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The algebra and geometry of Steiner and other quadratically parametrizable surfaces. (English) Zbl 0875.68860

Summary: Quadratically parametrizable surfaces \((x_1,x_2,x_3,x_4)=(\varphi_1({\mathbf u}), \varphi_{2}({\mathbf u}), \varphi_{3}({\mathbf u}), \varphi_{4}({\mathbf u}))\) where \(\varphi_{k}\) are homogeneous functions are studied in \({\mathbb{P}}^{3}({\mathbb{R}})\). These correspond to rationally parametrizable surfaces in \({\mathbb{R}}^{3}\). All such surfaces of order greater than two are completely catalogued and described. The geometry of the parametrizations as well as the geometry of the surfaces are revealed by the use of basic matrix algebra. The relationship of these two geometries is briefly discussed. The presentation is intended to be accessible to applied mathematicians and does not presume a knowledge of algebraic geometry.

MSC:

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
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