the issue with buckeye's solution is the uv coordinates won't be continuous except for the special case of a paralellogram/rectangle/square

a continous uv mapping could consist of the solutions to:

p = a + ux + vy + uvz

where the four coordinates of your rectangle in order are a,b,c,d, with x = b-a, y = d-a, z = a+c-d-b


i don't however know of a solution to this, and wolframalpha doesn't give you anything pretty for it!

(I come up with that by linearly interpolating accross top and bottom lengths with 'u', then linearly interpolating between those two points with 'v' which gives a unique, continous, but very complex mapping of p to (u,v) coordinates.

If you use two triangles to stretch a rectangle image like that, you might need to consider the problem described on the following page: http://home.xyzw.us/~cass/qcoord/

Don't know of a closed solution, but of course is it possible to "bilateral binary search" over (u,v) and checking each step in which quadrant p lies. It would even be sufficient to iterate over u only, until the resulting line will be close enough to p, and then solving for v analytically.

there is a closed solution, but it's very long!

ultimately you can reduce the problem to:


( u )
( 1 a b c )( uv ) = ( 0 )
( 0 1 d e )( v ) ( 0 )
( 1 )


which has a very complex closed solution of: http://www.wolframal...,+solve+for+u,v

of which only one of the two top solutions should be the correct one.

a,b,c,d,e being rather complex equatinos involving your coordinate p, and the coordinates of the 4 rectangle points starting with:



( u )
( bx-ax cx-bx dx-ax ax-px )( uv ) = ( 0 )
( by-ay cy-by dy-ay ay-py )( v ) ( 0 )
( 1 )


with rectangle being represented with top-left a, top-right b, bottom-right c, bottom-left d. u running left to right, v running top to bottom

There are lots of reasonable answers here.

We've had this discussion here in the past; if you search the old forums you can probably find those threads. The solutions always boiled down to one of,
1.) Split into triangles and do barycentric interpolation on each (buckeye's answer)
2.) Bilinear interpolation (luca-deltodesco's answer)
Details were given in those threads for both of these approaches.

Here's a third: [EDIT: Removed; was wrong.]

Yeah, I though of looking at it the following way:

The approach of the bilinear interpolation
p[sub]1[/sub] := a + ( b - a ) * u
p[sub]2[/sub] := d + ( c - d ) * u
p := p[sub]1[/sub] + ( p[sub]2[/sub] - p[sub]1[/sub] ) * v
means that there is a line between p[sub]1[/sub] and p[sub]2[/sub] that passes through p. Expressing this line the other way, namely
p + l = p[sub]1[/sub]
p + k * l = p[sub]2[/sub]
so that l + k * l (where k is obviously need to be a negative number) is the said line. Considering that this is done in 2D, we have 4 equations with 4 unkowns (k, l[sub]x[/sub], l[sub]y[/sub], and u) where k * l[sub]x[/sub] and k * l[sub]y[/sub] are the problems. Fortunately, from the lower of the both equations, we can isolate
k * l[sub]x[/sub] = d[sub]x[/sub] - p[sub]x[/sub] + ( c[sub]x[/sub] - d[sub]x[/sub] ) * u
k * l[sub]y[/sub] = d[sub]y[/sub] - p[sub]y[/sub] + ( c[sub]y[/sub] - d[sub]y[/sub] ) * u
and divide both of these, so that
l[sub]x[/sub] / l[sub]y [/sub]= [ d[sub]x[/sub] - p[sub]x[/sub] + ( c[sub]x[/sub] - d[sub]x[/sub] ) * u ] / [ d[sub]y[/sub] - p[sub]y[/sub] + ( c[sub]y[/sub] - d[sub]y[/sub] ) * u ]

Now, using the upper of the equations to get
l[sub]x[/sub] = a[sub]x[/sub] - p[sub]x[/sub] + ( b[sub]x[/sub] - a[sub]x[/sub] ) * u
l[sub]y[/sub] = a[sub]y[/sub] - p[sub]y[/sub] + ( b[sub]y[/sub] - a[sub]y[/sub] ) * u
and setting these into the above l[sub]x[/sub] / l[sub]y[/sub], we get a quadratic equation solely in u which can be solved using the famous p,q-formula.

This gives, as luca-deltodesco has mentioned, 0 or 1 real solutions in general. But due to the fact that p always lies inside the quad, I expect 1 real solution. With that, v can be determined easily.