How Can 2D Plane Coordinates Be Transformed into 3D Coordinates?

[jash7205]

hi,

so this is actually for a program I'm writing, but it's definitely more of a math question than a programming question. basically, i have an object that gets detected by a webcam attached to a computer. the object is just a piece of paper with a pattern on it, so it is, for the purpose of the program, a 2D plane. from the webcam i am able to obtain 2d coordinates of the four corners of the pattern. also, i know the 3d dimensions of the plane (length, width, etc).

the goal is to somehow get 3d coordinates, which i will use to find the orientation of the pattern, from that information. i spent hours searching the web, but haven't really found anything helpful (there was a lot of info on 3d to 2d projections, which is the opposite of what i want). does anyone know of any way of going about this? is there any necessary data that is missing?

thanks for your help,
-j

 

[fresh_42]

Wikipedia has an entry for this https://en.wikipedia.org/wiki/3D_projection with further sites to look at the specific subitems. Maybe you can even use the picture there which illustrates your situation quite well. Just put a 3D coordinate system in it, and then you can determine the linear projection $Px=y$ by solving for $P$ given the four points $x$ and parts of $y$.