Consider 2 arbitrary points, A and B, which are located on Earth's surface. We assume the Earth to be a perfect sphere for the purposes of my question. Each point is given by their latitude and longitude: $A=(\phi_A, \lambda_A)$, $B=(\phi_B, \lambda_B)$.
An orthodromic curve is defined as the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere. The condition of uniqueness for an orthodromic curve is that A and B are NOT antipodal, that is, if at least one of these conditions are met: $$\phi_B \neq \phi_A$$ or $$\lambda_B\neq\pi+\lambda_A$$
But what about a Loxodromic curve? A loxodromic curve, or rhumb line, is defined as arc that crosses all meridians of longitude at the same angle (rhumb).
I'm thinking hard to get the condition A and B have to meet so that there is only ONE loxodromic between them. Any hints that would lead me to the correct path will be more than welcome.