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What is a point at infinity?

What is a point at infinity? Here are some definitions.

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. (geometry, Euclidean projective geometry) Any point added to a space to achieve projective completion.
  3. (geometry, hyperbolic geometry) An ideal point.
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