# Curve Fitting and Curve Displaying

• Theo Pavlidis

## Abstract

In Section 7.6 we discussed how one can define a curve on a discrete grid, and in Section 9.7 we discussed how to derive curves from skeletons. The representation of a curve as a sequence of pixels may be adequate for some applications, but for others we would like to have a mathematical expression for it. Such an expression may be far more compact than the discrete forms, as a comparison between Figure 9.8 and Table 9.1 shows. Finding a curve that passes through a set of given points is the problem of interpolation, while finding a curve that passes near a set of given points is the problem of approximation. We shall use the term curve fitting to refer collectively to both of them. The problem of curve displaying is also of interest: given a mathematical expression, identify the pixels that must be marked so that an image of the curve described by the expression is displayed. The problem is by no means trivial, even in the case of straight line displays (Section 7.6).

### Keywords

Hull Sine Dinate Editing

## Preview

### Relevant Literature

1. [10.BE]
Bezier, P. Numerical Control New York: J. Wiley, 1972. (Transl. from the French by A. R. Forrest and A. F. Pankhurst.)
2. [10.BR]
Bresenham, J. “A Linear Algorithm for Incremental Display of Circular Arcs,” CACM, 20 (1977), pp. 100–106.
3. [10.CG]
Coueignoux, P. and Guedj, R. “Computer Generation of Colored Planar Patterns on TV-Like Rasters,” IEEE Proceedings, 68 (1980), pp. 909–922.
4. [10.CR]
Crow, F. C. “The Use of Grayscale for Improved Raster Display of Vectors and Characters” SIGGRAPH’78, pp. 1–5.Google Scholar
5. [10.DA]
Davis, P. J. Interpolation and Approximation, New York: Random House, Blaisdell, 1963.
6. [10.IK]
Isaacson, E. and Keller, H. B. Analysis of Numerical Methods, New York: J. Wiley, 1966.
7. [10.KU]
Kulpa, Z. “On the Properties of Discrete Circles, Rings, and Disks” CGIP, 10 (1979), pp. 348–365.Google Scholar
8. [10.LL]
Levy, H. and Lessman, F. Finite Difference Equations, New York: Macmillan, 1961.Google Scholar
9. [10.LR]
Lane, J. M. and Riesenfeld, R. F. “A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces,” IEEE Trans. on Pattern Analysis and Machine Intelligence, PAMI-2 (1980), pp. 35–46.
10. [10.RI]
Rice, J. R., The Approximation of Functions, vols. 1 and 2, Reading, Mass.: Addison-Wesley, 1965 and 1969.
11. [10.WA]
Warnock, J. F. “The Display of Characters Using Gray Level Sample Arrays,” SIGGRAPH’80, pp. 302–307.Google Scholar