Wikipedia
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 affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a projective plane, in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over any field, and more generally over any division ring.
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, CP, also called the Riemann sphere (when complex numbers are mapped to each point).
In the case of a hyperbolic space, each line has two distinct ideal points. Here, the set of ideal points takes the form of a quadric.
Usage examples of "point at infinity".
In projective geometry, the `point at infinity' was where two parallel lines met: the trick was to draw them in perspective, like railway lines heading off towards the horizon, in which case they appear to meet on the horizon.
In order words, an unlimited straight line has only one point at infinity.
And just beyond that red-white point at infinity she made out the dull glow of the South Pole.
And just beyond that red-white point at infinity she made out the dull glow of the South Pole&hellip.
It seemed to go out forever, a vast brown wall that narrowed to a cylinder, to a dark line with a gentle westward curve to it, to a point at infinity-and the point was tipped with green.