Data interpolation with planar quintic Pythagorean-Hodograph B-spline curves in revised form

This thesis focuses on interpolating data using planar quintic C^2 PH B-spline curves, a generalization of PH Bézier curves consisting of piecewise-polynomial parametric curves for which the Euclidean norm of the first derivative is a polynomial B-spline. The main contribution originates from a new representation of planar quintic C^2 PH B-splines that minimizes the number of knots and control points, and focuses on demonstrating how to constrain the degrees of freedom in order to interpolate a given sequence of input data. Thanks to the new formulation, the interpolation problem requires solving a system of nonlinear equations rather than a constrained optimization problem, thus significantly reducing the computational complexity. More precisely, the content of the thesis is structured as follows. Chapter 1 introduces the simplified representation of planar quintic PH B-splines, focussing on the two cases of major interest in applications, i.e., the so-called clamped and closed configurations. Chapter 2 examines interpolation with clamped curves, deriving the system of nonlinear equations and detailing three numerical strategies to solve it, each supported by pseudocode and examples. Chapter 3 adapts these techniques to closed quintic PH curves. Chapter 4 proposes an automatic strategy to select the two free control points of the preimage of a clamped PH B-spline, eliminating the need for user input and ensuring high-quality interpolation; this step is unnecessary for closed curves, where the interpolation conditions determine all control points. Overall, the proposed approach improves computational efficiency of the previously proposed strategies without reducing the quality of the results and shows potential for being extended to spatial quintic PH B-spline curves.