Research Papers

Quad Pillars and Delta Pillars: Algorithms for Converting Dexel Models to Polyhedral Models

[+] Author and Article Information
Masatomo Inui

Department of Intelligent Systems Engineering,
Ibaraki University,
4-12-1, Nakanarusawa,
Hitachi, Ibaraki 316-8511, Japan
e-mail: masatomo.inui.az@vc.ibaraki.ac.jp

Nobuyuki Umezu

Department of Intelligent Systems Engineering,
Ibaraki University,
4-12-1, Nakanarusawa,
Hitachi, Ibaraki 316-8511, Japan
e-mail: nobuyuki.umezu.cs@vc.ibaraki.ac.jp

Contributed by the Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received August 21, 2014; final manuscript received August 31, 2016; published online February 16, 2017. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 17(3), 031001 (Feb 16, 2017) (9 pages) Paper No: JCISE-14-1257; doi: 10.1115/1.4034737 History: Received August 21, 2014; Revised August 31, 2016

In the geometric simulation of multi-axis milling, a dexel representation solid model is frequently used. In this modeling method, the object shape is defined as a collection of vertical segments (dexels) based on a two-dimensional regular square grid in the XY plane. In this paper, the authors propose the quad pillars algorithm and its enhanced version named the delta pillars algorithm for converting a dexel model to an equivalent polyhedral stereolithography (STL) model. These algorithms define a series of vertical pillar shapes for each square cell of the grid to represent the object shape as a bundle of pillars. The final polyhedral model is obtained by performing a simplified Boolean union operation of the pillar shapes. Unlike prior methods, the proposed algorithms are simple and fast and are guaranteed to generate a watertight polyhedral model without holes, gaps, or T-junctions. An experimental system is implemented and conversion tests are performed. The system converted a dexel model based on a high-resolution grid to a polyhedral model in a practical amount of time.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


VanHook, T. , 1986, “ Real-Time Shaded Milling Display,” Comput. Graphics (Proc. of ACM Siggraph 86), 20(4) pp. 15–20. [CrossRef]
Huang, Y. , and Oliver, J. H. , 1994, “ NC Milling Error Assessment and Tool Path Correction,” Comput. Graphics (Proc. of ACM Siggraph 94) ACM, New York, pp. 287–294.
Stifter, S. , 1995, “ Simulation of NC Machining Based on the Dexel Model: A Critical Analysis,” Int. J. Adv. Manuf. Technol., 10(3), pp. 149–157. [CrossRef]
Menon, J. P. , Marisa, R. J. , and Zagajac, J. , 1994, “ More Powerful Solid Modeling Through Ray Representations,” IEEE Comput. Graphics Appl., 14(3), pp. 22–35. [CrossRef]
Zhu, W. , 2003, “ Virtual Sculpting and Polyhedral Machining Planning System With Haptic Interface,” Ph.D thesis, North Carolina State University, Raleigh, NC.
Zhu, W. , and Lee, Y.-S. , 2004, “ Virtual Sculpting and Multi-Axis Polyhedral Machining Planning Methodology With 5-DOF Haptic Interface,” EuroHaptics, Munich, Germany, June 5–7, pp. 267–272.
Zhu, W. , and Lee, Y.-S. , 2005, “ A Visibility Sphere Marching Algorithm of Constructing Polyhedral Models for Haptic Sculpting and Product Prototyping,” Rob. Comput. Integr. Manuf., 21(1), pp. 19–36. [CrossRef]
Yuksek, K. , Zhang, W. , Ridzalski, B. I. , and Leu, M. C. , 2008, “ A New Contour Reconstruction Approach From Dexel Data in Virtual Sculpting,” IEEE 3rd International Conference on Geometric Modeling and Imaging, July 9–11, pp. 82–86.
Herman, G. T. , and Liu, H. K. , 1979, “ Three-Dimensional Display of Organs From Computed Tomograms,” Comput. Graphics Image Process., 9(1), pp. 1–21. [CrossRef]
Dutta, D. , Prinz, F. B. , Rosen, D. , and Wess, L. , 2000, “ Layered Manufacturing: Current Status and Future Trends,” ASME J. Comput. Inf. Sci. Eng., 1(1), pp. 60–71. [CrossRef]
Chiu, W. K. , and Tan, S. T. , 1998, “ Using Dexels to Make Hollow Models for Rapid Prototyping,” Comput. Aided Des., 30(7), pp. 539–547. [CrossRef]
Wang, C. C. L. , 2011, “ Approximate Boolean Operations on Large Polyhedral Solids With Partial Mesh Reconstruction,” IEEE Trans. Visualization Comput. Graphics, 17(6), pp. 836–849. [CrossRef]
Qian, X. , Villarrubia, J. , Tian, F. , and Dixson, R. , 2007, “ Image Simulation and Surface Reconstruction of Undercut Features in Atomic Force Microscopy,” Proc. SPIE, 6518, p. 651811.
Shade, J. , Gortler, S. , He, L. W. , and Szeliski, R. , 1998, “ Layered Depth Image,” Comput. Graphics (Proc. of ACM Siggraph’98), ACM, New York, pp. 231–242.
Wang, C. C. L. , and Manocha, D. , 2013, “ GPU-Based Offset Surface Computation Using Point Samples,” Comput. Aided Des., 45(2), pp. 321–330. [CrossRef]
Wang, C. C. L. , 2011, “ Computing on Rays: A Parallel Approach for Surface Mesh Modeling From Multi-Material Volumetric Data,” Comput. Ind., 62(7), pp. 660–671. [CrossRef]
Zhao, H. , and Wang, C. C. L. , 2011, “ Parallel and Efficient Boolean on Polygonal Solids,” Visual Comput., 27(6–8) pp. 507–517. [CrossRef]
Benouamer, M. O. , and Michelucci, D. , 1997, “ Bridging the Gap between CSG and Brep via a Triple Ray Representation,” ACM Symposium on Solid Modeling and Applications, pp. 68–79.
Muller, H. , Surmann, T. , Stautner, M. , Albersmann, F. , and Weinert, K. , 2003, “ Online Sculpting and Visualization of Multi-Dexel Volumes,” 8th ACM Symposium on Solid Modeling and Applications, pp. 258–261.
Ren, Y. , Zhu, W. , and Lee, Y.-S. , 2008, “ Feature Conservation and Conversion of Tri-Dexel Volumetric Models to Polyhedral Surface Models for Product Prototyping,” Comput. Aided Des. Appl., 5(6), pp. 932–941.
Zhang, W. , and Leu, M. C. , 2008, “ NC Machining Simulation Based on Triple-Dexel Representation,” 2008 International Symposium on Flexible Automation, Paper No. ISFA2008U_100.
Koenig, A. H. , and Groller, E. , 1998, “ Real Time Simulation and Visualization of NC Milling Processes for Inhomogeneous Materials on Low-End Graphics Hardware,” Computer Graphics International 98, IEEE Computer Society, June 26, pp. 338–349.
Lorensen, W. E. , and Cline, H. E. , 1987, “ Marching Cubes: A High Resolution 3D Surface Construction Algorithm,” Comput. Graphics, 21(4) pp. 163–169. [CrossRef]
Ju, T. , Losasso, F. , Schaefer, S. , and Warren, J. , 2002, “ Dual Contouring of Hermite Data,” Comput. Graphics (Proc. of ACM Siggraph 2002), 21(3), pp. 339–346.
Gelden, A. V. , and Wilhelms, J. , 1994, “ Topological Considerations in Isosurface Generation,” ACM Trans. Graphics, 13(4), pp. 337–375. [CrossRef]
Meyers, D. , Skinner, S. , and Sloan, K. , 1992, “ Surface From Contours,” ACM Trans. Graphics, 11(3), pp. 228–258. [CrossRef]
Hoppe, H. , DeRose, T. , Duchamp, T. , McDonald, J. , and Stuetzle, W. , 1992, “ Surface Reconstruction From Unorganized Points,” Computer. Graphics (Proc. SIGGRAPH’92), pp. 71–78.
Kobbelt, L. P. , Botsch, M. , Schwanecke, U. , and Seidel, H.-P. , 2001, “ Feature Sensitive Surface Extraction From Volume Data,” Computer Graphics (SIGGRAPH’01), pp. 57–66.
Lauterbach, C. , Garland, M. , Sengupta, S. , Luebke, D. , and Manocha, D. , 2009, “ Fast BVH Construction on GPUs,” Comp. Graphics Forum, 28(2), pp. 375–384.
NVIDIA, 2012, “ CUDA C Programming Guide,” NVIDIA, Santa Clara, CA, Document No. PG-02929-001_v5.0.


Grahic Jump Location
Fig. 1

Dexel representation of a solid model

Grahic Jump Location
Fig. 2

Outline of quad pillars algorithm

Grahic Jump Location
Fig. 3

Dexel lists on four adjacent grid points

Grahic Jump Location
Fig. 4

Section of dexel models with overhangs and their conversion to pillars

Grahic Jump Location
Fig. 6

Expansion of gray zones by shifting borders of the black and white zones

Grahic Jump Location
Fig. 7

Dexels representing a horizontally placed box and their corresponding zones

Grahic Jump Location
Fig. 8

Selection of the highest endpoint in a WGB gray zone for each dexel list

Grahic Jump Location
Fig. 10

Removal of overlapping portions of the wall trapezoids in quad pillars algorithm

Grahic Jump Location
Fig. 11

Necessary Boolean subtraction of trapezoids in quad pillars algorithm

Grahic Jump Location
Fig. 12

Impossible configuration of trapezoids

Grahic Jump Location
Fig. 13

T-junctions on an edge

Grahic Jump Location
Fig. 14

Triangulation process of remaining trapezoid in sidewall

Grahic Jump Location
Fig. 15

Construction of a triangular pillar shape using three dexel lists

Grahic Jump Location
Fig. 16

Selection rules of type A, B, C, and D triangular pillars

Grahic Jump Location
Fig. 17

Removal of overlapping portions of wall-side trapezoids in delta pillars algorithm

Grahic Jump Location
Fig. 18

Conversion results using quad pillars and delta pillars algorithms

Grahic Jump Location
Fig. 19

Conversion result of sample model A

Grahic Jump Location
Fig. 20

Conversion result of sample model B

Grahic Jump Location
Fig. 21

Relationship between number of dexels and required time for conversion



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In