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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.
More precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded. The projectively extended real line may be identified with a real projective line in which three points have been assigned the specific values 0 , 1 and ∞ .
In hyperbolic geometry, an ideal point, omega point [1] or point at infinity is a well-defined point outside the hyperbolic plane or space. Given a line l and a point P not on l, right- and left-limiting parallels to l through P converge to l at ideal points. Unlike the projective case, ideal points form a boundary, not a submanifold. So, these ...
The line at infinity is thus a line like any other in the theory: it is in no way special or distinguished. (In the later spirit of the Erlangen programme one could point to the way the group of transformations can move any line to the line at infinity). The parallel properties of elliptic, Euclidean and hyperbolic geometries contrast as follows:
The points at infinity are the "extra" points where parallel lines intersect in the construction of the extended real plane; the point (0, x 1, x 2) is where all lines of slope x 2 / x 1 intersect. Consider for example the two lines
This embedding allows us to identify the point [x: y] either with the real number x / y if y ≠ 0, or with ∞ in the other case. The same construction may be done with the other chart. In this case, the point at infinity is [0: 1]. This shows that the notion of point at infinity is not intrinsic to the real projective line, but is ...
The infinity symbol may also be used to represent a point at infinity, especially when there is only one such point under consideration. This usage includes, in particular, the infinite point of a projective line, [13] and the point added to a topological space to form its one-point compactification. [14]
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