The Community for Technology Leaders
RSS Icon
Issue No.02 - March/April (1994 vol.14)
pp: 14-23
<p> A new, recursive, space-subdivision algorithm for rasterizing algebraic curves and surfaces gets its accuracy from a newly devised, computationally efficient, and asymptotically correct test. The approach followed is essentially the interval arithmetic method for rendering implicit curves. The author's contribution is a particularly efficient way to construct inclusion functions for polynomials. An ideal algorithm is given for rendering an algebraic curve Z(f)={(x,y):f(x,y)=0} in a square box of side n. The algorithm scans the square and paints only those pixels cut by the curve. This algorithm is ideal, because every correct algorithm should paint exactly the same pixels, but it is impractical. It requires n/sup 2/ test evaluations, one for each pixel in the square. However, since in general it will be rendering a curve on a planar region, the number of pixels it is expected to paint is only O(n). We need a more efficient algorithm. There are two issues to examine. The first is how to reduce the computational complexity by recursive subdivision. The second is how to test whether the curve Z(f) cuts a square.</p>
Gabriel Taubin, "Rasterizing Algebraic Curves and Surfaces", IEEE Computer Graphics and Applications, vol.14, no. 2, pp. 14-23, March/April 1994, doi:10.1109/38.267467
1. C.L. Bajaj et al., "Tracing Surface Intersections,"Computer Aided Geometric Design, Vol. 5, No, 4, Nov. 1988, pp. 285-307.
2. D.S. Arnon, "Topologically Reliable Display of Algebraic Curves,"Computer Graphics, Vol. 17, No. 3, July 1983, pp. 219-227.
3. E.L. Allgower and P.H. Schmidt, "An Algorithm for Piecewise-Linear Approximation of an Implicitly Defined Manifold,"SIAM J. Numerical Analysis, Vol. 22, No. 2, Apr. 1985, pp. 322-346.
4. W.E. Lorensen and H.E. Cline, "Marching Cubes: A High-Resolution 3D Surface Construction Algorithm,"Computer Graphics(Proc. Siggraph 87), ACM Press, New York, Vol. 21, No. 4, 1987, pp. 163-169.
5. E.L. Allgower and S. Gnutzmann, "Simplicial Pivoting for Mesh Generation of Implicitly Defined Surfaces,"Computer-Aided Geometric Design, Vol. 8, No. 4, Oct. 1991, pp. 305-325.
6. M. Hall and J. Warren, "Adaptive Polygonization of Implicitly Defined Surfaces,"IEEE CG&A, Vol. 10, No. 6, Nov. 1990, pp. 33-42.
7. G. Taubin, "Distance Approximations for Rasterizing Implicit Curves,"ACM Trans. Graphics, Vol. 13, No. 1, Jan. 1994 (to appear).
8. P. Pedersen,Counting Real Zeros, doctoral thesis, Computer Science Dept., New York Univ., New York, 1991.
9. R.T. Farouki and V.T. Rajan, "On the Numerical Condition of Poly-1 nomials in Bernstein Form,"Computer-Aided Geometric Design, Vol. 4, No. 3, Nov. 1987, pp. 191-216.
10. T. Duff, "Interval Arithmetic and Recursive Subdivision for Implicit Functions and Constructive Solid Geometry,"Computer Graphics(Proc. Siggraph), Vol. 26, No. 2, July 1992, pp. 131-138.
11. A. Borodin and I. Munro,The Computational Complexity of Algebraic and Numeric Problems, American Elsevier, New York, 1975.
12. R. Walker,Algebraic Curves, Princeton Univ. Press, Princeton, N.J., 1950.
14 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool