In general I´d say a point (x,y) inside P1 can end up anywhere within (maybe not even nessecarily within) P2 but this probably isn´t the answer you were looking for, so:

Assume linearity:
Split your polygon into triangles (two of them in this case). A point (x,y) in T1= {(x1,y1),(x2,y2),(x3,y3)} can be written as:
(x,y) = (x1,y1) + a*((x2,y2)-(x1,y1)) + b*((x3,y3)-(x1,y1)).
Let a and b remain unchanged under your transformation (which is what I meant by linearity):
(u,v) = (u1,v1) + a*((u2,v2)-(u1,v1)) + b* (...)

just a few thoughts; I won´t write anything more here this evening as I´m quite drunk now (have been offered a diploma thesis and thus had a good reason/excuse to open a bottle of wine) and don´t want to flood this forum with potentially stupid stuff ...

- What do you actually try to do? I thought something like you have a texture bitmap in your polygon that you want to transform.

- What do you mean by "distorted"?

- Did you actually check that the point you transform is within the triangle you are using (0<= a,b <=1) ?

- >> but the transformation depends only on the 3 points of the
>> triangle and not on the 4 points of the polygon.
right. The 4th point is covered by the 2nd triangle.

- to explicitely point it out, you:
1) check in which one of the triangles your point lies.
2) transform the point according to the rules of the triangle it lies in.