point at infinity

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point at infinity

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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.
   • 2005, John Stillwell, The Four Pillars of Geometry, page 92:

     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.

2. (geometry, Euclidean projective geometry) Any point added to a space to achieve projective completion.
   • 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.

   • 1990, Shreeram Shankar Abhyankar, Algebraic Geometry for Scientists and Engineers, page 10:

     As the degree form Y² or X² of a parabola has only a single root, it has only one point at infinity.

   • 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.

   • 2003, Ron Goldman, Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling, page 18:

     Projective space is, by definition, the collection of the points in affine space together with the points at infinity.

3. (geometry, hyperbolic geometry) An ideal point.
   • 1996, John Stillwell, Sources of Hyperbolic Geometry, page 82:

     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.

   • 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, .

   • 2012, Norbert A'Campo, Athanase Papadopoulos, “Notes on non-Euclidean Geometry”, in Athanase Papadopoulos, editor, Strasbourg Master Class on Geometry, page 126:

     To describe a parabolic transformation f of the hyperbolic plane, we consider two parallel lines l and l’ and we call ω their common point at infinity.

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 line, plane or hyperplane at infinity.

In hyperbolic geometry, points at infinity (more commonly called ideal points) 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 quadric.

• (hyperbolic geometry): ideal point, omega point

• line at infinity
• plane at infinity
• hyperplane at infinity

• vanishing point