Where parallels meet ...

In a previous post, I introduced projective geometry, directly,  by means of two interactive applications demonstrating the quality of projective phenomena.  In this post I want to introduce the basic concepts of projective geometry.  The story begins in renaissance Italy.

 In 1420 Massacio painted a fresco in a chapel in Florence that is considered the first example of perspective painting.   When one attempts to define what "perspective painting" means one is led naturally to the thought constructs that lie at the basis of projective geometry.  The French architect and mathematician  Rene Desargues (1591-1661) is considered the father of projective geometry based on his book Brouillon Projet, which represents the culmination of a gradual coming-into-consciousness of the new thought forms revealed by perspective painting.  This is a fascinating story which lies outside the scope of this post, which restricts itself to  the fundamental connection between the art of perspective painting and the science of projective geometry.

A perspective image of a scene is defined by a process, called central projection, involving the following ingredients:

• the lines passing through the eye of the painter (called the center of the projection), 
• the points at which those lines first intersect an object in the world, and finally,
• a flat screen that is inserted between the eye and the objects of the world.  

The perspective image is created by transferring, for each line through the center, the color of the intersection point with the world to the intersection point with the screen. In this way a colored image is created that reproduces the visual impression of an observer whose eye is positioned at the center.

This interactive application illustrates central projection with some simple scenes.  The center of the projection (the "eye") is on the left, the lines ray out to the checkerboard (the "world") and the image is created on the vertical screen in the middle by transferring the colors from the checkerboard to the screen.

Perspective images contain interesting features.  For example, consider a scene consisting of a strip of constant width leading away from the screen, like parallel train tracks disappearing in the distance.  (Use the '4' key in the application to obtain such a scene.) What is the perspective image of such a scene?  It's easy to see that the image of a line is again a line.  The parallel lines will be mapped to lines which are not parallel.   Indeed, consider the line through the eye which is parallel to the train tracks.  It's not hard to see that where this line intersects the screen will be where the train tracks appear to meet. It's called a vanishing point.   There is one vanishing point for every set of parallel lines.

The step from central projection to projective geometry is a small but significant one.  It occurs when one assumes that the train tracks themselves -- and not just their perspective images -- have a point in common. After all, I do see such a point -- the vanishing point!  Such a point is called an ideal point and they form the foundation of projective geometry.  One could characterize an ideal point as a point which one sees, but which one can never reach.  Here one sees how the tension between sight and touch -- mentioned at the end of the previous post --  is built into the foundation of projective geometry.


The set of ideal points is organized in a nice way.  In every plane, in every direction, there is an ideal point, where all lines having this direction meet.  Taken together, all these ideal points behave just like a line -- it's called the ideal line of the plane.  The horizon line is an example of such an ideal line.  This interactive application allows you to experiment with this concept.  It shows 4 sets of parallel lines in a plane; seen from above (the left image) one experiences the euclidean parallelism; when one rotates the scene away from the viewer (right image) one sees the four ideal points on the horizon line where the sets of parallel lines meet and experiences the horizon line as a real entity.


Finally, all the ideal points of all the planes in space form a plane, the ideal plane of space.  Projective geometry arises when one takes all the ordinary points of space and appends this ideal plane, with all its ideal lines and ideal points.  Further posts on this blog will explore the consequences of this inconspicious extension.