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Converting u,v to .MAP texture axes

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Hi folks, For our generic modeling package, I am trying to write a decent .MAP exporter (the 220 format, targeting the Torque map2dif tool to be precise). What I have is a polygon representation, including u, v coordinates, of a convex solid. What I need to output for each plane (polygon) is
( x1 y1 z1 ) ( x2 y2 z2 ) ( x3 y3 z3 ) TEXTURE_NAME [ tx1 ty1 tz1 toffs1 ] [ tx2 ty2 tz2 toffs2 ] rotation scaleX scaleY
I tackle the conversion from polygons to plane by simple writing out 3 vertex coordinates. However, I simply don't "grok" the calculation of texture axes. So far I have been postponing this by generating a simple planar mapping, but now is the time to face my lack of math skills and get this solved. :D The conversion from axes to coordinates is straightforward (as found in a different post on this forum):
// assuming rot_angle = 0, and scale is 1
u = (DotProduct(A, U_Axis) + U_shift) / texture_width;
v = (DotProduct(A, V_Axis) + V_shift) / texture_height;
Now how do I reverse this to solve for the U and V axes? Writing this out yields 2 sets of 3 equation with 4 unknowns, which implies an infinite set of solutions, but I am sure I am missing the point here. Should I assume U_Axis and V_Axis to be orthogonal (which makes sense for any texture mapping our tool can generate)? I don't really need the rot_angle, now do I? So, to close off, a small recap: How do I calculate the texture axes U and V (including shift, and possibly rotation and scale), given 3 vertices and 3 sets of (u,v) coordinates? Any help is most appreciated!

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Okidoki, I stumpled upon a lucky google-hit, did the math from there using trial and error, and it looks like I got things up and running now. For future reference, here is what I found:

Quoting Matt McLelland in a post on alt.math:


If I may generalize your problem a bit, lets assume that your texture has its
origin at some point O with U and V vectors along the unit u and v axes. So the
texel at (u,v) would be at P = O + u*U + v*V in 3-space. Now, to determine O,
U, and V from known locations of texels we will need three such points. So,
suppose we know that (x_i, y_i, z_i) corresponds to the texel (u_i, v_i) for
i=1,2,3. Then, the equations involving the x-coordinate would be:
x_i = O_x + u_i*U_x + v_i*V_x for i=1,2,3.
Three equations and three unknowns. For your convenience, here is the matrix
[x1] [1 u1 v1] [O.x]
[x2] = [1 u2 v2] * [U.x]
[x3] [1 u3 v3] [V.x]
Equations in the other coordinates are analogous.

Thanks Matt! Solving using cramers rule:
D := Det(
1, u1, v1,
1, u2, v2,
1, u3, v3 );

If D = 0, then no solution exists.
O.x = Det(
x1, u1, v1,
x2, u2, v2,
x3, u3, v3 ) / D;
U.x = Det(
1, x1, v1,
1, x2, v2,
1, x3, v3 ) / D;
V.x = Det(
1, u1, x1,
1, u2, x2,
1, u3, x3 ) / D;

Analogous for Y and Z. This gives us the U and V axis (we disregard the O for now).

However, the map file format uses axis that represent a fraction of the texture, not the distance covered in 3D space, so we need to scale:
U = U * TEXTURE_WIDTH / Length( U ) ^ 2;
V = V * TEXTURE_WIDTH / Length( U ) ^ 2;

And finally, we need to calculate our offsets along those axes
offsetU = u1 * TEXTURE_WIDTH - Dot( P1 , U )
offsetV = v1 * TEXTURE_HEIGHT - Dot ( P1, V )

Rotation and scale are redundant and can always be left to 0, 1, 1.

Hope it helps someone. Oh, and any comments are most welcome.

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