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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).

rhumb-line-wikipedia

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.

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I suspect that uniqueness might happen in the case of two points with the same latitude if you argue that a line of latitude is a loxodromic curve and you argue that all loxodromic curves other than lines of latitude start at one pole and finish at the other. You would still need to say that going round the earth westwards and going round eastwards was following the same line of latitude and so same loxodromic curve.

For points at different latitudes, there will be many loxodromic curves passing through both of them, though most of these will wind round the earth several times in a sort of spiral. If they have different longitudes, there will also be non-spirals eastwards and westwards.

The easiest way to see this is to imagine several Mercator projections stuck together, extending horizontally. Plot a single starting point but several copies of the finishing point: a loxodromic curve is then any straight line joining the start point with a finishing point

Try this for five loxodromic curves between London and Auckland

enter image description here

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