Friday, June 3, 2011

Are We In Need of a Reality Check?

It may seem that we are expending an excessive amount of time on studying a piece of paper but what we are interested in is the coordinate transformation which takes us from the plane of the paper to the plane of the camera image. If one compares the lines in the image with a straight edge there is a noticable curvature. This seems to run counter to the rules used in perspective drawing. Parallel lines there remain straight and converge on vanishing points. It is likely that this is actually a lower order of approximation. One sometimes encounters a discrepancy between appearance and reality. Parallel lines do not actually converge in the real world since their separation remains constant. But if we consider two parallel lines one on either side of us they will appear to converge in both directions that they move away from us. If we turn 90° the two lines angle appears to be 90° to this direction. We find ourselves in a bit of a quandary and so we need to reexamine our initial assumptions.

This is a position that idealists often get themselves into. But their assumptions do not get past the naive impressionist. The real world is his source of knowledge. He takes that as it is. False assumptions can make the world appear surreal and perhaps we should take this as an indication that we should be a little more critical in our thinking.

Some related Wikipedia articles on what we have been doing:

Ruled surface


Geodesic curvature

Perspective (graphical)


Real Projective Plane

History of manifolds and varieties

Supplemental: There may be some trickery involved in perspective drawings but they are still an effective illusion.

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