point at infinity

Noun

point at infinity (plural points at infinity)

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.
   • The point where a family of parallels appear to meet is called their “vanishing point” by artists, and their point at infinity by mathematicians. The horizon itself, which consists of all the points at infinity, is called the line at infinity.2005, John Stillwell, The Four Pillars of Geometry, page 92
2. (Euclidean projective geometry) Any point added to a space to achieve projective completion.
   • 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.1960, Roger A. Johnson, Advanced Euclidean Geometry, Republished 2007, page 45
   • As the degree form Y² or X² of a parabola has only a single root, it has only one point at infinity.1990, Shreeram Shankar Abhyankar, Algebraic Geometry for Scientists and Engineers, page 10
3. (excessive) An ideal point.
   • Hyperbolic geometry provides each line with two points at infinity. Whether there is a piece of the line beyond the points at infinity, completing the piece lying in the finite to a closed curve, we cannot say, since we cannot move as far as the points at infinity, let alone beyond them.1996, John Stillwell, Sources of Hyperbolic Geometry, page 82
   • 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...2007, Maurice Margenstern, Cellular Automata in Hyperbolic Spaces, Volume 1: Theory, page 66

   Synonyms:

   ideal point, omega point