Projective Infinity

According to [H. Eves, p. 59], infinity has been introduced into geometry by Johann Kepler (1571-1630), but it was Gérard Desargues (1593-1662) who began using the idea systematically. Addition of the points and the line at infinity metamorphoses the Euclidean plane into the projective plane and Desargues was one of the founders of projective geometry.

In projective geometry, no two lines are parallel, but the essential features of points and lines are inherited from Euclidean geometry, to boot,

  1. Any two distinct points determine a unique line,
  2. Any two distinct (and, in Euclidean geometry, non-parallel) lines determine a unique point.

So how do we do that, i.e., how do we metamorphose the common Euclidean plane into a projective one? We start with introducing new elements known as ideal - points and lines - so as to make the elements of the augmented plane satisfy the above two conditions. First, assign a point at infinity to every line in the plane. Let's experiment (that is, let's imagine an experiment). Given a line l, a point M not on l and a line m through M crossing l at P. Permit P move in any direction. The farther P moves from its initial position, the smaller becomes the angle between the lines l and m. In the limit, the two become parallel. The purpose of having those ideal points at infinity is to insure that even two parallel lines cross, even if that happens outside the "observable" region of the plane. But, if l and m intersect, they share the point of intersection, so that, when parallel, they intersect at the same point at infinity. It follows that, to satisfy the two forgoing requirements, our assignment of the ideal elements can't be arbitrary. Any two parallel lines must be assigned the same ideal point meaning that a point at infinity is associated with each bunch of parallel lines rather than with individual lines.

Now collect all the points of infinity into a single (ideal) element, a line at infinity, and see that the two conditions above hold for the augmented set of points and lines.

Any two distinct points determine a unique line. We should consider three cases:

  1. The two points are both finite.
  2. One of the points is finite, the other is ideal.
  3. Both points are ideal.

In the first case there, indeed, is a unique line that passes through the two given points. No ideal elements come into a play.

In the second case, the ideal point is associated with a bunch of parallel lines of which only one passes through the given finite point.

In the third case, only the line at infinity may pass through two points at infinity since every regular line passes a single such point.

Any two distinct lines determine a unique point. The verification is similar. Any two regular non-parallel lines meet at a unique finite point and any two parallel lines meet at a unique point at infinity. In addition, any regular line meets the line at infinity at a single point at infinity defined by the bunch of parallel lines to which the given one belongs.

The addition of the ideal elements allows us to embed the Euclidean plane into the projective plane and shows the relation between the two. However, in projective geometry per se, all points are the same as are all the lines. In projective geometry, there are no ideal elements. The augmented Euclidean plane, in fact, just serves one possible model of projective geometry. Of course, there are others. For example, in analytic geometry, a point in the projective plane is identified with a triple of homogeneous coordinates (x, y, z) which, to distinguish them from the Cartesian coordinates, are often written as x : y : z. But then any such triple can be associated with a straight line passing through the origin in the 3D space. In this model, there is no natural way to designate some elements regular and some ideal.

References

  1. H. Eves, A Survey of Geometry, Allyn and Bacon, Inc.1972

|Contact| |Front page| |Contents| |Algebra| |Up|

Copyright © 1996-2018 Alexander Bogomolny

 63714227

Search by google: