Abstract
The main purpose of this paper is to study projective classes of bidegree (2,1) parametrizable surfaces in a real projective 3-space. It turns out that the implicit degree of such surfaces is two, three, or four, and singular curves have degree three. We describe all possibilities for singular curves and pinch points on such surfaces. The presentation has been made from the point of view of analytic geometry and does not presume a deep knowledge of algebraic geometry.
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References
A. Coffman, A. J. Schwartz, and C. Stanton, The algebra and geometry of Steiner and other quadratically parametrizable surfaces,Computer Aided Geometric Design,13, 257–286 (1996).
W. L. Edge,The Theory of Ruled Surfaces, Cambridge (1931).
W. Fulton,Intersection Theory, Springer, Berlin (1984).
P. Griffiths and J. Harris,Principles of Algebraic Geometry, Wiley, New York (1978).
M. Peternell,Rational parametrizations for envelopes of quadric families, Thesis, Vienna University of Technology, Vienna (1997).
G. Salmon,Analytische Geometrie des Raumes, 2 Teil, Leipzig (1880).
R. Sturm,Liniengeometrie in synthetischer Behandlung, 1 Teil, Leipzig (1892).
Additional information
Partially supported by the Lithuanian State Science and Studies Foundation.
Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 38, No. 3, pp. 379–402, July–September, 1998.
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Zubė, S. Bidegree (2,1) parametrizable surfaces in projective 3-space. Lith Math J 38, 291–308 (1998). https://doi.org/10.1007/BF02465903
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DOI: https://doi.org/10.1007/BF02465903