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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 polar cone of a convex cone is the set := { : , } This definition gives a duality on points and hyperplanes, writing the latter as the intersection of two oppositely-oriented half-spaces. The polar hyperplane of a point x ∈ X {\displaystyle x\in X} is the locus { y : y , x = 0 } {\displaystyle \{y~:~\langle y,x\rangle =0\}} ; the dual ...
Stereographic projection of a spherical cone's generating lines (red), parallels (green) and hypermeridians (blue). Due to conformal property of Stereographic Projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles or straight lines. The generatrices and parallels generates a 3D dual cone.
In mathematics, the bipolar theorem is a theorem in functional analysis that characterizes the bipolar (that is, the polar of the polar) of a set. In convex analysis, the bipolar theorem refers to a necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel ...
These two lines (sometimes called the null cone) are perpendicular in and have slopes ±1. Split-complex numbers z and w are said to be hyperbolic-orthogonal if z, w = 0. While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle.
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the reference plane (sometimes fundamental plane).
In Euclidean space, the dual of a polyhedron is often defined in terms of polar reciprocation about a sphere. Here, each vertex (pole) is associated with a face plane (polar plane or just polar) so that the ray from the center to the vertex is perpendicular to the plane, and the product of the distances from the center to each is equal to the square of the radius.
When using polar coordinates, the eccentricity of the hyperbola can be expressed as where is the limit of the angular coordinate. As φ {\displaystyle \varphi } approaches this limit, r approaches infinity and the denominator in either of the equations noted above approaches zero, hence: [ 19 ] : 219