tailieunhanh - Real parametric surfaces of bidegree (1, 2) and inverse problem of singularity

In this article we present a complete classification for the parametric surfaces of bidegree (1, 2) over the real field. We also provide some results for the inverse problem: given a segment of a line or twisted cubic curve, we look for a patch (1, 2) which includes this segment as a subset of its singular locus. | JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci. 2012 Vol. 57 No. 7 pp. 20-34 This paper is available online at http REAL PARAMETRIC SURFACES OF BIDEGREE 1 2 AND INVERSE PROBLEMS OF SINGULARITY Le Thi Ha Faculty of Mathematics Hanoi National University of Education Abstract. In this article we present a complete classification for the parametric surfaces of bidegree 1 2 over the real field. We also provide some results for the inverse problem given a segment of a line or twisted cubic curve we look for a patch 1 2 which includes this segment as a subset of its singular locus. For instance we characterize the ruled surfaces containing a twisted cubic curve such that all generating lines cut twice the cubic curve which are indeed parametric surfaces of bidegree 1 2 . Keywords Parametric surface 1 2 normal form implicit equation singular locus. 1. Introduction In Computer Aided Geometric Design and Geometric Modeling patches of parametric real surfaces of low degrees are commonly used. The common representation of surfaces is via parametrized patches . images of maps Φ 0 1 0 1 R3 Φ1 t u Φ2 t u Φ3 t u t u 7 Φ t u Φ0 t u Φ0 t u Φ0 t u where Φ0 Φ1 Φ2 Φ3 are polynomials in the two variables t and u with real coefficients. Surface patches are encountered in many applications. However a precise description of the geometry of the whole real surface is generally difficult to master. Therefore it is worthwhile to study systematically parametrized surfaces of low degree in order to have Received September 8 2012. Accepted October 5 2012. Mathematics Subject Classification 14J10 14J17 14J26 14Q10. Contact Le Thi Ha e-mail address lethiha@ 20 Real parametric surfaces of bidegree 1 2 and inverse problems of singularity at our disposal mastered geometric models together with their singular loci. Surfaces of total degree 1 . max0 i 3 deg Φi 1 are planes while surfaces with parametrization of bidegree 1 1 are planes or quadrics. The surfaces