Introduction to Projective Geometry

Juan Carlos Álvarez Paiva

Points and lines at infinity

The method of projection and section is peculiar in that there are points on one representation that seem to have no image in another. For example, in the following two-dimensional picture, the point B seems to have no image.

The classical way of dealing with this difficulty is to decree the image of B to the the point at infinity on the line m. This notion of ideal points can be made rigorous, but we will leave that for future lectures (see chapter two of the interactive course on projective geometry). The same phenomenom occurs in following figure, but there a whole line of points on the source plane have no image on the target plane. This image of this line is said to be the line at infinity on the target plane. The line together with its ideal point at infinity will be termed the projective line, and the plane with its ideal line at infinity will be termed the projective plane.


Whenever we have a class of transformations, in our case those obtained through projection and section, it is interesting to study what properties are or are not preserved by them. The properties that are preserved are called invariants and most of mathematics and physics consists in their study.

The problem posed by Alberti is to determine the invariants of perspective. It is clear that distances are not preserved, nor angles, nor areas. Parallel lines are not taken to parallel lines. This capacity for distorting images seems to make perspective hard to study. But not all is distorted: lines go to lines and if a point lies on a line, then in any other point of view the corresponding point will lie on the corresponding line. We say that incidence is preserved. This may not seem like much, but it is enough to develop a mathematical theory of great power and beauty.

The two classical result depending solely on the notion of incidence are the theorems of Pappus and Desargues. Before stating them, let us fix some notation. The line joining two points X and Y is denoted by XY while the point of intersection of the lines a and b is denoted by a.b.

Pappus' theorem. If A,B,C and A',B',C' are two triples of collinear points, then the points AB'.BA', AC'.CA', and BC'.CB' are collinear.

Click here for animation
Click here for applet.

Desargues theorem. If two triangles are in perspective, then their corresponding sides intersect in three collinear points.

Click here for applet.

[Previous page] [Next page] [References] [Projective geometry] [J.C. Alvarez's home page]