Suppose I have a set of $k$ points $\{x_1,x_2,\ldots,x_k\}$ in $\mathbb{R}^n$ that I can project into $\mathbb{R}^m$ with the linear operator $\mathcal{P}$, with $\alpha, \beta, \ldots$ parameters of the projection operator. Is there a well known best method for determining the class of functions $f:(\mathbb{R}^m)^k \rightarrow \mathbb{R}$ such that $f(\mathcal{P}x_1,\mathcal{P}x_2,\ldots,\mathcal{P}x_k)$ is invariant under changes in $\alpha, \beta, \ldots$?

As an example, a situation I'm interested in has $\mathcal{P}: \mathbb{R}^3 \rightarrow \mathbb{R}^2, x \mapsto A[\phi, \theta, \eta]x$, where:

$A[\phi, \theta, \eta] = \eta \left( \begin{matrix} \cos\phi\cos\theta & \sin\phi\cos & \sin\theta \\ -\sin\phi & \cos\phi & 0 \end{matrix}\right)$

i.e. the projection of points in 3-D space onto the plane with unit normal $(\cos\phi\sin\theta,\sin\phi\sin\theta,\cos\theta)$, with an additional scaling parameter of $\eta$.

Setting derivatives of $f$ with respect to the parameters equal to zero yields a set of equations with no obvious general solution, and a feeling that I'm missing a certain set of tools.