Computer-Based Automation pp 617-634 | Cite as

# A Graph-Theoretic Approach to the Hidden Line Elimination Problem

## Abstract

We consider the hidden line elimination problem for displaying 3D solid objects represented as polyhedra on 2D display devices. The objects are assumed to be composed of polygonal faces. The algorithm presented is based on a graph-theoretic approach in that a certain graph representing the projections of the edges of the polyhedron is constructed and visible edges are found using depth-first search technique in the graph. The way in which visible edges are found is different from the previous ones in that it first assumes all the edges of the polygonal faces are invisible from the viewer and based on some criteria seeks out visible ones, whence the efficiency of the algorithm is, to some extent, independent of the number of hidden edges. In this sense it is more appropriate to refer to the algorithm as visible line determination algorithm.

The worst case time complexity of the algorithm is shown to be 0(n_{b}nlog n) + F, where n is the total number of edges, n_{b} the number of boundary edges and F the number of visible edges displayed. However, if the number n_{b} of boundary edges is much smaller than n, which is true in most cases, the algorithm is more efficient than previous approaches whose worst case complexity may be 0(n^{2}).

## Keywords

View Plane Front Face Intersection Graph Boundary Edge Back Face## Preview

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## References

- [1]J. K. Aggarwal and R. O. Duda, “Computer analysis of moving polygonal images,” IEEE Trans. Comput., C-24, 10, Oct. 1975.Google Scholar
- [2]P. Atherton, K. Weiler and D. Greenberg, “Polygon shadow generation,” Computer Graphics, 12,3, pp. 275–281, August 1978.Google Scholar
- [3]J. L. Bentley and T. Ottmann, “Algorithms for reporting and counting geometric intersections,” IEEE Trans. Comput., C-28, 9, pp. 643–647, Sept. 1979.CrossRefGoogle Scholar
- [4]D. K. Brotz, “Intersecting polyhedra with successive planes,” Computer & Graphics, 2, 1976.Google Scholar
- [5]J. H. Clark, “Hierarchical geometric models for visible surface algorithm,” Comm. ACM, 19,10, pp. 547–554, Oct. 1976.Google Scholar
- [6]M. Cyrus and J. Beck, “Generalized two-and three-dimensional clipping,” Computer & Graphics, 3, 1978.Google Scholar
- [7]C. Eastman, J. Livdini and D. Stoken, “A database for designing large physical systems,” AFIPS44, pp. 603–611, 1975.Google Scholar
- [8]W. R. Franklin, “A linear time exact hidden surface algorithm techniques,” Computer Graphics, 14,3, pp. 117–123, July 1980.Google Scholar
- [9]H. Freeman and P. P. Loutrel, “An algorithm for the solution of the two-dimensional hidden-line problem,” IEEE Trans. Electronic Comput., EC-16, 6, pp. 784–790, Dec. 1967.CrossRefGoogle Scholar
- [10]H. Fuchs, Z. Kedem and B. Naylor, “On visible surface generation by a priori tree structures,” Computer Graphics, 14,3, pp. 124–133, July 1980.Google Scholar
- [11]R. Galimberti and U. Montanari, “An algorithm for hidden line elimination,” Comm. ACM, 12,4, pp. 206–211, April 1969.Google Scholar
- [12]G. Hamlin, Jr. and C. W. Gear, “Raster-scan hidden surface algorithm techniques, Computer Graphics, 11,2, pp. 206–213, Summer 1977.Google Scholar
- [13]M. Lee and R. V. DiMarco, “Computer graphics with hidden surfaces,” Computer & Graphics, 3, 1978.Google Scholar
- [14]P. P. Loutrel, “A solution to the hidden line problem for computer-drawn polyhedra,” IEEE Trans. Comput., C-19, 3, pp. 205–213, March 1970.CrossRefGoogle Scholar
- [15]L. G. Roberts, “Machine perception of three-dimensional solids,” Optical and Electro-Optical Information Processing, MIT Press, Cambridge, Mass., pp. 159, 1964.Google Scholar
- [16]S. Sechrest and D. P. Greenberg, “A visible polygon reconstruction algorithm,” Computer Graphics, 15,3, pp. 17–27, Aug. 1981.Google Scholar
- [17]I. E. Sutherland and G. W. Hogman, “Reentrant polygon clipping,” Comm. ACM, 17,1, pp. 32–42, Jan. 1974.Google Scholar
- [18]I. E. Sutherland, R. F. Sproull and R. A. Shumacker, “A characterization of ten hidden surface algorithms,” Computing Surveys, 6, pp. 1–55, Mar. 1974.MATHCrossRefGoogle Scholar
- [19]G. S. Watkins, “A real time visible surface algorithm,” Univ. of Utah, Comput. Sci. Dept., UTEC-CSc-70–101, NTIS AD 762004, June 1970.Google Scholar
- [20]K. Weiler and P. Atherton, “Hidden surface removal using polygon area sorting,” Computer Graphics, pp. 214–222, Summer 1977.Google Scholar