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Projective geometry.

The EYE sees all!

If we're to paint a building well or learn projective geometry well, we must be able to answer the question: What does the eye see when the eye looks at a building? One way to answer this is to open up the eyeball itself and study the retina to get a feeling for what is seen. This extreme measure is unnecessary, though. In order to understand what is seen, we must recognize that the world around us is just that--around us. There are all kinds of different directions in which we can look, and all of these directions equally surround us. The world as we see it is just a big spherical painted ball surrounding us. This suggests that to understand the geometry of the eye, we should study the geometry of the sphere.

"Hey! What's going on in there?"

We have seen before that the geometry of the eye is the geometry of a plane (with infinite points). Here, we see that it is the geometry of the sphere. How can these two different systems both describe the correct geometry of the eye? Which one is the right one?

Of course, both descriptions of the geometry of vision are correct. This leaves us with the duty of reconciling them, and figuring out how to understand each system in terms of the other. Essentially, we need to make a Sphere-to-Plane dictionary which will help us translate understanding about each system back and forth to the other.

An Alternate View

This is a schematic of you looking at a plane. You're in the center of the large sphere, of course. The sphere represents the world as seen by you. The plane represents the actual world (rather flat). Each point on the plane corresponds to a point on the sphere, via your line of sight. The "real object" forming the red circle produces an image on your sphere, as does the object drawn as a blue square. The straight lines on the plane (radiating out from the point where you're standing) produce straight lines (of a sort) on the sphere. Of course they're not really straight lines--they're great circles going all the way (or is it only half way?) around the circle. The circles on the plane (surrounding you) produce circles on the sphere. In other words, lines can "look like" great circles, while circles can "look like" circles.

Puzzles to try

1. Which points on the sphere correspond to infinite points on the plane? (of course the infinite points aren't drawn in the picture!) <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*">
2. If you look at some straight line in the plane (other than the ones you're standing on), what shape on the sphere will it make? Be precise! <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*">
3. Are there points on the sphere which don't correspond to any point on the plane? <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*">
4. We've seen circles (on the plane) which look like circles (on the sphere). Can anything else (on the plane) look like a circle? If so, what? <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*"> <img align="bottom" src="/mmss/courses/infinity/images/pencil.gif" height="21" width="20" alt="*">

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The page last modified Thursday, 13-May-2004 15:03:01 EDT