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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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[1] Apery, F., Models of the real projective plane: computer graphics of Steiner and boy surfaces, (1987), Vieweg Berlin · Zbl 0623.57001
[2] Bocher, M., Introduction to higher algebra, (1964), Dover Braunschweig · Zbl 0131.24804
[3] Boehm, W.; Prautsch, H., Geometric concepts for geometric design, (1994), AK Peters New York
[4] Borsuk, K.; Spalinska, H., Multidimensional analytic geometry, (1969), Polish Scientific Publishers Wellesley, MA
[5] Coolidge, J.L., A long way from euclid, (1963), T.Y. Crowell Warsaw
[6] Cox, D., Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra, (1992), Springer New York · Zbl 0756.13017
[7] Francis, G.K., A topological picturebook, (1987), Springer New York · Zbl 0612.57001
[8] Hilbert, D., Geometry and the imagination, (1990), Chelsea New York
[9] Kummer, M., Über die flächen vierten grades, auf welchen scharen von kegelschnitten liegen, J. reine angew. math., 64, 66-96, (1865)
[10] McLeod, R.J.Y., The Steiner surface revisted, (), 157-174, no. 1737 · Zbl 0425.65007
[11] Meyer, W.F.; Meyer, W.F., Spezielle algebraische flächen, (), 1647-1660, Teubner, Leipzig, III C 10 · JFM 57.0840.02
[12] Rath, W., Kinematische erzeugung der Römerflächen im flaggernraum, J. geometry, 36, 129-142, (1989)
[13] Reid, M., Undergraduate algebraic geometry, London math. soc., London, (1988) · Zbl 0701.14001
[14] Salmon, G., A treatise on the analytic geometry of three dimensions, (1965), Chelsea New York
[15] Schreier, O.; Sperner, E., Projective geometry of n dimensions, (1961), Chelsea New York
[16] Schwartz, A. and Stanton, C. (unpublished), Classification of Steiner surfaces.
[17] Sederberg, T.W.; Anderson, D.C., Steiner surface patches, IIEE comput. graph. appl., 5, 23-36, (1985)
[18] Semple, J.G.; Roth, L., Introduction to algebraic geometry, (1949), Clarendon Press New York · Zbl 0041.27903
[19] Sommerville, D., Analytical geometry of three dimensions, (1934), The University Press Oxford · JFM 60.1251.10
[20] Steiner, J.; Steiner, J., (), 741-742, Berlin
[21] Todd, J.A., Projective and analytical geometry, (1946), Pitman Publishing Cambridge · Zbl 0061.30608
[22] Wunderlich, W., Römerflächen mit ebenen fallinien, Ann. di. mat. pure ed appl., 57, 97-108, (1962) · Zbl 0107.15601
[23] Wunderlich, W., Durch schiebung erzeugbare Römerflächen, Sitzungsber. österr. wissensch., 176, 473-497, (1968) · Zbl 0174.24502
[24] Wunderlich, W., Kinematisch erzeugbar Römerflächen, J. reine angew. math., 236, 67-78, (1969) · Zbl 0174.53003
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