Geometry of root-related parameters of PH curves

1. I N T R O D U C T I O N Rational parametrization of curves and surfaces is a very interesting problem in mathematics. For curves, it is shown that a curve of genus zero is a rational by Noether. As for surfaces, there is Noether 's theorem that an algebraic surface S is rational if and only if it contains an irreducible rational curve C with dim ICI >1. Although there exist some techniques to find a rational parametrization, such as Abhyankar 's method for a quartic curve with three double points, moving line/surface method, and adjoint method, finding a rational parametrization in practice is not an easy task. The main difficulties in rational parametrization of geometrically driven objects which are widely used in CAGD such as offset, pipe/canal surface, and swept volume are caused by the irrationality of unit normal vector fields. Once we restrict ourselves to rational curves and surfaces whose unit normal vectors are rational, many troubles disappear. This is the motivation of Pythagorean hodograph curves and Pythagorean normal surfaces. By its definition, a polynomial/rat ional curve is PH if its speed function is a polynomial/rational. After Farouki introduced Pythagorean hodograph curves in [1], there has been vast research on this class of curves by him and others [2-7]. As Hilgarter et al. [8] pointed out, the offset with varying distance function r(t) given by the spine curve m(t) admits a rational parametrization over R in accordance with m(t) if and only 0893-9659/02/$ see front matter (~) 2002 Elsevier Science Ltd. All rights reserved. Typeset by .4.AdS-TEX PII: S0893-9659(02)00143-X