Unprojecting 2D Screen Coordinates to 3D Coordinates

Question

I was wondering how I would go about mapping 2D screen coordinates to a 3D world (specifically the xz plane) knowing:

-position of the camera

-equation of the screen plane

-equation of the xz plane

All I want to do is have the land on the xz plane light up when I hover the mouse over it.

Any help is greatly appreciated!

Thanks!

Answer 1

If your world is rotated and shifted such that the camera is at x=y=z=0 (world coordinates), the world z coordinate increases away from the viewer (into the screen), and the projection plane is parallel to the screen and is at z=d (world coordinate; d > 0), then you determine screen coordinates from world coordinates this way:

xs = d * xw / zw
ys = d * yw / zw

And that's pretty intuitive: the farther the object from the viewer/projection plane, the bigger its zw and the smaller xs and ys, closer to the vanishing point of xw=yw=0 and zw=+infinity, which projects onto the center of the projection plane xs=ys=0.

By rearranging each of the above you get xw and zw back:

xw = xs * zw / d
zw = d * yw / ys

Now, if your object (the land) is a plane at a certain yw, then, well, that yw is known, so you can substitute it and get zw:

zw = d * yw / ys

Having found zw, you can now get xw by, again, substitution:

xw = xs * zw / d = xs * (d * yw / ys) / d = xs * yw / ys

So, there, given the setup described in the beginning and screen coordinates xs and ys of the mouse pointer (0,0 being the screen/window center), the distance between the camera and the projection plane d, and the land plane's yw you get the location of the land spot the mouse points at:

xw = xs * yw / ys
zw = d * yw / ys

Of course, these xw and zw are in the rotated and shifted world coordinates and if you want the original absolute coordinates in the "map" of the land, you un-rotate and un-shift them.

That's the gist of it.