ArticlesNo Access

BEZIER AND B-SPLINE CURVES WITH KNOTS IN THE COMPLEX PLANE

KONSTANTINOS I. TSIANOS

Department of Computer Science, Rice University, 6100 Main Street, Houston, TX, 77005, USA

Search for more papers by this author

and

RON GOLDMAN

Department of Computer Science, Rice University, 6100 Main Street, Houston, TX, 77005, USA

Search for more papers by this author

https://doi.org/10.1142/S0218348X11005221Cited by:6 (Source: Crossref)

Abstract

We extend some well known algorithms for planar Bezier and B-spline curves, including the de Casteljau subdivision algorithm for Bezier curves and several standard knot insertion procedures (Boehm's algorithm, the Oslo algorithm, and Schaefer's algorithm) for B-splines, from the real numbers to the complex domain. We then show how to apply these polynomial and piecewise polynomial algorithms in a complex variable to generate many well known fractal shapes such as the Sierpinski gasket, the Koch curve, and the C-curve. Thus these fractals also have Bezier and B-spline representations, albeit in the complex domain. These representations allow us to change the shape of a fractal in a natural manner by adjusting their complex Bezier and B-spline control points. We also construct natural parameterizations for these fractal shapes from their Bezier and B-spline representations.

Keywords:

References

R. Goldman, The fractal nature of Bezier curves, Proceedings of Geometric Modeling and Processing 2004 Theory and Applications (Beijing, China, 2004) pp. 3–11. Google Scholar
S. Schaefer, R. Goldman and D. Levin, Subdivision schemes and attractors, Proceedings of Third Eurographics Symposium on Geometric Processing 2005, eds. M. Desbrun and H. Pohmann (Vienna, Austria, 2005) pp. 171–180. Google Scholar
H. Prautzsch and C. A. Micchelli, Comput. Aided Geom. D 4, 133 (1987), DOI: 10.1016/0167-8396(87)90030-6. Crossref, Google Scholar
C. A. Micchelli and H. Prautzsch, Comput. Aided Geom. D 4, 321 (1987). Crossref, Google Scholar
B. Foster, T. Blu and M. Unser, Appl. Comp. Harmon. Anal. 20, 281 (2006). Google Scholar
G. Farin , Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide ( Academic Press , London , 2002 ) . Google Scholar
R. Goldman , An Integrated Approach to Computer Graphics and Geometric Modeling ( Francis and Taylor , New York , 2009 ) . Google Scholar
M. F. Barnsley , Fractals Everywhere , 2nd edn. ( Academic Press , Boston , 1993 ) . Google Scholar
H. Abelson and A. DiSessa , Turtle Geometry: The Computer as a Medium for Exploring Mathematics ( MIT Press , MA , 1986 ) . Crossref, Google Scholar
T. Ju, S. Schaefer and R. Goldman, Comput. Graph. 28, 991 (2004), DOI: 10.1016/j.cag.2004.08.016. Crossref, Web of Science, Google Scholar
R. Goldman , Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling ( Morgan Kaufmann , San Diego , 2002 ) . Google Scholar
L. V. Ahlfors , Complex Analysis ( McGraw-Hill , 1979 ) . Google Scholar
L. Ramshaw, Theoretical Foundations of Computer Graphics and CAD, NATO ASI 40, ed. R. A. Earnshaw (Springer-Verlag, New York, 1988) pp. 757–776. Crossref, Google Scholar
L. Ramshaw, Comput. Aided Geom. D 6, 323 (1989), DOI: 10.1016/0167-8396(89)90032-0. Crossref, Web of Science, Google Scholar
W. Boehm, Comput. Aided D 12, 99 (1980). Web of Science, Google Scholar
S. Schaefer and R. Goldman, Comput. Aided Geom. D 26, 75 (2009). Crossref, Web of Science, Google Scholar
J. Lane and R. Riesenfeld, IEEE Trans. Pattern Anal. 2, 35 (1980). Web of Science, Google Scholar