New algorithms and techniques for computing with geometrically continuous spline curves of arbitrary degree
ESAIM: Modélisation mathématique et analyse numérique, Topics in computer aided geometric design , Tome 26 (1992) no. 1, pp. 149-176. 
@article{M2AN_1992__26_1_149_0,
     author = {Seidel, H.-P.},
     title = {New algorithms and techniques for computing with geometrically continuous spline curves of arbitrary degree},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {149--176},
     publisher = {AFCET - Gauthier-Villars},
     address = {Paris},
     volume = {26},
     number = {1},
     year = {1992},
     mrnumber = {1155005},
     zbl = {0752.65008},
     language = {en},
     url = {http://archive.numdam.org/item/M2AN_1992__26_1_149_0/}
}
TY  - JOUR
AU  - Seidel, H.-P.
TI  - New algorithms and techniques for computing with geometrically continuous spline curves of arbitrary degree
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 1992
SP  - 149
EP  - 176
VL  - 26
IS  - 1
PB  - AFCET - Gauthier-Villars
PP  - Paris
UR  - http://archive.numdam.org/item/M2AN_1992__26_1_149_0/
LA  - en
ID  - M2AN_1992__26_1_149_0
ER  - 
%0 Journal Article
%A Seidel, H.-P.
%T New algorithms and techniques for computing with geometrically continuous spline curves of arbitrary degree
%J ESAIM: Modélisation mathématique et analyse numérique
%D 1992
%P 149-176
%V 26
%N 1
%I AFCET - Gauthier-Villars
%C Paris
%U http://archive.numdam.org/item/M2AN_1992__26_1_149_0/
%G en
%F M2AN_1992__26_1_149_0
Seidel, H.-P. New algorithms and techniques for computing with geometrically continuous spline curves of arbitrary degree. ESAIM: Modélisation mathématique et analyse numérique, Topics in computer aided geometric design , Tome 26 (1992) no. 1, pp. 149-176. http://archive.numdam.org/item/M2AN_1992__26_1_149_0/

[1] B. A. Barsky, The Beta-spline : a local représentation based on shape parameters and fundamental geometric measures, PhD Dissertation, Univ. of Utah, Salt Lake City, USA, 1981.

[2] B. A. Barsky and J. C. Beatty, Local control of bias and tension in Beta-splines, ACM Trans. Graph. 2, 109-134, 1983. | Zbl

[3] B. A. Barsky, Computer Graphics and Geometric Modelling Using Beta-splines, Springer, 1988. | MR | Zbl

[4] B. A. Barsky, Introducing the rational Beta-spline, Proc. 3rd Int. Conf. Eng. Graphics Descr. Geometry, Vienna, 1988. | MR

[5] B. A. Barsky and T. D. Derose, Geometric continuity of parametric curves : Three equivalent characterizations, IEEE Comput. Graph. Appl 9(5), 60-68, 1989.

[6] B. A. Barsky and T. D. Derose, Geometric continuity of parametric curves : Constructions of geometrically continuous splines, IEEE Comput. Graph. Appl. 60-68, 1990.

[7] R. H. Bartels and J. C. Beatty, Beta-splines with a difference, Technical Report CS-83-40, Dept. of Computer Science, Univ. of Waterloo, 1983.

[8] R. H. Bartels, J. C. Beatty and B. A. Barsky, An Introduction to Splines for Use in Computer Graphics and Geometric Modeling, Morgan Kaufmann Publishers, 1987. | MR | Zbl

[9] W. Boehm, Inserting new knots into a B-spline curve, Comput. Aided Design, 12, 50-62, 1980.

[10] W. Boehm, G. Farin and J. Kahmann, A survey of curve and surface methods in CAGD, Comput. Aided Geom. Design 1, 1-60, 1984. | Zbl

[11] W. Boehm, Curvature continuous curves and surfaces, Comput. Aided Geom. Design 2, 313-323, 1985. | MR | Zbl

[12] W. Boehm, Smooth curves and surfaces, in : Farin, G. (ed.), Geometric Modeling, Algorithms and New Trends, SIAM, 1987. | MR

[13] W. Boehm, Rational geometric splines, Comput. Aided Geom. Design 4, 67-77, 1987. | MR | Zbl

[14] C. De Boor, On calculating with B-splines, J. Approx. Theory 6, 50-62, 1972. | MR | Zbl

[15] C. De Boor, A Pratical Guide to Splines, Springer, New York, 1978. | MR | Zbl

[16] P. De Casteljau, Formes à pôles, Hermes, Paris, 1985. | Zbl

[17] P. De Casteljau, Shape Mathematics and CAD, Kogan Page Ltd, London, 1986.

[18] B. W. Char et al., Maple Reference Manual, 5th ed., Watcom Publ. Ltd, Waterloo, 1988.

[19] E. Cohen, T. Lyche and R. F. Riesenfeld, Discrete B-splines and subdivision techniques in computer aided geometric design and computer graphics, Comput. Graph. Image Process. 14, 87-111, 1980.

[20] E. Cohen, A new local basis for designing with tensioned splines, ACM Trans. Graph. 6(2), 81-122, 1987.

[21] H. S. M. Coxeter, Introductin to Geometry, Wiley, New York, 1961. | MR | Zbl

[22] T. D. Derose, Geometric continuity : a parametrization independent measure of continuity for computer aided geometric design, PhD Dissertation, UC Berkeley, Berkeley, U.S.A., 1985.

[23] T. D. Derose and B. A. Barsky, Geometric continuity, shape parameters, and geometric constructions for Catmull-Rom splines, ACM Trans. Graph. 7, 1-41, 1988. | Zbl

[24] P. Dierckx and B. Tytgat, Inserting new knots into Beta-spline curves, in : Lyche, T. and Schumaker, L. L. (eds.), Mathematical Methods in Computer Aided Geometric Design 195-206, Academic Press, 1989. | MR | Zbl

[25] P. Dierckx and B. Tytgat, Generating the Bezier points of a β-spline curve, Comput. Aided. Geom. Design 6, 279-291, 1989. | MR | Zbl

[26] N. Dyn, A. Edelmann and C. A. Micchelli, A locally supported basis function for the representation of geometrically continuous curves, Analysis 7, 313-341, 1987. | MR | Zbl

[27] N. Dyn and C. A. Micchelli, Piecewise polynomial spaces and geometric continuity of curves, IBM Res. Rep. Mathematical Sciences Dept., IBM T. J. Watson Research Center, Yorktown Heights, N.Y., 1985. | Zbl

[28] M. Eck and D. Lasser, B-spline-Bezier representation of geometric spline curves, Preprint 1254, FB. Mathematik, TH. Darmstadt, 1989. | Zbl

[29] M. Eck, Algorithms for geometric spline curves, Preprint 1309, FB Mathematik, TH. Darmstadt, 1990. | MR | Zbl

[30] G. E. Farin, Visually C2-cubic splines, Comput. Aided Design. 14, 137-139, 1982.

[31] G. E. Farin, Some remarks on V2-splines, Comput. Aided Geom. Design 2, 325-328, 1985. | MR | Zbl

[32] G. E. Farin, Curves and Surfaces for Computer Aided Geometric Design, Academic Press, 1988. | MR | Zbl

[33] G. Geise, Über berührende kegelschnitte einer ebenen Kurve, Z. Angew Math. Mech. 42(7/8), 297-304, 1962. | Zbl

[34] R. N. Goldman and C. A. Micchelli, Algebraic aspects of geometric continuity, in Lyche, T. And Schumarker, L. L. (eds.), Mathematical Methods in Computer Aided Geometric Design, 313-332, Academic Press, 1989. | MR | Zbl

[35] R. N. Goldman and B. A. Barsky, On Beta-continuous functions and their application to the construction of geometrically continuous curves and surfaces, in : Lyche, T. and Schumaker, L. L. (eds.), Mathematical Methods in Computer Aided Geometric Design, 299-312, Academic Press, 1989. | MR | Zbl

[36] R. N. Goldman, Blossoming and knot algorithms for B-spline curves, to appear in Comput. Aided Geom. Design. | MR

[37] T. N. T. Goodman, Properties of Beta-splines, J. Approx. Theory 44, 132-153, 1985. | MR | Zbl

[38] T. N. T Goodman and K. Unsworth, Generation of Beta-spline curves using a recurrence relation, in : Earnashaw, R. (ed.), Fundamental Algorithms for Computer Graphics, 325-357, Springer, 1985.

[39] T. N. T Goodman and C. A. Micchelli, Corner cutting algorithms for the Bézier representation of free from curves, IBM Research Report RC 12139, IBM T. J. Watson Research Center, Yorktown Heights, N. Y., 1986. | Zbl

[40] T. N. T Goodman and K. Unsworth, Manipulating shape and producing geometric continuity in Beta-spline curves, IEEE Comput. Graph. Appl. 6(2), 50-56, 1986.

[41] T. N. T Goodman, Constructing piecewise rational curves with Frenet frame continuity, to appear, in Comput. Aided. Geom. Design. | MR | Zbl

[42] J. Gregory, Geometric continuity, in : Lyche, T. and Schumaker, L. L. (eds.), Mathematical Methods in Computer Aided Geom. Design, Academic Press, 1989. | MR | Zbl

[43] H. Hagen, Geometric spline curves, Comput. Aided Geom. Design 2, 223-227, 1985. | MR | Zbl

[44] M. E. Hohmeyer and B. A. Barsky, Rational Continuity : Parametric, Geometric, and Frenet Frame Continuity of Rational Curves, ACM Trans. Graph. 8(4), 1989. | Zbl

[45] J. Hoschek and D. Lasser, Grundlagen der geometrischen Datenverarbeitung, Teubner, 1989. | MR | Zbl

[46] B. Joe, Rational Beta-spline curves and surfaces and discrete Beta-splines, Technical Report TR 87-04, Dept. of Computing Science, Univ. of Alberta, 1987.

[47] B. Joe, Quatric Beta-splines, Technical Report TR 87-11, Dept. of Computing Science, Univ. of Alberta, 1987.

[48] B. Joe, Discrete Beta-splines, Computer Graphics 21(4) (Proc. SIG-GRAPH'87), 137-144, 1987. | MR

[49] B. Joe, Multiple-knot and rational cubic β-splines, ACM Trans. Graph. 8(2), 100-120, 1989. | Zbl

[50] D. Lasser and M. Eck, Bézier representation of geometric spline curves, Technical Report NPS-53-88-004, Naval Postgraduate Schoo, Monterey, 1988.

[51] G. M. Nielson, Some piecewise polynomial alternatives to splines under tension, in : Barnhill, R. E. and Riesenfeld, R. F. (eds.), Computer Aided Geometric Design, Academic Press, 1974. | MR

[52] H. Pottmann, Curves and tensor product surfaces with third order geometric continuity, Proc. 3rd Int. Conf. Eng. Graphics Descr. Geometry, Vienna, 1988. | MR

[53] H. Pottmann, Projectively invariant classes of geometric continuity, Comput. Aided Geom. Design 6, 307-322, 1989. | MR | Zbl

[54] H. Prautzsch, A round trip to B-splines via de Casteljau, ACM Trans. Graph. 8(3), 243-254, 1989. | Zbl

[55] L. Ramshaw, Blossoming : A connect-the-dots approach to splines, Digital Systems Research Center, Palo Alto, 1987.

[56] L. Ramshaw, Béziers and B-splines as multiaffine maps, in : Theoretical Foundations of Computer Graphics and CAD, 757-776, Springer, 1988. | MR

[57] L. Ramshaw, Blossoms are polar forms, Comput. Aided Geom. Design 6, 323-358, 1989. | MR | Zbl

[58] L. L. Schumaker, Spline Functions : Basic Theory, John Wiley & Sons, New York, 1981. | MR | Zbl

[59] H.-P. Seidel, Knot insertion from a blossoming point of view, Comput. Aided Geom. Design 5, 81-86, 1988. | MR | Zbl

[60] H.-P. Seidel, A new multiaffine approach to B-splines, Comput. Aided Geom. Design 6, 23-32, 1989. | MR | Zbl

[61] H.-P. Seidel, Polynome, Splines und symmetrische rekursive Algorithmen im Computer Aided Geometric Design, Habilitationsschrift, Tübingen, 1989.

[62] H.-P. Seidel, Geometric Constructions and Knot Insertion for Geometrically Continuous Spline Curves of Arbitrary Degree, Research Report CS-90-24, Department of Computer Science, University of Waterloo, Waterloo, 1990.

[63] M. C. Stone and T. D. Derose, A geometric characterization of parametric cubic curves, ACM Trans. Graph. 8, 147-163, 1989. | Zbl