Multi-Degree Splines
Multi degree splines (MD-splines, for short) are piecewise functions comprised of
polynomial segments of different degrees.
They were proposed in the seminal paper [1] and they have been a subject of study
in several works [2], [3], [4], [5], [6] and [7] and other more recent ones [8] and [9].
To model a shape, the MD-splines use, in addition to the knot interval and control points, an additional parameter, the degree.
The degree can be chosen locally to get the best shape fitting,
thus allowing to use less control points than those necessary with conventional splines (the latter being intended as spline
spaces where every piece is spanned by polynomials of the same degree).
At the same time, MD-splines reduce to conventional splines when all segments are of the same degree, thus generalizing the
traditional approach.
With C1 MD-splines we mean a subclass where, between two segments of different degrees, is allowed at most C1 continuity.
[1]Nurnberger G., Schumaker L.L., Sommer M., Strauss H. Generalized chebyshevian splines. SIAM J. Math Anal 1984; 15(4): 790–804.
[2]Buchwald B,Muhlbach G. Construction of B-splines for generalized spline spaces generated from local ECT-systems. J.Comput.Appl.Math.
2003;159(2):249–67.
[3] Sederberg, T.W., Zheng, J. and Song, X., 2003,
Knot Intervals and Multi-degree Splines, CAGD, 20 (7)
[4] Wang, G. and Deng, C., 2007, On the degree elevation of B-spline curves and corner cutting,
CAGD, 24 (2) pp.90-98.
[5] Shen, W. and Wang, G., 2010, Changeable degree spline basis functions,
JCAM, 234 (8) pp.2516-2529.
[6] Shen, W. and Wang, G., 2010, A basis of multi-degree splines,
CAGD, 27 (1) pp.23-35.
[7] Li, X., Huang, Z.J. and Liu, Z., 2012, A Geometric Approach for Multi-Degree Spline,
Journal of Computer Science and Technology, 27 (4) pp.841-850.
[8] Shen, W., Wang, G. and Yin, P., 2013, Explicit representations of changeable degree spline basis functions,
JCAM, 238 (1), pp.39-50.
[9] Shen W., Yin P. and Tan C., 2016, Degree elevation of changeable degree spline,
JCAM, 300 pp.56-67.
[10] Toshniwal D., Speleers H., Hiemstra R.R., Hughes T.J.R, 2017, Multi-degree smooth polar splines: A framework
for geometric modeling and isogeometric analysis, Comput. Methods Appl. Mech. Engrg. 316 pp.1005-1061.
[11] Beccari C.V., Casciola G., Morigi S., On multi-degree splines, 2017, CAGD, 58 pp.8-23.
[12] Speleers H., Algorithm 999: Computation of multi-degree B-splines. ACM Transactions on Mathematical Software 2019; 45(4).
[13] Beccari C.V., Casciola G., A Cox-de Boor-type recurrence relation for C1 multi-degree splines, 2019, CAGD, 75.
[14] Toshniwal D., Speleers H., Hiemstra R.R., Manni C., Hughes T.J.R, 2018, Multi-degree B-splines: Algorithmic computation
and properties, 2020, CAGD, 76.
[15] Beccari C.V., Casciola G., Matrix representations for multi-degree B-splines, 2021, JCAM, 381.
[16] Beccari C.V., Casciola G., Stable numerical evaluation of multi-degree B-splines, 2022, JCAM, 400.
[17] Ma X., Shen W., Generalized de Boor-Cox Formulas and Pyramids for Multi-Degree Spline Basis Functions, 2023, Mathematics, 11, 367.