Search results
Results From The WOW.Com Content Network
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.
A star in the night sky is an intuitive example of a "point at infinity", in the sense that it defines some direction, but practically speaking it is impossible to reach. The milky way forms a hazy stripe of stars across the sky; it behaves, in some sense, like a "line at infinity". The sky itself is a "plane at infinity".
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 ...
More recently the mathematical structure of inversive geometry has been interpreted as an incidence structure where the generalized circles are called "blocks": In incidence geometry, any affine plane together with a single point at infinity forms a Möbius plane, also known as an inversive plane. The point at infinity is added to all the lines.
Parallel lines in the plane intersect at the vanishing point in the line at infinity.. The notion of a projective plane arises out of the idea of perspection in geometry and art: that it is sometimes useful to include in the Euclidean plane an additional "imaginary" line that represents the horizon that an artist, painting the plane, might see.
The complex plane extended by a point at infinity is called the Riemann sphere. If f is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros.
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 .