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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 conical hull of a finite or infinite set of vectors in is a convex cone. The tangent cones of a convex set are convex cones. The set is a cone but not a convex cone. The norm cone is a convex cone. The intersection of two convex cones in the same vector space is again a convex cone, but their union may fail to be one.
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though it was sometimes called as a fourth type.
In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities. Definitions in nonlinear analysis [ edit ] In nonlinear analysis, there are many definitions for a tangent cone, including the adjacent cone , Bouligand 's contingent cone , and the Clarke tangent ...
The cone over a closed interval I of the real line is a filled-in triangle (with one of the edges being I), otherwise known as a 2-simplex (see the final example). The cone over a polygon P is a pyramid with base P. The cone over a disk is the solid cone of classical geometry (hence the concept's name). The cone over a circle given by
Frustum. In geometry, a frustum ( Latin for 'morsel'); [a] ( pl.: frusta or frustums) is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting the solid. In the case of a pyramid, the base faces are polygonal and the side faces are trapezoidal. A right frustum is a right pyramid or a right cone ...
In geometry, a conical surface is a three-dimensional surface formed from the union of lines that pass through a fixed point and a space curve. Definitions [ edit ] A ( general ) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some ...
The polar of the closed convex cone C is the closed convex cone Co, and vice versa. For a set C in X, the polar cone of C is the set [4] It can be seen that the polar cone is equal to the negative of the dual cone, i.e. Co = − C* . For a closed convex cone C in X, the polar cone is equivalent to the polar set for C. [5]
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