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In mathematics, the cone condition is a property which may be satisfied by a subset of a Euclidean space. Informally, it requires that for each point in the subset a cone with vertex in that point must be contained in the subset itself, and so the subset is "non-flat".
Blunt cones can be excluded from the definition of convex cone by substituting "non-negative" for "positive" in the condition of α, β. A cone is called flat if it contains some nonzero vector x and its opposite −x, meaning C contains a linear subspace of dimension at least one, and salient otherwise.
The axis of a cone is the straight line passing through the apex about which the cone has a circular symmetry. In common usage in elementary geometry, cones are assumed to be right circular, i.e., with a circle base perpendicular to the axis. [1] If the cone is right circular the intersection of a plane with the lateral surface is a conic section.
It can be shown that any developable surface is a cone, a cylinder, or a surface formed by all tangents of a space curve. [5] Developable connection of two ellipses and its development. The determinant condition for developable surfaces is used to determine numerically developable connections between space curves (directrices).
In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X , the relative Spec C = Spec X R {\displaystyle C=\operatorname {Spec} _{X}R}
A cone C in a vector space X is said to be self-dual if X can be equipped with an inner product ⋅,⋅ such that the internal dual cone relative to this inner product is equal to C. [3] Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual.
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
The normal cone C X Y or / of an embedding i: X → Y, defined by some sheaf of ideals I is defined as the relative Spec (= / +).. When the embedding i is regular the normal cone is the normal bundle, the vector bundle on X corresponding to the dual of the sheaf I/I 2.