Suppose I am located at fixed point $P$ in $3$-space. However, the coordinates of point $P$ are unknown.
I have a map with $N$ landmark points around me, and the world space coordinate of each landmark is known.
I establish an arbitrary local spherical coordinate system around $P$, and am able to look at each of the $N$ landmark points and determine $\phi$ and $\theta$ from $P$, Note that $\rho$, the distance from $P$ to each landmark, is unknown. I then use $\phi$ and $\theta$ to find the unit direction vector from $P$ to each landmark in local coordinates.
Is there a solution to determine
- Position $P$ in world coordinates
- Matrix $M$ to convert world coordinates to local coordinates
- Matrix $M^{-1}$ to convert local coordinates to world coordinates
and how many landmarks $N$ are needed?