• Pythagorean-hodograph
• PH curves
• Bézier curves
• Computer-aided Geometric Design
• Hermite Interpolation

The aim of this thesis is to introduce the class of Pythagorean-hodograph (PH) curves, to study their properties and explore some applications, mainly to CNC (Computer Numerical Control) machining.
PH curves are polynomial parametric curves for which their hodograph (i.e., derivative with respect to the curve parameter) satisfies a Pythagorean condition in the n-dimensional euclidean space; our attention is however focused on the particular cases n=2,3, since they are of greater practical interest.
The main appeal of these curves is their property of a polynomial parametric speed, which allows exact computation of the arc-length parameter. This property is especially useful in some CAM (Computer-Aided Manufacturing) applications, where an error-free representation of velocity and acceleration are essential to a reliable implementation.\par
After formally defining PH curves and giving some initial properties, we observe that these curves can be more efficiently handled by using appropriate algebraic tools: instead of relying on the classical real representation, a complex representation is preferred in the two-dimensional case, while quaternions need to be employed in the three-dimensional case.
Interpolation algorithms are then developed to build PH splines of various degrees and continuity orders, for both the two- and three-dimensional cases. Several of the proposed methods operate locally, which is essential for the application to CNC machining, since in that context the interpolation points are not known a priori.
Finally, we introduce and solve the problem of five-axis machining using PH curves by employing the previously presented algorithms.