We study the Pythagorean hodograph (PH) curves in the Euclidean or Minkowski spaces of various dimensions in the Clifford algebra framework. After the introductory chapter (Chapter 1) and the preliminaries on the basic setup (Chapter 2), we study, in Chapter 3, the topological selection problem of the planar quintic Hermite interpolants, culminating in the complete resolution of the problem. In Chapter 4, we continue the study of the Hermite interpolation problem of the space quintic PH curves. This study heavily relies on the Clifford algebra framework within which a complete description of the length-minimizing space PH quintic curve is obtained. In Chapter 5, we study the rotation-minimizing rational parametrization of the canal surfaces. It turns out that the geometry of canal surface is intricately tied with the Lorentzian geometry of the 4-dimensional Minkowski space, which we exploit in full detail. Utilizing the Minkowski space and the Clifford algebra thereof, we are able to reformulate the problem as a simpl
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