Smoothing of Polygonal Chains for 2D Shape Representation Using a G2-Continuous Cubic A-Spline
We have developed a G 2-continuous cubic A-spline, suitable for smoothing polygonal chains used in 2D shape representation. The proposed A-spline scheme interpolates an ordered set of data points in the plane, as well as the direction and sense of tangent vectors associated to these points. We explicitly characterize curve families which are used to construct the A-spline sections, whose members have the required interpolating properties and possess a minimal number of inflection points. The A-spline considered here has many attractive features: it is very easy to construct, it provides us with convenient geometric control handles to locally modify the shape of the curve and the error of approximation is controllable. Furthermore, it can be rapidly displayed, even though its sections are implicitly defined algebraic curves.
Mathematics Subject Classification: 65D07(splines), 65D05 (interpolation), 65D17 (Computer Aided Design).
KeywordsAlgebraic cubic splines polygonal chain data interpolation and fitting 2D shape representation
- 3.Behar, S., Hernández, V., Alvarez, L., Estrada, J.: Computing a revolution shell using a G 2-continuous A-spline and a semidiscrete method for the EDPs. In: Proceedings IV, ITLA, pp. 241–250 (2001) ISBN: 959-7056-13-5Google Scholar
- 4.Estrada, J., Martínez, D., León, D., Theisel, H.: Solving Geometric Problems using Subdivision Methods and Range Analysis. In: Daehlen, M., Morken, K., Shumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces: Tromso 2004, pp. 101–114. Nashboro Press, Brentwood (2004)Google Scholar
- 5.Hernández, V., Martínez, D., Estrada, J.: Fitting a conic A-spline to contour image data. Revista Investigación Operacional 29, 55–64 (2002)Google Scholar
- 6.Hernández, V., Behar, S., Estrada, J.: Geometric design by means of a G 2 continuous A-spline, Approximation. Optimization and Mathematical Economics, pp. 133–145. Physica, Heidelberg (2001)Google Scholar
- 7.Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. International. J. Comput. Vision, 321–331 (1988)Google Scholar
- 8.Paluszny, M., Patterson, R.: G 2-continuous cubic algebraic splines and their efficient display. In: Laurent, P.J., Le Méhauté, A., Schumacker, L.L. (eds.) Curves and Surfaces II, pp. 353–359 (1994)Google Scholar