Construction of Optimal Bézier Splines

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.