point at infinity

Noun

 * 1) An asymptotic point in 3-dimensional space, viewed from some point, at which parallel lines appear to meet and which in perspective drawing is represented as a vanishing point.
 * 2)  Any point added to a space to achieve projective completion.
 * 3) * 1960, Roger A. Johnson, Advanced Euclidean Geometry, Republished 2007, page 45,
 * In the geometry of inversion, therefore, it is usual to sacrifice the line of points at infinity which is so useful in other fields, and adopt the convention of a single point at infinity, the inverse of the center of the circle of inversion.
 * 1) * 1997, Susan Addington, Stuart Levy, Lost in the Fun House: An Application of Dynamic Projective Geometry, James King, Doris Schattschneider, Geometry Turned On: Dynamic Software in Learning, Teaching, and Research, page 159,
 * To unify the treatment of concurrent and parallel lines, define a point at infinity for each family of parallel lines, and declare each point at infinity to lie on each of the parallel lines that define it.Now parallel lines meet at infinity, and vanishing points are the images of points at infinity.
 * 1)  An ideal point.
 * 2) * 2007, Maurice Margenstern, Cellular Automata in Hyperbolic Spaces, Volume 1: Theory, page 66,
 * As we have seen, points at infinity behave as ordinary points: a single line passes through two distinct points at infinity or through a point in the plane and a point at infinity.
 * This representation of the points at infinity is a difference between the disc and the half-plane models of Poincaré,.
 * 1) * 2007, Maurice Margenstern, Cellular Automata in Hyperbolic Spaces, Volume 1: Theory, page 66,
 * As we have seen, points at infinity behave as ordinary points: a single line passes through two distinct points at infinity or through a point in the plane and a point at infinity.
 * This representation of the points at infinity is a difference between the disc and the half-plane models of Poincaré,.
 * This representation of the points at infinity is a difference between the disc and the half-plane models of Poincaré,.

Usage notes
In Euclidean or affine spaces, depending on the dimensionality and nature of the space, the projective completion may comprise a single point at infinity (such as in the cases of the real projective line and the Riemann sphere) or a set called the, or  at infinity.

In hyperbolic geometry, points at infinity (more commonly called ) are not regarded as belonging to the space, but are bounding points, each line in the space having two distinct such points. The set of ideal points forms a.