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3d point - perspective projection

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I wrote an equation to calculate a 3d point's projection
98% of the time this equation works, but it's crucial that it will work 100% of the time.
here's what I got:
first we define some vars.
eye -> 3D point for our point of view of the scene
point -> 3D point that we want to project on the screen
p2d -> our point after projection
I assume that on screen, Z = 0, therefore:
0 = point.z + u * (eye.z - point.z)
u = -point.z / ( eye.z - point.z )
p2d.x = point.x + u * (eye.x - point.x)
p2d.y = point.y + u * (eye.y - point.y)

this is an image when it's ok:
this is an image when it's bad:

just to make sure you see it, in the "bad" picture, you can see that one corner of the table moves to a different place
which makes the table look bad...

any idea why this happens?

also, is there another way of doing this without changing my axes?

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See here for a derivation of the perspective projection formulas.

I tried to read your derivation, but could not understand it very well. One thing I noted is that u = -point.z / ( eye.z - point.z ) can cause a division by zero, and that you don't seem to be using the concepts of a near plane and an eye (observer) point (or they are somehow hidden implicitly in your derivation).

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what I did is a 3D-Line equation (and I split it into x,y,z )

(x, y, z) = point + t * ( eye - point )
to solve the equation I decided that Z=0 (since Z=0 on screen)

btw thanks for the link, I think I'll manage from here

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