A surface in 3-space can be parameterized by two variables (or coordinates) and such that(1)(2)(3)If a surface is parameterized as above, then the tangentvectors(4)(5)are useful in computing the surface area and surfaceintegral.
A ruled surface is a normal developable of a curve if can be parameterized by , where is the normal vector (Gray 1993, pp. 352-354; first edition only).
There are nine possible types of isolated singularities on a cubic surface, eight of them rational double points. Each type of isolated singularity has an associated normal form and Coxeter-Dynkin diagram (, , , , , , , and ).The eight types of rational double points (the type being the one excluded) can occur in only 20 combinations on a cubic surface (of which Fischer 1986a gives 19): , , , , , , , , , , , , , (), , , , , and (Looijenga 1978, Bruce and Wall 1979, Fischer 1986a).In particular, on a cubic surface, precisely those configurations of rational double points occur for which the disjoint union of the Coxeter-Dynkin diagram is a subgraph of the Coxeter-Dynkin diagram . Also, a surface specializes to a more complicated one precisely when its graph is contained in the graph of the other one (Fischer 1986a)...
Let be a regular surface and a unit tangent vector to , and let be the plane determined by and the normal to the surface . Then the normal section of is defined as the intersection of and .
A ruled surface parameterization is called noncylindrical if is nowhere . A noncylindrical ruled surface always has a parameterization of the formwhere and , where is called the striction curve of and the director curve.
A Monge patch is a patch of the form(1)where is an open set in and is a differentiable function. The coefficients of the first fundamental form are given by(2)(3)(4)and the second fundamental form by(5)(6)(7)For a Monge patch, the Gaussian curvature andmean curvature are(8)(9)
The 27 real or imaginary lines which lie on the general cubic surface and the 45 triple tangent planes to the surface. All are related to the 28 bitangents of the general quartic curve.Schoute (1910) showed that the 27 lines can be put into a one-to-one correspondence with the vertices of a particular polytope in six-dimensional space in such a manner that all incidence relations between the lines are mirrored in the connectivity of the polytope and conversely (Du Val 1933). A similar correspondence can be made between the 28 bitangents and a seven-dimensional polytope (Coxeter 1928) and between the tritangent planes of the canonical curve of genus four and an eight-dimensional polytope (Du Val 1933).