Mathematics Colloquia and Seminars

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It is impossible to parameterize any plane curve, other than a straight line, by rational functions of its arc length. Despite its apparently simple and fundamental nature, the proof of this fact is rather subtle, and involves ideas about Pythagorean triples of polynomials, the integration of rational functions, and the calculus of residues. As a corollary to this proof, one is naturally drawn to consider the family of special curves called Pythagorean-hodograph (PH) curves. These curves incorporate a special algebraic structure that allows the arc length to be computed exactly (i.e., without numerical quadrature), a property that is extremely useful in real-time control applications such as robotics and NC machining. The representation of plane curves as complex- valued functions of a real parameter can greatly facilitate the construction and analysis of PH curves. This approach also leads to an interest in the Minkowski geometric algebra of complex sets, which has unexpected connections with topics in numerical analysis, geometrical optics, and other disciplines.