a) Two bridge triangles from 〈i,j〉. (b) Visibility vertex Pk of rung 〈i,j〉.

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Fig. 4

A multiple connected region (a) and a cut (b)

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Fig. 5

Two Catmull–Rom splines with τ = 0.5 in (a) and τ = 1.0 in (b)

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Fig. 6

A tent model is generated by different methods. (a) Wang and Tang’s result [6]. (b) The result of Liu et al. [4]. (c) Our result.

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Fig. 7

Modify a polygon boundary by moving its vertices within a bound. Adjusting the vertex Pi finally obtains Pi'.

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Fig. 8

The diagram of adding point rule

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Fig. 9

(a) An umbrella model. (b) Wire umbrella model. (c) Unfolded planar mesh of an optimal triangulation.

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Fig. 10

(a) A tower model is made up of five optimal triangulation strips. (b) The solid model of the tower. Right column: unfolded planar meshes of several optimal triangulation strips.

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Fig. 11

(a) The top surface of a mouse is made up of five optimal triangulation strips. (b) The variational developable surface interpolation, and the black points represent the varying points. Right column: unfolded planar meshes.

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Fig. 12

(a) A hat model consists of three optimal triangulation strips (except the feather model and top surface). (b) The variational developable surface interpolation, and the black points represent the varying points. Right column: unfolded planar meshes.

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Fig. 13

(a) A flower model. (b) The flower model after varying several points. (c)–(j) Unfolded planar meshes corresponding to the optimal triangulation strips in (b). (k) The solid flower model.

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Fig. 14

(a) The top surface of a mouse is made up of two optimal triangulation strips. (b) Two strips in (a) are converted into two quadrilateral strips. (c) The top surface is approximated by two G1 continuous quasi-Coons surfaces.



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