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Research Papers

# A Randomized Approach to Volume Constrained Polyhedronization Problem

[+] Author and Article Information
Jiju Peethambaran, Amal Dev Parakkat, Ramanathan Muthuganapathy

Department of Engineering Design,
Indian Institute of Technology,

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received September 25, 2013; final manuscript received October 31, 2014; published online January 30, 2015. Assoc. Editor: Xiaoping Qian.

J. Comput. Inf. Sci. Eng 15(1), 011009 (Mar 01, 2015) (9 pages) Paper No: JCISE-13-1187; doi: 10.1115/1.4029559 History: Received September 25, 2013; Revised October 31, 2014; Online January 30, 2015

## Abstract

Given a finite set of points in $R3$, polyhedronization deals with constructing a simple polyhedron such that the vertices of the polyhedron are precisely the given points. In this paper, we present randomized approximation algorithms for minimal volume polyhedronization (MINVP) and maximal volume polyhedronization (MAXVP) of three dimensional point sets in general position. Both, MINVP and MAXVP, problems have been shown to be NP-hard and to the best of our knowledge, no practical algorithms exist to solve these problems. It has been shown that for any point set S in $R3$, there always exists a tetrahedralizable polyhedronization of S. We exploit this fact to develop a greedy heuristic for MINVP and MAXVP constructions. Further, we present an empirical analysis on the quality of the approximation results of some well defined point sets. The algorithms have been validated by comparing the results with the optimal results generated by an exhaustive searching (brute force) method for MINVP and MAXVP for some well chosen point sets of smaller sizes. Finally, potential applications of minimum and maximum volume polyhedra in 4D printing and surface lofting, respectively, have been discussed.

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Topics: Algorithms

## Figures

Fig. 1

A polyhedron which illustrates different types of faces

Fig. 2

Illustration of RAA_MINVP algorithm. (a) Point set S, (b) initial random tetrahedron P0 and S\P={p,q}, (c) selected random point q lies outside P0 and the associated nonintersecting tetrahedra, (d) the smallest volume tetrahedron that q forms with a face of P0 has been attached to form next polyhedron, P1, (e) next point selected, p lies inside P1 and associated nonintersecting tetrahedra, and (f) the largest volume tetrahedron is removed to obtain the final polyhedron, which is RAND_MINVP of S.

Fig. 3

RAND_MINVP generated for convex point sets. (a) Tetrahedral, (b) octahedral, (c) cube, (d) icosahedral, and (e) dodecahedral platonic point sets. (f) Triangular, (g) square, and (h) pentagonal prismatic point sets. (i) Square, (j) hexagonal, and (k) octagonal pyramid point sets.

Fig. 4

RAND_MINVPs and RAND_MAXVPs generated for some nonconvex point sets taken from PSB. Figures 4(a), 4(d), 4(g), and 4(j) show the models, Figs. 4(b), 4(e), 4(h), and 4(k) show the RAND_MINVPs, and Figs. 4(c), 4(f), 4(i), and 4(l) show corresponding RAND_MAXVPs.

Fig. 5

RAND_MINVPs generated for large sized random point sets. Point set sizes have been mentioned along with the subfigures.

Fig. 6

Optimal minimal volume polyhedronizations generated by RAA_MINVP for various point sets of sizes 5 (Figs. 6(a)6(c)), 6 (Fig. 6(d)), and 7 (Fig. 6(e)). All these results have been verified for the optimality using BFA.

Fig. 7

Illustration of trap regions in two and three dimensions. Figure 7(c) shows the shape of the trap region of the configuration in Fig. 7(b). It is to be noted that planes in Fig. 7(b) do not intersect each other.

Fig. 8

Illustration of local rearrangement in 3D with minimal set of blocking points, B={c,d} and the trapped point q. (i) Bird's eye view of the polyhedron with few faces from the top and bottom removed, (ii) top view, and (iii) bottom view. Figures 8(iv)–8(vi) illustrate the rearrangement from the top view with the removed/added tetrahedra zoomed in.

Fig. 9

Optimal minimal volume polyhedronizations of various point sets. (a) Octahedral, (b) cube, (c) square pyramid, and (d) triangular prismatic point sets.

Fig. 10

Illustration of how minvp can reduce the physical space in 4D printing. Assuming a volume of V to each cube, unfolded tesseract in Fig. 10(b) occupies a volume of 8 V whereas the one in Fig. 10(c) occupies a volume of 6.6 V.

Fig. 11

Illustration of surface lofting from point set sampled from contours of teapot. (a) Point set, (b) points with additional segmentation (indicated by green colored polygons) and processing direction (shown in red lines with black arrow head) information, (c) result after lofting few segments, and (d) union of all lofted segments representing the teapot model.

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