Tensorial Rational Surfaces with Base Points via Massic Vectors

Abstract

Let S be a tensorial rational surface defined by a rational function from $\left[ 0,1\right] ^{2}$ onto $\mathbb{R} ^{3}$ with a base point at $\left( u,v\right) =\left( 0,0\right) $. We demonstrate that the image of this base point is a set of rational curves; a base point is a parameter value for which the rational parametrization takes the value $\left( \frac{0}{0},\frac{0}{0},\frac{0}% {0}\right) $. This result was established by Clebsch [Ueber die abbildung algebraischer flächen, insbesondere der vierten und fünften ordnung, Math. Ann., 1 (1869), pp. 253--316]. Base points were first introduced in the context of geometric design by Chionh [Base Points, Resultants and the Implicit Representation of Rational Surfaces, Ph.D. thesis, Department of Computer Science, University of Waterloo, Waterloo, Ontario, 1990] and Manocha and Canny [Implicitizing Rational Parametric Surfaces, Tech. report 90/592, Computer Science Division, University of California, Berkeley, 1990] and were used by Warren [ACM Trans. Graphics, 11 (1992), pp. 127--139] to define multisided rational Bézier patches. We give here a constructive approach of this result to exploit it directlyin the industrial scope of computer aided design. We show that these rational curves are placed end to end by using the formalism of massic vectors introduced by Fiorot and Jeannin [Courbes et Surfaces Rationnelles. Applications à la CAO, R.M.A. 12, Masson, Paris, 1989]. Furthermore, we give the relations between the massic vectors which define these curves and the massic vectors which define S. Finally, we give an algorithm to draw on a computer a surface having base points.