I'm reading the Wiki page now, but so far all I've come across is "oblique projection" (which is anything but orthogonal projection.)
Projection looks like this:
proj(u, v) = (u . v) v
= proj(proj(u, v), v)
What I've been doing is this:
w = unproj(u, v) = u - proj(u, v)
= proj(u - proj(u, v), u)
proj(w, v) = 0
(u - proj(u, v)) x (v) = 0
I considered using cross product somehow to replace it, mostly because using the dot product / projection seemed wrong when the intent is the opposite, but also because u x u = 0 should help eliminate the parallel components. Unfortunately it transforms the orthogonal components too, but on paper at least it looks like you can follow up the forward cross with an inverse cross (as in cross product with the multiplicative inverse of the right operand, so generally conjugate of inverse magnitude.)
If you look up the definition of cross product, one of the factors is a sin(theta) derived from the dot product anyway.
Any ideas what this might be called? If it matters I'm programming so I have to call it something. "Unproj" to my "Proj" makes some sense to me, but if I am going to name it I want to steer clear of "Aproj"/"Arcproj"/etc. because it's definitely not an inverse.
Best Answer
This is called vector rejection. A common concept in Geometric algebra.