If you take a square and look at it from some point in space it looks like a quadrilateral. What are the possible shapes of this quadrilateral?

The simplified view would be what's approached as one gets further and further from the square: an orthographic view.

Then the square would in general "look like" a parallelogram. There'd be certain orientations such that this parallelogram was a rectangle, or even, if looked at straight on, a square. Of course, edge on it would look like a line. But of course infinitely far away, you'd need an infinitely powerful telescope to see it.

But if the square were closer than infinity, then you need a perspective view. But what in fact is a perspective view?

Even taking the simplest case, where the eye is exactly on a line perpendicular to the square intersecting the square at its center. The corners are viewed obliquely, so the angles appear obtuse. In fact the situation is that of a pyramid with the eye at the apex and the square being the base. The angle perceived at a corner is just the dihedral angle of two adjacent sides of the pyramid.

Suppose, however, the perpendicular that goes through the eye is one that is perpendicular at a corner of the square. That corner will appear as a right angle as that dihedral angle is 90°. But the opposite corner will be obtuse. In fact, that opposite corner will be the only one that's other than 90°.  This is hard to describe in plane geometry.

Perhaps what's sought is some sort of plane figure that would produce the same angles if viewed from a point on a line perpendicular to the "looked-like" shape at its own center. That would require working backwards from the obtained spherical angles on the "vision sphere". I think this can get quite messy.

Posted by Charlie on 2009-11-06 12:20:10