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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
This is a list of surfaces in mathematics. They are divided into minimal surfaces , ruled surfaces , non-orientable surfaces , quadrics , pseudospherical surfaces , algebraic surfaces , and other types of surfaces.
In geometry a conoid (from Greek κωνος 'cone' and -ειδης 'similar') is a ruled surface, whose rulings (lines) fulfill the additional conditions: (1) All rulings are parallel to a plane, the directrix plane. (2) All rulings intersect a fixed line, the axis. The conoid is a right conoid if its axis is perpendicular to its directrix ...
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.
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 ...
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 ...
In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). Conversely, it is a surface which can be made by transforming a plane (i.e. "folding
For example, the conic x 2 + y 2 + z 2 = 0 in P 2 over the real numbers R is uniruled but not ruled. (The associated curve over the complex numbers C is isomorphic to P 1 and hence is ruled.) In the positive direction, every uniruled variety of dimension at most 2 over an algebraically closed field of characteristic zero is ruled.
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