Locus of all points

Question

What is the locus of all points equidistant from a fixed point and a fixed circle on a sphere? (By examining an "extreme" case, i.e. the fixed point being the North Pole and the fixed circle being a Great Circle that ALMOST passes through the North Pole, it is obvious to me that the locus can not be a circle. But I can't find the problem dealt with on this or any other site.)

Answer 1

Let choose the spherical coordinate system in such a way that center of the circle lies on the polar axis. Then the circle represents a circle of latitude with polar angle $\theta_c$. Let $\theta_p$ be the polar angle of the fixed point, its azimuthal angle $\phi_p$ being $0$. Then the locus of points in question (with distance being assumed spherical) is determined by the equation: $$L(\theta,\phi)=0$$ with $$ L(\theta,\phi)=\sin\theta_p\sin\theta\cos\phi+\cos\theta_p\cos\theta-\cos(\theta-\theta_c). $$