Astrolabes
By Spencer Connor
When looking back at the history of astronomical instruments, there is perhaps nothing that that looks quite like an astrolabe. There is something undeniably artistic about these device and yet they are clearly scientific in basis, it’s clear that there is a high level of craftsmanship associated with them beyond just functionality. But what are they used for and how? Astrolabes are often quoted as having more than 1000 uses but fundamentally they really only do one, like a modern calculator may only have a handful of functions but each can serve a multitude of uses. That fundamental function is a coordinate transform between the celestial sphere and the observer reference frame.
The celestial sphere can be imagined as a transparent shell around the earth upon which all the stars are fixed. In antiquity this was believed to have have been a physical construct floating out around the globe, the true form of the heavens. But even in modern day we use it as a convenient analog, as it accurately represents the results of calculations for astrolabes and similar devices to the precision required.
The first step of the observer frame is just the earth. It is considered fixed, as we will make the celestial sphere rotate about it.
Building on Armillary Spheres
The image of the celestial sphere shown here may look familiar, it is reminiscent of another ancient astronomical device called an armillary sphere. Armillary spheres are simply a three-dimensional model of the earth and celestial sphere, a most direct representation. While more intuitive to understand, the armillary sphere suffers from two major impracticalities. First, armillary spheres are not particularly portable. A large, delicate sphere would quickly become a annoyance for travel let alone being used for navigation. Secondly and likely most importantly, it is difficult or impossible to take meaningful measurements off of them.
The astrolabe solves both of these issues by being a two-dimensional representation of an armillary sphere. It is flat and robust, making it well suited for travel and adverse conditions. And the method by which the 3D to 2D “flattening” is achieved makes angular measurements very straightforward.
Stereographic Projection
That method of representing a three-dimensional system in a two-dimensional model is key. Those familiar with cartography will know the difficulty in representing the surface of a sphere in a flat format for a map. Various projections like Mercator and Boggs eumorphic can be used to optimize certain properties, but nothing works for all cases. Areas get distorted, straight lines become curved, or some combination of the two result. And that’s just for a single static sphere! Our armillary is showing two spheres, and one is rotating relative to the other. There is a type of projection that works perfectly well for this, as it happens. That is a stereographic projection. Like other projections it doesn’t preserve all parameters (notably the size of features is drastically varied by their location), but those things have little use compared to the parameters that ARE preserved. Specifically,
