Question

Assume that the camera is located at (0,0,1) point looking into the origin. -z direction is going into the screen. Objects beyond z = -100 are not visible ( far viewing plane). At the far viewing plane x and y are clipped at 100 at right and -100 at left for x-axis and likewise for y-axis.

The viewport window is 600 pixel wide and 300 pixel height. On the 2d pixel coordinates (x2d, y2d), (0,0) is the top left corner, x2d increases to right and y2d increases going downward.

Given above parameters, what are the formulas that calculates (x2d,y2d) given a point (x,y,z)? Each 3d point maps to a pixel unless 3d point is clipped ( not visible in the viewport).
Assume perspective projection.

Please don’t provide a link that talks about the theory of 3d projections. I am looking for specific solution to this specific problem with the given parameters.

Answer 1

If all you want to do is find 2D screen locations from 3D locations its easy if you know similar triangles. No triggers are required. just make up some variable that is about the distance in pixels out of the screen to your eyes – call that the focal_length. You can adjust that figure to make it look more realistic (too low looks really stretchy)

This solution is for looking forward in x, looking backwards just needs and extra line to negative the z which is just needless complexity in my humble opinion but i don’t know your purpose.

first clip out pixels outside of your range (just use a couple of if statements. I’m not gonna pseudocode that out)

then position yourself in the 3D world:

x0 = x - yourx
y0 = y - youry
z0 = z - yourz

(in your example yourx = 0, youry = 0, yourz = -1)

Then project away:

x2d = focal_length * x0 / z0
y2d = focal_length * y0 / z0

Simple huh? I just made this up by thinking about train tracks and am now trying to make a basic 3d game 😉

If you want to get into the math of 3D rotation or hiding faces etc… that’s when things gets … tricky.

By the way depending on how your screen coordinates are, you might need negative signs on one or the other or both of x2d or y2d equations. Meh you figure it out.