An important tool in the field of Computer Aided Geometric Design are Pythagorean hodograph (PH) curves, as they are smooth, computationally efficient, and allow the straightforward analytical computation of some of their properties. Moreover, these curves can be considered in both Euclidean and Minkowski spaces of arbitrary dimension. Although their construction is based on the same underlying idea, their characterization varies significantly. For example, PH curves in the Euclidean plane ${\mathbb R}^2$ are most conveniently expressed using complex numbers, while PH curves in ${\mathbb R}^3$ are typically expressed with quaternions. To address these differences, we present a unified characterization of general PH curves within the framework of Clifford algebra. We define these curves with the PH representation map and verify the validity of the definition through the construction of already known PH curves. Special attention is given to the less frequently studied case of PH curves in the four-dimensional Minkowski space ${\mathbb R}^{3,1}$. The practical utility of the unified framework is demonstrated through numerical construction of various types of PH curves using Bézier curves. It is shown that differences in the construction are solely in the choice of the preimage of the PH representation map and in the degree of the Bézier curve.
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