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Space Square (Posted on 2009-11-06) Difficulty: 3 of 5
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?

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Some Thoughts a few thoughts | Comment 1 of 3

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
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