I have a set of points on the surface of a sphere specified in one coordinate system (specifically, the equatorial coordinate system), and for each point I need to work on all its neighbouring points as if it were on the equator (i.e. as if its elevation were zero).
Specifically, each point is specified by a longitude $\alpha\in[0,2\pi]$ around the system's equator and latitude $\delta\in[-\frac{\pi}{2},\frac{\pi}{2}]$ along a meridian. What I want to do is rotate the equator such that it intersects the point under consideration $(\alpha_0,\delta_0)$ while preserving its longitude; i.e. the point's position becomes $(\alpha_0,0)$.
How should I go about calculating the coordinates of the neighbouring points in the new coordinate system?
P.S. My reason for doing this is to allow me to approximate the space as Euclidean in the vicinity of each point, which only works close to the equator.