A Review of Curve Drawing Algorithms
A variety of curve drawing methods have now been devised for computer graphics and CAD. These range from simple, piecewise interpolating curves through a set of points in a plane, to complex, smoothing curves for CAD, and approximation curves which allow error bounds on individual points to be specified and incorporated in the curve definition.
Single-valued or multi-valued in either coordinate
Shape and axis-independence on transformation (e.g. rotational invariance)
Smoothness and fairness — mathematical, aesthetic, or model-based
Global and local control of shape
The importance of a particular characteristic is often related to the requirements of the application area under consideration, so it is possible to select an approach which satisfies the prime requirement. However, there are advantages and disadvantages to the different methods.
The interface between curve generation and curve display necessitates a consideration of how best to translate curve specifications into drawn curves (e.g. by some optimal vector sequence) and also what primitive functions it is desirable for graphical output devices to accommodate or emulate, in order to facilitate specification of this interface at a high level.
The development of curve drawing algorithms from the early ad hoc approaches in the 1960’s to the sophisticated polynomials of the 1980’s will be reviewed, concentrating on those methods and algorithms which are believed to be fundamental, and the basis for future development.
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- 1.‘Some Algol Plotting Procedures’, J.A.Th.M. Van Berckel and B.J. Mailloux, MR73, Mathematisch Centrum, Amsterdam, 1965.Google Scholar
- 2.‘Methods for Curve Drawing’, B.R. Heap, National Physical Laboratory, 1970.Google Scholar
- 3.‘Numerical Methods for Curve and Surface Fitting’, J.G. Hayes, Bulletin of the Institute of Mathematics and its Applications, Vol 10, No 5 /6. 1974, pp 144–152.Google Scholar
- 6.‘Achievements in Computer-Aided Design’, C.A. Lang, IFIP Proceedings, 1974.Google Scholar
- 7.‘The Theory of Splines and their Applications’, J.H. Ahlberg, E.N. Nilson, J.L. Walsh, New York: Academic Press, 1967.Google Scholar
- 8.‘Theory and Application of Spline Functions’, T.N.E. Greville ( Ed ), Academic Press, 1969.Google Scholar
- 9.‘Mathematical Principles for Curve and Surface Representation’, A.R. Forrest, IPC Science and Technology Press, Proceedings of the Conference ‘Curved Surfaces in Engineering’, Churchill College Cambridge, 1972, pp 5–13.Google Scholar
- 10.‘An Algorithm for generating Spline-Like Curves’, D.V. Ahuja, IBM Systems Journal, Nos 3,4, 1968, pp 206–217.Google Scholar
- 11.‘Numerical Algorithms Group Library’, NAG Ltd, Oxford, 1976.Google Scholar
- 14.‘Procedure Curve’, P.J. Le Riche, Computer Journal, 1969, p 291.Google Scholar
- 15.‘Optimizing Curve Segmentation in Computer Graphics’, K. Reumann and A.P.M. Witkam, Proceedings of the International Computing Symposium 1973, A. Gunther et al. ( Eds ), North Holland, 1974, pp 467–472.Google Scholar
- 16.‘Principles of Interactive Computer Graohicsp, J.D. Foley and A. van Dam, Addison Wesley, 1982, pp 514–536.Google Scholar
- 17.‘An Introduction to the Use of Splines in Computer Graphics’, R.H. Bartels, J.C. Beatty, and B.A. Barsky, University of Waterloo TR CS-83–09, UC Berkeley, TR UCB/CSD 83–136, Revised May 1984.Google Scholar
- 18.‘Computer-Aided Geometric Design’, B.A. Barsky, IEEE CG &A, July 1981, pp 67–109.Google Scholar
- 19.‘A Description and Evaluation of Various 3-D Models’, B.A. Barsky, IEEE CG & A, January 1984, pp 38–52.Google Scholar
- 21.‘Algorithms for the Evaluation and Perturbation of Beta-Splines’, B.A. Barsky, IEEE CG & A, 1984.Google Scholar
- 22.‘User Interfaces for Free-Form Surface Design’, A.R. Forrest, University of East Anglia, CGP 82/4, 1982.Google Scholar
- 24.‘Some Mathematical Tools for a Modeller’s Workbench’, E. Cohen, IEEE CG & A, Vol 3, No 7, 1983, pp 63–66.Google Scholar
- 25.‘Superquadrics and Angle-Preserving Transformations’, A.H. Barr, IEEE CG & A, Vol 1, No 1, 1981, pp 11–23.Google Scholar
- 27.‘The Algebraic Properties of Homogeneous Second Order Surfaces’, J.F. Blinn, SIGGRAPH 84 Tutorial Notes 15 ‘The Mathematics of Computer Graphics’.Google Scholar