Pythagorean-Hodograph curves are a special class of parametrically given curves, whose key property is the exact calculation of the arc length of the curve, i.e. without using numerical tools. Parametrically given curves are represented in a B-spline basis, which allows efficient construction of Pythagorean-Hodograph curves and provides local shape control, continuity and numerical stability. In order to perform appropriate operations on curves, planar parametric curves are represented with complex valued functions, while spatial curves are represented using quaternions. Pythagorean-Hodograph curves are constructed from so-called preimage, arbitrary parametric curve, which is transformed into a hodograph of the Pythagorean-Hodograph curve using appropriate operation over complex or quaternion valued functions. The resulting hodograph is further integrated to obtain the final Pythagorean-Hodograph curve.
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