The line at infinity

 

A point at infinity is the meeting point of parallel lines.

Consider all lines parallel to a given line. They all have the same direction. We often say that they all meet at a point at infinity. We often call each point at infinity a direction. The collection of all points at infinity is the line at infinity.

A point at infinity is often called an ideal point. It exists logically as the meeting point of a pair of parallel lines, or as a direction, but we do not need to assume that ideal points actually exist. Their logical existence is independent of their physical existence.

This page is to familiarize you with the line at infinity, and a strange one it is!

The circle double counts lines.

Consider all the lines through a point. There are 360° in a circle, but only half of them are needed to count all directions.Think of a circle surrounding a point. Each point on the circle is described by a unique angle. But the same line goes through two opposite points on the circle, so opposite angles (separated by 180°) represent the same line. Topologists would say that opposite points are identified, creating a strange, Möbius-like curve. The line at infinity, being the line where each direction represents a point, has no endpoint. As you increase the angle representing direction, the angle, and (twice as often) the direction of the line, repeats itself.

A common attempt to visualize the line at infinity is to think of an ever expanding circle. As the circle gets larger any section of it gets straighter. When the circle expands to infinite radius, it is in fact a straight line. This is all well and good, but, although locally correct, this is not quite the line at infinity, which is a projective line as described above, not really a circle.

Asymptotes

Now that we have these points at infinity, we notice that they have been staring us in the face all along. If we travel along a line, we eventually get to infinity, which is the same point at both ends of the line! Most asymptotes behave this way. They go out in one direction and come in on the other side (the same direction!). y = 1/x is a nice example. The asymptote exits in the first quadrant, goes through infinity, and comes back in in the 4th quadrant. In the projective plane y = 1/x is a coninuous curve.

The equation of the line at infinity

We are taught to think about lines using ideas like slope, x intercept, and y intercept. Slope is not meaningful for the line at infinity (it either has all slopes, or no undefined slope) but the x and y intercepts are useful. They are known and are both infinite. Using the intercept form of the straight line, we see that the equation of the line at infinity is 1 = 0.  (!)  Yes, it really is, and in this form becomes very useful in graphing equations (see here).

The line at infinity has no scale

Several properties of the infinite line can be seen from this form. 1 = 0, 2 = 0, 3 = 0, are all the same line; it is independent of scale and does not really exist in the metric Euclidean plane where distances are measurable.

All circles go through the same two points on the line at infinity

Consider the circle x 2 + y 2 = r 2 , which at infinity is the same as x 2 + y 2 = 0. This solves to x = ± iy, so that (1, i) and (1, -i) are the solutions. These two point are called the circular points at infinity. Note that (-1, i) and (-1, -i) also solve the equations, but being 180° opposite, they are the same points on the line at infinity as the other two.

All circles go through these two points.

The lines y = ix and y = -ix go through the circular points at infinity. They are each self perpendicular, which you can see by checking slopes.

The sections on building conics and cubics from lines have nice examples of using the line at infinity.

This web page was written by a professor from Ga. Tech and is a nice read.