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 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 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.
Projective geometry can be modeled by the affine plane (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary". [5] An algebraic model for doing projective geometry in the style of analytic geometry is given by homogeneous coordinates.
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 "ideal surface" projection below is a plane-at-infinity, in the Poincaré half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle. Poincaré disk model
Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. Given a point (,) on the Euclidean plane, for any non-zero real number , the triple (,,) is called a set of homogeneous coordinates for the point. By this definition, multiplying the three homogeneous coordinates by a ...
The points with coordinates [x : y : 1] are the usual real plane, called the finite part of the projective plane, and points with coordinates [x : y : 0], called points at infinity or ideal points, constitute a line called the line at infinity. (The homogeneous coordinates [0 : 0 : 0] do not represent any point.)