1. Introduction
Curves and surfaces which possess rational offsets are important for many practical applications in robotics, CAD/CAM systems, animations, manufacturing, etc. In the curve case, rational offsets can be assigned to the so-called Pythagorean hodograph (PH) curves. These curves have first been introduced in Farouki and Sakkalis (1990) and have been widely examined since then (see Farouki, 2008 and the references therein). The condition that characterizes a PH curve is a (piecewise) polynomial norm of its hodograph. Although, this condition connects the coefficients of the polynomial curve in a nonlinear way, an elegant construction that uses univariate polynomials with quaternion (complex) coeffi- cients in a spatial (planar) case enables us to construct PH curves in a simple way. Moreover, interpolation schemes with these curves are easier to handle if the quaternion (complex) representation is used (see, e.g. Farouki, 1994; Farouki and Neff, 1995; Farouki et al., 2003, 2002; Pelosi et al., 2005; Kwon, 2010; Han, 2008; Choi et al., 2008; Bastl et al., 2014b, 2014a). Surfaces with rational offsets are much less investigated than their curve counterparts. A surface with a rational field of unit normal vectors is called a Pythagorean normal vector (PN) surface, and such a surface clearly has rational offsets. Based on a dual approach PN surfaces were derived in Pottmann (1995) as the envelope of a two-parametric family of tangent planes with unit rational normals. Unfortunately, dual construction leads in general to rational surfaces and no algebraic criteria for a reduction of rational PN surfaces to polynomial ones is known yet. Also, to design a curve from its dual representation is not very intuitive and it is hard to avoid singularities and points at infinity.
