Projective Geometry

Projective geometry began with Renaissance artists who wanted to portray a scene as someone actually on the scene might see it. A painting is a central projection of the points in the scene onto a canvas or wall, with the artist's eye as the center of the projection (the fact that the rays are converging on the artist's eye instead of emanating from it does not change the principles involved), but the scenes, usually Biblical, existed only in the artists' imagination. The artists needed some principles of perspective to help them make their projections of these imagined scenes look real.

Among those who sought such principles was Gerard Desargues (1593-1662). One of the many things he discovered was the remarkable theorem which now bears his name:

If two triangles ABC and A'B'C' are perspective from a point (i.e., if the lines drawn through the corresponding vertices are concurrent at a point P), then the extensions of their corresponding sides will intersect in collinear points X, Y, and Z.

The converse of this theorem is also true: If two triangles are drawn so that the extensions of their corresponding sides intersect in three collinear points, then the lines drawn through the corresponding vertices will be concurrent.

It is not obvious what this theorem has to do with perspective drawing or with projections. If the two triangles were in separate planes, however, (in which case the theorem is not only true, it is easier to prove) one of the triangles could be a triangle on the ground and the other its projection on the artist's canvas.

If, in Figure 1, BC and B'C' were parallel, they would not intersect. If one imagines a "point at infinity," however, they would intersect and the theorem would hold true. Kepler is credited with introducing such an idea, but Desargues is credited with being the first to use it systematically. One of the characteristics of projective geometry is that two coplanar lines always intersect, but possibly at infinity.

Another characteristic of projective geometry is the principle of duality. It is this principle that connects Desargues' theorem with its converse, although the connection is not obvious. It is more apparent in the three postulates which Eves gives for projective geometry:

I. There is one and only one line on every two distinct points, and there is one and only one point on every two distinct lines.

II. There exist two points and two lines such that each of the points is on just one of the lines and each of the lines is on just one of the points.

III. There exist two points and two lines, the points not on the lines, such that the point on the two lines is on the line on the two points.

These postulates are not easy to read, and to really understand what they say, one should make drawings to illustrate them. Even without drawings, one can note that writing "line" in place of "point" and vice versa results in a postulate that says just what it said before. This is the principle of duality. One can also note that postulate I guarantees that every two lines will intersect, even lines which in Euclidean geometry would be parallel.