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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 graphical perspective where a parallel projection arises as a central projection where the center C is a point at infinity, or figurative point. [5]
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 ...
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 mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity.The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a projective plane meet in exactly one point (there is no "parallel" case).
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 pole of the x-axis is the point of infinity of the vertical lines and the pole of the y-axis is the point of infinity of the horizontal lines. The construction of a correlation based on inversion in a circle given above can be generalized by using inversion in a conic section (in the extended real plane).
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
Mathematically, points at infinity have the advantage of allowing one to not consider some special cases. For example, in a projective plane, two distinct lines intersect in exactly one point, whereas without points at infinity, there are no intersection points for parallel lines. So, parallel and non-parallel lines must be studied separately ...