Point at infinity
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In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an
In the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the complex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, CP1, also called the Riemann sphere (when complex numbers are mapped to each point).
In the case of a
Affine geometry
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.[2]
As a projective space over a field is a
Perspective
In artistic drawing and technical perspective, the projection on the picture plane of the point at infinity of a class of parallel lines is called their vanishing point.[3]
Hyperbolic geometry
In
All points at infinity together form the
Projective geometry
A symmetry of points and lines arises in a projective plane: just as a pair of points determine a line, so a pair of lines determine a point. The existence of parallel lines leads to establishing a point at infinity which represents the intersection of these parallels. This axiomatic symmetry grew out of a study of
Though a point at infinity is considered on a par with any other point of a
Other generalisations
This construction can be generalized to
See also
- Division by zero
- Sphere at infinity
- Midpoint § Generalizations
- Asymptote § Algebraic curves
References
- ^ Weisstein, Eric W. "Point at Infinity". mathworld.wolfram.com. Wolfram Research. Retrieved 28 December 2016.
- Coxeter, H. S. M. (1987). Projective Geometry(2nd ed.). Springer-Verlag. p. 109.
- ISBN 978-0262062206.
- ^ Kay, David C. (2011). College Geometry: A Unified Development. CRC Press. p. 548.
- ^ Halsted, G. B. (1906). Synthetic Projective Geometry. p. 7.