دانلود رایگان مقاله رویکرد چهارگانه برای سطوح PN چند جمله ای

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.

8. Conclusion

Although polynomial surfaces with rational field of unit normal vectors are important in practical applications, not much results about these surfaces is known. The present paper introduces a new approach for a construction based on bivariate quaternion polynomials. Particular relations between the quaternion coefficients are derived that allow us to construct polynomial PN surfaces of degrees 2n + 1 and 2n + 2 from degree n quaternion polynomial. As in the curve case such a representation could be particularly useful if the interpolation with PN surfaces is considered. The analysis of the interpolation scheme with quartic and quintic PN surfaces, proposed in Section 7, is left for a future research. Also the extension to interpolating PN G1 splines over triangulations might be a useful topic to study. It would also be interesting for the future work to examine whether polynomials g1 and g2, defined in (3), that satisfy (4), could be constructed using ϕi being rational functions or polynomials of higher degrees. This may lead to find a full description of all polynomial PN parameterizations at least for some low degrees, which is still an open problem.