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Since the plane at infinity is a projective plane, it is homeomorphic to the surface of a "sphere modulo antipodes", i.e. a sphere in which antipodal points are equivalent: S 2 /{1,-1} where the quotient is understood as a quotient by a group action (see quotient space).
In an affine or Euclidean space of higher dimension, the points at infinity are the points which are added to the space to get the projective completion. [citation needed] The set of the points at infinity is called, depending on the dimension of the space, the line at infinity, the plane at infinity or the hyperplane at infinity, in all cases a projective space of one less dimension.
In the projective space, each projective subspace of dimension k intersects the ideal hyperplane in a projective subspace "at infinity" whose dimension is k − 1. A pair of non- parallel affine hyperplanes intersect at an affine subspace of dimension n − 2 , but a parallel pair of affine hyperplanes intersect at a projective subspace of the ...
In mathematics, the classical Möbius plane (named after August Ferdinand Möbius) is the Euclidean plane supplemented by a single point at infinity. It is also called the inversive plane because it is closed under inversion with respect to any generalized circle , and thus a natural setting for planar inversive geometry .
In mathematics, the Riemann sphere, named after Bernhard Riemann, [1] is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers , that is, the complex numbers plus a value ∞ {\displaystyle \infty } for infinity .
A projective plane is defined axiomatically as an incidence structure, in terms of a set P of points, a set L of lines, and an incidence relation I that determines which points lie on which lines. As P and L are only sets one can interchange their roles and define a plane dual structure. By interchanging the role of "points" and "lines" in C ...
The plane P 1 defines a projective line which is called the line at infinity of P 2. By identifying each point of P 2 with the corresponding projective point, one can thus say that the projective plane is the disjoint union of P 2 and the (projective) line at infinity.
The circular points at infinity are the points at infinity of the isotropic lines. [2] They are invariant under translations and rotations of the plane.. The concept of angle can be defined using the circular points, natural logarithm and cross-ratio: [3]