Let's say I have some coplanar points in R^3. I want to describe these points by coordinates on the plane they share, in R^2. This way I can do some tests faster. How should I do this?

Thanks! Am I right when I think that the final projection could be a matrix multiplication by the 2x3 vectors we obtained?

If you have an orthonormal basis of your subspace, you can find the coefficients of the projection by simply computing the inner product of the vector to be projected and the basis vectors. So yes, the computation boils down to a matrix multiplication that turns three numbers into two.

In order to get an orthonormal basis for the sub-space, you can use the Gram-Schmidt process.

[EDIT: I am not saying anything new, just rephrasing what others have said, in the hopes that my description might be easier to understand to some.]

I'm wondering if there's a way to avoid having to subtract the subspace's origin before doing projection, by encoding the origin in the matrix. Is it possible using homogenous coordinates?

I'm wondering if there's a way to avoid having to subtract the subspace's origin before doing projection, by encoding the origin in the matrix. Is it possible using homogenous coordinates?

Yes, you can encode a translation into the matrix by adding a row and a column and by appending a "1" to the coordinates of the point. This is related to homogeneous coordinates, but I am not going to try to describe the details here.

I haven't tried but wouldn't I have to subtract the origin BEFORE doing the projection? or does the order not matter?