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Preserving straight lines
Posted:
Mar 17, 2006 12:05 PM
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Hello to all you sophisticated geometers: I need help with this question.
I have a plane P containing various points and straight lines. I have another plane R with
the same number of points and lines. I want to "distort" P to another plane Q, so that "n" of Q's
points coincide with "n" points on R, where n is as large as possible. (Either n=3 or 4, I suspect.)
For example by translating P to Q I can make n=1 of Q's points coincide with one point on R.
By translating, scaling, and rotating P to Q, I can make n=2 of Q's points coincide with two points
in R.
I need to know the most general transformation of P to Q, such that a straight line in P is
also straight in Q and any point on the left side of a directed line in P is on the left side in Q
(etc.). The goal is to make as many Q points as possible coincide with points in R.
I think it might be a central projection with a variable point of view and a variable image
plane, but I'm not sure. Does the central projection allow skewing of P by an arbitrary
amount? Does preservation of line straightness alone assure that points on one side of a directed
line in P stay on that side in Q? Does allowing the image plane to have a variable angle to the POV
line give any more generality?
How many degrees of freedom can I have? (That should determine n.)
Thank you for any information, leads, insights, etc. I'd appreciate any replies also emailed
to me at stevebg@adelphia.net .
Steve Gray
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