The line at infinity

 

Two lines in the Euclidean plane are either parallel or intersect. Often we remove the “or” part of the statement by introducing ideal points known as points at infinity, so that two parallel lines meet at a single point which is infinitely far away. The collection of these infinite points is the line at infinity. This page is to familiarize you with this line, and a strange one it is!

A quick consequence of this: all lines parallel to a given line go through the same point at infinity. We often call each point at infinity a direction.

A common, but not entirely correct, way 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 this is not quite the line at infinity, which is a projective line, not really a circle.

To see this, we are mapping the space of lines onto the space of directions, each direction being a point at infinity. But this is a two to one mapping. Consider all the lines through a point. Each line represents a different direction and all directions are present. But if you rotate a line by 180°, it is the same line. There are 360° in a circle, but only 180° is needed for all directions. If we think of a circle surrounding this point, each point on the circle has an angle but the same line goes through two opposite points on the circle, so these two points represent different angles but the same direction. This line at infinity has no endpoint, as you continue the directions, it wraps around on itself. Topologists would say that these opposite points are identified, creating a strange, Möbius-like shape. This strange shape is the shape of the line at infinity.

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.

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, no undefined slope) but the x and y intercepts 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). Several properties of the infinite line can be seen from this form: it is homogeneous, meaning that 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 everything has a measurable distance.

 

A nice link to this topic is here from a Ga Tech professor.