Abstract
A integer version of the well-known subdivision algorithm of NURBS curves is presented here. The algorithm is used to render NURBS curves of any degree on a raster device using a piece-wise linear approximation. The approximation is independent of the parametrization, that is, it is independent of the weights used. The maximum deviation between a precisely sampled curve and that of the subdivisionbased rendering is one pixel — an inherent feature of the subdivision technique. The algorithm works entirely in the screen coordinate system and produces smooth rendering of curves without oversampling. The integer arithmetic allows the rendering of relatively complex curves of 2 to 8 degrees within a fraction of a second on an i80286/386 processor, and is a good candidate for hardware implementation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Bartels RH, Beatty JC, Barsky BA (1987) An introduction to splines for use in computer graphics and geometric modeling. Morgan Kaufmann, Los Altos, CA
Boehm W (1990) Inserting new knots into B-spline curves and surfaces. Computer-Aided Design 12(4):199–201
Boehm W (1981) Generating the Bézier points of B-spline curves and surfaces. Computer-Aided Design 13(6):365–366
Boehm W, Prautzsch H (1985) The insertion algorithm. Computer-Aided Design 17(2):58–59
Boehm W, Farin G, Kahmann J (1984) A survey of curve and surface methods in CAGD. Computer Aided Geometric Design 1(1):1–60
Botting RJ (1978) A theory of parametric curve plotting. comput Graph and Image Proc 7:139–145
Chang S-L, Shantz M, Rocchetti R (1989) Rendering cubic curves and surfaces with integer adaptive forward differencing. Comput Graph 23(3):157–166
Cohen E, Lyche T, Riesenfeld RF (1980) Discrete B-splines and subdivision techniques in computer-aided geometric design and computer graphics. Comput Graph and Image Processing 14:87–111
Crampin M, Guifo R, Read GA (1985) Linear approximation of curves with bounded curvature and data reduction algorithm. Computer-Aided Design 17(6):257–261
de Boor C (1978) A practical guide to splines. Springer, New York Berlin Heidelberg
Casteljau P de (1963) Courbes et surfâces a pôles. Andrè Citroën Automobiles SA, Paris
Casteljau P de (1986) Mathematics and CAD, vol 2: Shape mathematics and CAD. Kogan Page, London
Hartly PJ, Judd CJ (1980) Parametrization and shape of B-spline curves for CAD. Computer-Aided Design 12(5):235–238
Imai H, Iri M (1986) Computational-geometric methods for polygonal approximation of a curve. Comput Graph and Image Proc 36:31–41
Koparkar PA, Mudur SP (1983) The development of programs for the processing of parametric curves. Comput Graph Forum 2(2–3):135–144
Koparkar PA, Mudur SP (1983) A new class of algorithms for the processing of parametric curves. Computer-Aided Design 15(1):41–45
Lane JM, Riesenfeld RF (1980) A theoretical development for the computer generation and display of piecewise polynomial surfaces. IEEE Trans on Pattern Analysis and Machine Intelligence, PAMI 2(1):35–46
Lien S-L, Shantz M, Pratt V (1987) Adaptive forward differencing for rendering curves and surfaces. Comput Graph 21(4):111–118
Mastin CW (1986) Parametrization in grid generation. Computer-Aided Design 18(1):22–24
Maxwell PC, Baker PW (1979) The generation of polygons representing circles, ellipses and hyperbolas. Comput Graph and Image Proc 10:84–93
Piegl L (1983) A fast algorithm for parametric curve plotting. Comput Graph 7(3–4):361–367
Piegl L (1991) On NURBS: a survey. IEEE Comput Graph Appl 11(1):55–71
Piegl L, Tiller W (1987) Curve and surface constructions using rational B-splines. Computer-Aided Design 19(9):485–498
Piegl L, Tiller W (1991) Storage efficient decomposition of B-spline curves, TR CSE 91-01, Department of Computer Science and Engineering University of South Florida, Tampa, FL 33620, USA
Ramer U (1972) An interactive procedure for polygonal approximation of plane cuves. Comput Graph and Image Proc 1:244–256
Rockwood AP (1987) A generalized scanning technique for display of parametrically defined surfaces. IEEE Comput Graph Appl 7(8):15–26
Rogers DF, Adams JA (1990) Mathematical elements for computer graphics, 2nd ed. McGraw-Hill, New York
Shantz M, Chang S-L (1988) Rendering trimmed NURBS with adaptive forward differencing. Comput Graph 22(4):189–198
Sharpe RJ, Thorne RW (1982) Numerical method for extracting an arc length parametrization from parametric curves. Computer-Aided Design 14(2):79–81
Smith LB (1971) Drawing ellipses, hyperbolas and parabolas with a fixed number of points and maximum inscribed area. Comput J 14(1):81–86
Stone H (1961) Approximation of curves by line segments. Math Comput 15:40–47
Tiller W (1983) Rational B-splines for curve and surface representation. IEEE Comput Graph Appl 3(10):61–69
Williams CM (1981) An efficient algorithm for piecewise linear approximation of planar curves. Comput Graph and Image Proc 8:286–293
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Anantakrishnan, N., Piegl, L.A. Integer subdivision algorithm for rendering NURBS curves. The Visual Computer 8, 149–161 (1992). https://doi.org/10.1007/BF01902135
Issue Date:
DOI: https://doi.org/10.1007/BF01902135