Interpolating B\'ezier spline surfaces with local control

A. P. Pobegailo

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2017), pp. 51-62.

Résumé

This paper presents an approach to construct interpolating spline surfaces over a bivariate network of curves with rectangular patches. Patches of the interpolating spline surface are constructed by means of blending their boundaries with special polynomials. In order to ensure a necessary parametric continuity of the designed surface the polynomials of the corresponding degree are used. The constructed interpolating spline surfaces have local shape control. If the surface frame is determined by means of Bézier curves then patches of the interpolating spline surface are Bézier surfaces.

Bibliographie

[1] Barnhill R. E., Brown J. H., Klucewicz I. M., “A new twist in computer aided geometric design”, Computer Graphics and Image Processing, 8:1 (1978), 78–91 | DOI

[2] Barnhill R. E., Farin G., Fayard L., Hagen H., “Twists, curvatures and surface interrogation”, Computer-Aided Design, 20:6 (1988), 341–346 | DOI | Zbl

[3] Chiyokura H., Kimura F., “Design of solids with free-form surfaces”, Computer Graphics, 17:3 (1983), 289–298 | DOI

[4] Comninos P., “An interpolating piecewise bicubic surface with shape parameters”, Computers Graphics, 25:3 (2001), 463–481 | DOI

[5] Coons S. A., Surfaces for computer-aided design of space forms, Report MAC-TR-41, Project MAC, Massachusetts Institute of Technology, 1967

[6] Farin G., Hansford D., “Agnostic $G^1$ Gregory Surfaces”, Graphical Models, 76:4 (2012), 346–350 | DOI

[7] Faux I. D., Pratt M. J., Computational Geometry for Design and Manufacturing, Ellis Horwood Ltd., New York, 1979 | MR

[8] Forrest A. R., “On Coons and other methods for the representation of curved surfaces”, Computer Graphics and Image Processing, 4:1 (1972), 341–359 | DOI | MR

[9] Goldman R., “On the Algebraic and Geometric Foundations of Computer Graphics”, ACM Transactions on Graphics, 21:1 (2002), 52–86 | DOI

[10] Gordon W. J., “Spline-blended surface interpolation through curve networks”, J. Math. and Mech., 18:10 (1969), 931–957 | MR

[11] Gregory J., “Smooth interpolation without twist constraints”, Computer Aided Geometric Design, eds. R. E. Barnhill, R. F. Riesenfeld, Academic Press, 1974, 71–88 | DOI | MR

[12] Juhásza I., Hoffmann M., “Surface interpolation with local control by linear blending”, Annales Mathematicae et Informaticae, 36 (2009), 77–84 | MR

[13] Kallay M., Ravani B., “Optimal twist vectors as a tool for interpolating a network of curves with a minimum energy surface”, Computer Aided Geometric Design, 7:6 (1990), 465–473 | DOI | MR | Zbl

[14] Peters J., “Local smooth surface interpolation: a classification”, Computer Aided Geometric Design, 7:1–4 (1990), 191–195 | DOI | MR | Zbl

[15] Pobegailo A. P., “Local interpolation with weight functions for variable-smoothness curve design”, Computer-Aided Design, 23:8 (1991), 579–582 | DOI | Zbl

[16] Pobegailo A. P., “Interpolating rational Bézier spline curves with local shape control”, International Journal of Computer Graphics and Animation, 3:4 (2013), 1–14 | DOI | MR

[17] Pobegailo A. P., “Interpolating Bézier spline curves with local control”, Bul. Acad. Ştiinţe Repub. Moldova, Mat., 2014, no. 2(75), 18–28 | MR | Zbl

[18] Shirman L. A., Sequin C. H., “Local surface interpolation with Bézier patches”, Computer Aided Geometric Design, 4:4 (1987), 279–295 | DOI | MR | Zbl

[19] Shirman L. A., Sequin C. H., “Local surface interpolation with shape parameters between adjoining Gregory patches”, Computer Aided Geometric Design, 7:5 (1990), 375–388 | DOI | MR | Zbl

[20] Vida J., Martin R. R., Várady T., “A survey of blending methods that use parametric surfaces”, Computer Aided Design, 26:5 (1994), 341–365 | DOI | Zbl