3D coordinate of 2D point given camera and view plane

Question

I wish to generate rays from the camera through the viewing plane. In order to do this, I need my camera position ("eye"), the up, right, and towards vectors (where towards is the vector from the camera in the direction of the object that the camera is looking at) and P, the point on the viewing plane. Once I have these, the ray that's generated is:

ray = camera_eye + t*(P-camera_eye);

where t is the distance along the ray (assume t = 1 for now).

My question is, how do I obtain the 3D coordinates of point P given that it is located at position (i,j) on the viewing plane? Assume that the upper left and lower right corners of the viewing plane are given.

NOTE: The viewing plane is not actually a plane in the sense that it doesn't extend infinitely in all directions. Rather, one may think of this plane as a widthxheight image. In the x direction, the range is 0-->width and in the y direction the range is 0-->height. I wish to find the 3D coordinate of the (i,j)th element, 0

Answer 1

When I directly plugged in suggested formulas into my program, I didn't obtain correct results (maybe some debugging needed to be done). My initial problem seemed to be in the misunderstanding of the (x,y,z) coordinates of the interpolating corner points. I was treating x,y,z-coordinates separately, where I should not (and this may be specific to the application, since the camera can be oriented in any direction). Instead, the solution turned out to be a simple interpolation of the corner points of the viewing plane:

• interpolate the bottom corner points in the i direction to get P1
• interpolate the top corner points in the i direction to get P2
• interpolate P1 and P2 in the j direction to get the world coordinates of the final point

Answer 2

General solution of the itnersection of a line and a plane see http://local.wasp.uwa.edu.au/~pbourke/geometry/planeline/

Your particular graphics lib (OpenGL/DirectcX etc) may have an standard way to do this

edit: You are trying to find the 3d intersection of a screen point (eg a mouse cursor) with a 3d object in you scene?

Answer 3

To work out P, you need the distance from the camera to the near clipping plane (the screen), the size of the window on the near clipping plane (or the view angle, you can work out the window size from the view angle) and the size of the rendered window.

1. Scale the screen position to the range -1 < x < +1 and -1 < y < +1 where +1 is the top/right and -1 is the bottom/left
2. Scale normalised x,y by the view window size
3. Scale by the right and up vectors of the camera and sum the results
4. Add the look at vector scaled by the clipping plane distance

In effect, you get:

p = at * near_clip_dist + x * right + y * up

where x and y are:

x = (screen_x - screen_centre_x) / (width / 2) * view_width
y = (screen_y - screen_centre_y) / (height / 2) * view_height