Projecting from one set of dimensions to another

• Systems can project from one dimensionality (say 3D) to another (say 2D).
• How would you project from 3D to 2D?
• If there are points in 3D space, and you shine a light on them, the shadow is projected onto a 2D shape (say the floor or a wall).
• Does it matter where the light is? The surface that it is being projected onto? Of course it does.
• Can you project from 2D to 3D?
• People do this all the time when they take a 2D image (close one eye and that's what you get) and infer depth.
• Programmatically, you can just write a function that takes the two inputs (x and y), and makes a third (z).
• You can of course do this from any N-D to any X-D.
• In high dimensions, you often use principle component analysis or independent component analysis to project data in high dimensions to 1 or a small number of dimensions.