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How does trapezoid drawing with perspective correct works?

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Recently I've read a few articles about drawing trapezoid on the Internet, which it solved the problem of  wrong  perspective over texture mapping between two triangles by using homogeneous coordinate  stq (leave r zero).

 

It's said that the default 2d uv(0,0) ==> 3d (0,0,width)  homogeneous coordinate,  and 2d uv(0,1) ==> (0, width,width), which the width is length that the texture to be mapped, it 's clear that after divide by the the w(or q) the interpolated texture coordinate  is the same as  the original uv coordinate.  i need to know why is it? how does it works? cause i can't figure out the subtle difference in between, or math behind it.

 

Could your guys explain in detail. i really appreciate it! thank you!

Edited by qxsl2000

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