Concept
Ruled surface
Related concepts (17)
Surface (mathematics)
In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line. There are several more precise definitions, depending on the context and the mathematical tools that are used for the study. The simplest mathematical surfaces are planes and spheres in the Euclidean 3-space. The exact definition of a surface may depend on the context.
Three-dimensional space
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, the Euclidean n-space of dimension n=3 that models physical space. More general three-dimensional spaces are called 3-manifolds. Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n-dimensional Euclidean space.
Helicoid
The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid, there is a helix contained in the helicoid which passes through that point.
Cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines connecting a common point, the apex, to all of the points on a base that is in a plane that does not contain the apex. Depending on the author, the base may be restricted to be a circle, any one-dimensional quadratic form in the plane, any closed one-dimensional figure, or any of the above plus all the enclosed points.
Cylinder
A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology. The shift in the basic meaning—solid versus surface (as in ball and sphere)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces.
Paraboloid
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plane section of a paraboloid by a plane parallel to the axis of symmetry is a parabola. The paraboloid is hyperbolic if every other plane section is either a hyperbola, or two crossing lines (in the case of a section by a tangent plane).
Hyperboloid
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation. A hyperboloid is a quadric surface, that is, a surface defined as the zero set of a polynomial of degree two in three variables.
Developable 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", "bending", "rolling", "cutting" and/or "gluing"). In three dimensions all developable surfaces are ruled surfaces (but not vice versa). There are developable surfaces in four-dimensional space \mathbb{R}^4 which are not ruled.
Intersection theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem on curves and elimination theory. On the other hand, the topological theory more quickly reached a definitive form. There is yet an ongoing development of intersection theory. Currently the main focus is on: virtual fundamental cycles, quantum intersection rings, Gromov-Witten theory and the extension of intersection theory from schemes to stacks.
Hyperboloid structure
Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet. Often these are tall structures, such as towers, where the hyperboloid geometry's structural strength is used to support an object high above the ground. Hyperboloid geometry is often used for decorative effect as well as structural economy. The first hyperboloid structures were built by Russian engineer Vladimir Shukhov (1853–1939), including the Shukhov Tower in Polibino, Dankovsky District, Lipetsk Oblast, Russia.