David P. Dobkin
Abstract
We present a method for tracing a curve that is represented as the contour of a function in Euclidean space of any dimension. The method proceeds locally by following the intersections of the contour with the facets of a triangulation of space. The algorithm does not fail in the presence of high curvature of the contour; it accumulates essentially no round-off error and has a well-defined integer test for detecting a loop. In developing the algorithm, we explore the nature of a particular class of triangulations of Euclidean space, namely, those generated by reflections.
- 1 At.t.aowp.R, E. L., ANn (',~.nRa, K. Simplieiml And emntinllnf.imn mpthnd.~ fnr ApprnYirngf.ina fixed points and solutions to systems of equations. SIAM Rev. 22 (dan. 1980), 28-85.Google Scholar
- 2 ALLGOWER, E. L., AND SCHMIDT, P.H. An algorithm for piecewise linear approximation of an implicitly defined manifold. SIAM d. Numer. Anal. 22 (April 1985), 322-346.Google Scholar
- 3 ARNON, D.S. Topologically reliable display of algebraic curves. ACM SIGGRAPH Comput. Gr. 17 (duly 1983), 219-227. Google Scholar
- 4 BLANCHARD, P. Complex analytic dynamics on the Riemann sphere. Bull. Am. Math. Soc. I1 (duly 1984), 85-141.Google Scholar
- 5 CONWAY, J. n., AND SLOaNE, N. d.A. Fast quantizing and decoding algorithms for lattice quantizers and codes. IEEE Trans. In{. Theory 28 (Mar. 1982), 227-232.Google Scholar
- 6 COXETER, H. S. M. Discrete groups generated by reflections. Ann. Math. 35 (duly i934), 588-621.Google Scholar
- 7 COXETER, H. S.M. Regular Polytopes. 3rd ed. Dover, New York, 1973.Google Scholar
- 8 DONGARRA, J. J., BUNCH, J. R., MOLER, C. B., AND STEWART, G. W. LZNPACK Users'Guide. SIAM, Philadelphia, Pa., 1979.Google Scholar
- 9 DUNEAU, M., AND KATZ, A. Quasiperiodic patterns and icosahedral symmetry. J. Physique 47 (Feb. 1986), 181-196.Google Scholar
- 10 FOLEY, J. D., AND VAN DAM, A. Fundamentals of Interactive Computer Graphics. Addison- Wesley, Reading, Mass., 1982. Google Scholar
- 11 GEISOW, A. Surface interrogations. Ph.D. thesis, Dept. of Computer Science, Univ. of East Anglia, Norwich, U.K., 1983Google Scholar
- 12 GUILLEMIN, V., AND POLLACK, A. Differential Topology. Prentice-Hail, Englewood Cliffs, N.J., 1974.Google Scholar
- 13 HOFFMAN, K. Analysis in Euclidean Space. Prentice-Hall, Englewood Cliffs, N.J., 1975.Google Scholar
- 14 KARAMARDIAN, S., ED. Fixed Points: Algorithms and Applications. Academic Press, New York, 1977.Google Scholar
- 15 ORTEGA, J., AND RHEINBOLDT, W. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970. Google Scholar
- 16 OSTROWSKI, A.M. Solutions of Equations and Systems of Equations. 2nd ed. Academic Press, New York, 1966Google Scholar
- 17 REQUICHA, A., AND VOELCKER, S. Solid modeling: Current status and research directions. IEEE Comput. Graph. Appl. 3 (Oct. 1983), 25-37.Google Scholar
- 18 SABIN, M. A. Contouring~The state of the art. In Fundamental Algorithms for Computer Graphics, R. A. Earnshaw, Ed. Springer-Verlag, New York, 1985.Google Scholar
- 19 SCARF, H. The approximation of fixed points of a continuous mapping. SIAM J. Appl. Math. 15 (Sept. 1967), 1328-1343.Google Scholar
- 20 SPERNER, E. Neuer Beweis fur die Invarianz der Dimensionszahl und des Gebietes. Abh. a. d. Math. Sem. D. Univ. Hamburg 6 (1928), 265-272.Google Scholar
- 21 TODD, M.J. The Computation of Fixed Points with Applications. Lecture Notes in Economics and Mathematical Systems, 124. Springer-Verlag, New York, 1976.Google Scholar
- 22 TUKEY, P. A., AND TUKEY, J.W. Data-driven view selection; agglomeration and sharpening. In Interpreting Multivariate Data, V. Barnett, Ed. Wiley, New York, 1981, 215-243.Google Scholar
Index Terms
- Contour tracing by piecewise linear approximations
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Reviews
Andrew Timothy Thornton
The authors present the mathematics of a technique for contouring a given function, using a local technique. Given a starting point, the contour is followed by a process of localized triangulation, evaluation of the vertices, and then linear interpolation; a new triangulation is then performed by reflection, and the process is continued until the contour either exits the space being studied or forms a closed loop. As with all local triangulation methods, accuracy is restricted by the size of the triangulation chosen; the authors use a fixed triangulation and present some useful work comparing different triangulation schemes. The benefits, though, lie in the absence of the need to find the derivative of the function and in the assurance that it is not possible to lose the curve in areas of high curvature. Although the paper focuses on generating a linear contour, the technique can deal with an n-dimensional contour, which may be of use in visualizing multidimensional functions. The paper is to be followed up by one giving the details of an implementation of the algorithm, which should appeal to those with a pragmatic rather than mathematical turn of mind.
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