Designing Curves with Complex Numbers
by
Rida T. Farouki
University of California, Davis

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 special algebraic structures that allow the arc length to be computed exactly (ie, without numerical quadrature), a property that proves 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 greatly facilitates the construction and analysis of PH curves. For PH space curves, an analogous quaternion representation can be used. We survey the theory of PH curves, present algorithms for their construction and manipulation, and describe their use in real-time interpolators for CNC machines with variable feedrate capability.

Date received: February 7, 2002