Line at infinity Totally Explained

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:"Ideal line" redirects here. For the ideal line in racing, see Line (racing). In geometry and topology, the line at infinity is a line which is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line.

Geometric formulation

In projective geometry, any pair of lines always intersect at some point. But parallel lines don't intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on the line at infinity. The point at which the parallel lines intersect depends only on the slope of the lines, not at all on their y-intercept. Also, if any pair of lines intersect at a point on the line at infinity, then the pair of lines is parallel.
Every line intersects the line at infinity at some point. The point at which a line intersects the line at infinity determines the slope of the line, but not at all its y-intercept.
In the affine plane, a line runs off in two opposite directions. In the projective plane, the two opposite directions of a line meet each other at a point on the line at infinity. Therefore lines in the projective plane are closed curves: they're cyclical rather than linear. This is true of the line at infinity itself: it meets itself at its two endpoints (which are therefore not endpoints at all) and so it's actually cyclical.

Topological perspective

The line at infinity can be visualized as a circle which surrounds the affine plane. However, this circle is actually like cross-cap, which is homeomorphic to a Möbius strip: diametrically opposite points of the circle are equivalent -- they're the same point. The combination of affine plane and line at infinity makes the real projective plane,

mathbbP^2

.
A hyperbola can be seen as a closed curve which intersects the line at infinity in two different points. These two points are specified by the slopes of the two asymptotes of the hyperbola. Likewise, a parabola can be seen as a closed curve which intersects the line at infinity in a single point. This point is specified by the slope of the axis of the parabola. If the parabola is cut by its vertex into a symmetrical pair of "horns", then these two horns become more parallel to each other further away from the vertex, and are actually parallel to the axis and to each other at infinity, so that they intersect at the line at infinity.
The analogue for the complex projective plane is a 'line' at infinity that's (naturally) a complex projective line. Topologically this is quite different, in that it's a Riemann sphere, which is therefore a 2-sphere, being added to a complex affine space of two dimensions over C (so four real dimensions), resulting in a four-dimensional compact manifold. The result is orientable, while the real projective plane is not.

History

The complex line at infinity was much used in nineteenth century geometry. In fact one of the most applied tricks was to regard a circle as a conic constrained to pass through two points at infinity, the solutions of ť X² + Y² = 0.

This equation is the form taken by that of any circle when we drop terms of lower order in X and Y. More formally, we should use homogeneous coordinates ť [X:Y:Z]

and note that the line at infinity is specified by setting ť Z = 0.

Making equations homogeneous by introducing powers of Z, and then setting Z = 0, does precisely kill off terms of lower order.
Solving the equation, therefore, we find that all circles 'pass through' the circular points at infinity ť I = [1:i:0] and J = [1:−i:0].

These of course are complex points, for any representing set of homogeneous coordinates. Since the projective plane has a large enough symmetry group, they're in no way special, though. The conclusion is that the three-parameter family of circles can be treated as a special case of the linear system of conics passing through two given distinct points P and Q. This idea was used so often that a schoolmasterly joke arose, naming the circular points at infinity Isaac and Jacob, respectively.

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