Cubic Bézier-Spline Curves: Interpolation and Maximum Curvature

TL;DR

This work tackles the inverse interpolation problem for Bézier-spline curves and the computation of maximum curvature for Bézier-spline space curves. It introduces relaxed uniform $B$-splines and periodic variants to obtain closed $C^2$ interpolants for a given data set, deriving closed-form expressions for the control points and a piecewise cubic, $C^2$ parametrization with explicit end conditions. To locate the curvature extrema, the authors derive $K=\kappa^2$ and the nonlinear condition $K'=0$, implementing a Maple-based algorithm that per interval solves a degree-$7$ polynomial to find $\kappa_{\max}$. The approach is demonstrated on numerous 2D and 3D datasets and a closed curve, illustrating practical utility for design, graphics, and geometric analysis of tubular surfaces, with extensions to higher dimensions and differential-equation solvers.

Abstract

In this paper, we propose a closed-form solution to the inverse problem in interpolation with periodic uniform B-spline curves. This solution is obtained by modifying the one we have established to a similar problem with relaxed uniform B-spline curves. Then we use these solutions to determine the maximum curvature of a Bézier-spline curve. Our computational and graphical examples are presented with the aid of Maple procedures.

Figures (6)

• Figure 1: Bézier curves of degree $5$ in $\mathbb{R}^2$ (left) and $\mathbb{R}^3$ (right).
• Figure 2: The part $\mathscr{C}_k$ of $\mathscr{C}$ with its control points $S_{k-1}$, $P_{k-1}$, $Q_k$, $S_k$.
• Figure 3: Bézier-spline curves in $\mathbb{R}^2$ (left) and $\mathbb{R}^3$ (right).
• Figure 4: $\mathscr{C}$ on the left, $\mathscr{L}$ on the right.
• Figure 5: Curves obtained from given control points $B_k$.

Theorems & Definitions (2)

• Theorem 2.1
• Theorem 2.2