OT: fitting a plane model to 2d points 2003-12-05 - By Danny Kodicek
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-- -- Original Message -- --
From: "pat hammond" <ziloguser@(protected)>
To: <dir3d-l@(protected)>
Sent: Thursday, December 04, 2003 2:13 PM
Subject: RE: [Dir3d-l] OT: fitting a plane model to 2d points
> thanks :) good luck!
> it seemed straight-forward enough to me at first but after getting some
> nasty trig flashbacks I had to quit.
You're right that it's a lot harder than it sounds. I haven't fleshed out
the details but I did get it to the stage of a set of relatively simple
simultaneous equations...
You have points p_1, p_2, p_3, p_4 (numbered clockwise around the shape).
For each of these use spritespacetoworldspace to calculate four points in 3d
space q_1, q_2, q_3, q_4. Now you know that each of the actual points in
space is equal to c+k_i*q_i, for some set of coefficients k_i, and where c
is the location of the camera.
The vectors along the lines of the rectangle are therefore equal to
v_1=k_2q_2-k_1q_1, v_2=k_3q_3-k_2q_2, v_3=k_3q_3-k_4q_4 and
v_4=k_4q_4-k_1q_1.
Because it's a rectangle, we know that v_1=v_3 and v_2=v_4. We also know
that v_1.v_2=0 (and the same for the other vectors, but these are equivalent
because the sides are parallel: knowing a shape is a parallelogram, you only
need one right angle to know it's a rectangle). Finally, we know that
*either* v_1.v_1=a^2 and v_2.v_2=b^2 *or* v_1.v_1=b^2 and v_2.v_2=a^2, where
a and b are the sides of the rectangle.
Now, this is a wealth of information: far more than we should need to
determine the four unknowns k_i. That's good, because of course for an
arbitrary set p_i we can't find a solution. It's only because we know the
points were generated by the particular method that we know it's solveable
(and it's because it's so unlikely that a set of points should accidentally
be the same as those which were projected by a rectangle that optical
illusions and 3d displays work at all!)
Apologies for the scary-looking formulae. I wouldn't have used the
underscores if it wasn't for needing to write p_i rather than pi.
Danny
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