Preserving straight lines
Steve Gray
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 ste...@adelphia.net .
Steve Gray
Ken Pledger
Steve Gray <ste...@adelphia.net> wrote:
> ....
> 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.) ....
Yes, in general n = 4.
In your plane P, choose any four points A, B, C, D, of which no
three are collinear. In Q, choose E, F, G, H similarly. Then there's a
unique projective mapping (preserving straightness of lines) which takes
A to E, B to F, C to G and D to H.
The easiest way to find that mapping uses coordinates in the two
planes, and a 4 x 4 matrix M representing the mapping:
(x y z 1) -> (x y z 1).M.
Feeding in the coordinates of the points A, E, etc. gives four equations
to determine M. (The appropriate linear independence is guaranteed by
the conditions that no three of A, B, C, D colline, and no three of E,
F, G, H colline.)
Ken Pledger.