Geodesic
Construction of optimal Bézier splines
Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 3, pp. 57-72 
V. V. Borisenko. Construction of optimal Bézier splines. Fundamentalʹnaâ i prikladnaâ matematika, Tome 21 (2016) no. 3, pp. 57-72. http://geodesic.mathdoc.fr/item/FPM_2016_21_3_a3/
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     title = {Construction of optimal {B\'ezier} splines},
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     url = {http://geodesic.mathdoc.fr/item/FPM_2016_21_3_a3/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a construction of a smooth curve by a set of interpolation nodes. The curve is constructed as a spline consisting of cubic Bézier curves. We show that if we require the continuity of the first and second derivatives, then such a spline is uniquely defined for any fixed parameterization of Bézier curves. The control points of Bézier curves are calculated as a solution of a system of linear equations with a four-diagonal band matrix. We consider various ways of parameterization of Bézier curves that make up a spline and their influence on its shape. The best spline is computed as a solution of an optimization problem: minimize the integral of the square of the second derivative with a fixed total transit time of a spline.

[1] A primer on Bézier curves: A free, online book for when you really need to know how to do Bézier things, http://pomax.github.io/bezierinfo/

[2] Bartels R. H., Beatty J. C., Barsky B. A., “Bézier Curves”, An Introduction to Splines for Use in Computer Graphics and Geometric Modelling, Morgan Kaufmann, San Francisco, 1998, 211–245 | MR

[3] De Boor C., A Practical Guide to Splines, Springer, 1978 | MR | Zbl

[4] Hill climbing, https://en.wikipedia.org/wiki/Hill_climbing

[5] Knott G. D., Interpolating Cubic Splines, Springer, 2012 | MR

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[7] Shikin E. V., Plis A. I., Handbook on Splines for the User, CRC Press, 1995 | MR | Zbl