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Ruled surface generated by two Bézier curves as directrices (red, green) A surface in 3-dimensional Euclidean space is called a ruled surface if it is the union of a differentiable one-parameter family of lines. Formally, a ruled surface is a surface in is described by a parametric representation of the form
Castelnuovo's theorem implies that to construct a minimal model for a smooth surface, we simply contract all the −1-curves on the surface, and the resulting variety Y is either a (unique) minimal model with K nef, or a ruled surface (which is the same as a 2-dimensional Fano fiber space, and is either a projective plane or a ruled surface ...
They are divided into minimal surfaces, ruled surfaces, non-orientable surfaces, quadrics, pseudospherical surfaces, algebraic surfaces, and other types of surfaces. Minimal surfaces [ edit ]
The surface of a polyhedron is a topological surface, which is neither a differentiable surface nor an algebraic surface. A hyperbolic paraboloid (the graph of the function z = xy) is a differentiable surface and an algebraic surface. It is also a ruled surface, and, for this reason, is often used in architecture.
A ruled surface is one which can be generated by the motion of a straight line in E 3. [46] Choosing a directrix on the surface, i.e. a smooth unit speed curve c(t) orthogonal to the straight lines, and then choosing u(t) to be unit vectors along the curve in the direction of the lines, the velocity vector v = c t and u satisfy
Helicoid as translation surface with identical generatrices , Helicoid as translation surface: any parametric curve is a shifted copy of the purple helix. A helicoid is a special case of a generalized helicoid and a ruled surface. It is an example of a minimal surface and can be represented as a translation surface.
Chern numbers of minimal complex surfaces. The Enriques–Kodaira classification of compact complex surfaces states that every nonsingular minimal compact complex surface is of exactly one of the 10 types listed on this page; in other words, it is one of the rational, ruled (genus > 0), type VII, K3, Enriques, Kodaira, toric, hyperelliptic, properly quasi-elliptic, or general type surfaces.
Labs surface, a certain septic with 99 nodes; Endrass surface, a certain surface of degree 8 with 168 nodes; Sarti surface, a certain surface of degree 12 with 600 nodes; Quotient surfaces, surfaces that are constructed as the orbit space of some other surface by the action of a finite group; examples include Kummer, Godeaux, Hopf, and Inoue ...