I am trying to come up with a way to find the transformation parameters between two sets of planes and would like to get a cost function C which incorporates two functions C1 and C2 where
C1 = dot(n1,n2)
C2 = dist(p1,p2)
n1 and n2 are normal vectors and p1 and p2 are means of planes. However, I am not sure if
C = C1 + C2
can be used as a cost function to minimize the angle and distance together? If not, do I need to maximize them separately?
If two planes must be fitted to be equal, for a given unitary normal and distance from the origin: $$ n_i\cdot (x - n_i\rho_i)=0 $$
with $n_1 \cdot n_2 \in [-1 \ 1]$, and $\rho \in$ , lets say, $[0 \ \rho_0]$.
Please confirm the previous boundary.
If so, the cost function would be: $$ C(\mathcal P_1,\mathcal P_2)=\frac12n_1\cdot n_2 + \frac1{\rho_0}(\rho_1-\rho_2)^2 $$
In here both factors are equally penalized over the domain lengths.