The last category is the surfaces of general type, and most surfaces are in this category. Some techniques similar to those used in the study of ruled surfaces are possible, and since genus one curves are very well understood, again a rather detailed description of these surfaces, called elliptic surfaces, is available. The third category consists of surfaces with a family of genus one curves on them. These are the so-called abelian surfaces, K 3 surfaces, Enriques surfaces, and hyperelliptic surfaces. The second category consists of surfaces with a nowhere-vanishing regular 2-form, or finite quotients of such surfaces. The prototype is a product surface X × ℙ 1 for a curve X. Since genus zero curves are all isomorphic to lines, such surfaces are known as ruled surfaces, and a detailed understanding of them is possible. The first category consists of those surfaces with a family of genus zero curves on them. The Enriques classification of smooth surfaces essentially breaks up all surfaces into four categories. Even surfaces, for which a fairly satisfactory classification exists due to Enriques, Kodaira, and others, presents many open problems. The classification of higher-dimensional varieties is not anywhere near as complete. The construction and understanding of the moduli spaces ℳ g for smooth curves is tantamount to the successful classification of curves and their properties. Rick Miranda, in Encyclopedia of Physical Science and Technology (Third Edition), 2003 III.H Surfaces and Higher Dimensions , where line geometric counterparts to subdivision algorithms for curves and surfaces, like de Casteljau's algorithm, are developed. Line geometry applied to CAD has also been considered by Ravani et al. have formulated algorithms for the approximation of ruled surfaces by low degree algebraic ruled surfaces (ruled quadrics, cubic and quintic ruled surfaces) and have presented a G 1 Hermite interpolation scheme resulting in piecewise quadratic ruled surfaces. Approximation and Hermite interpolation algorithms for ruled surfaces amount to corresponding algorithms for curves on the quadric M 2 4 (see chapter 31 on quadrics, and ,).įor example, Peternell et al. The point model may be advantageous, because for some applications it is easier to deal with curves, even in projective 5-space, than working with ruled surfaces. In the Klein model of line space they appear as curves on the Klein quadric M 2 4. Ruled surfaces are generated by moving a straight line in 3-space. Helmut Pottmann, Stefan Leopoldseder, in Handbook of Computer Aided Geometric Design, 2002 3.3.3 Ruled surfaces
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