I have a collection of ratios (they are all the same degree) where the sum of their parts equate to $0$; and I need a way to represent these ratios as points in space (to perform k-means clustering on them). I can't think of some intuitive relationship between these two ideas.
The simplest approach would be to convert each part into a fraction, and then treat these fractions as the $x,y,z\dots$ components of the point: but this doesn't work for ratios that summate to zero (of which my data is exclusively constructed from)
An idea I had was to use barycentric coordinates, but from what I can tell, this only applies to ratios that consist of positive parts.
Another approach I considered was graphing the ratio as a line with equation $ax=by=cz=\dots$ where the ratio is $a:b:c:\dots$ calculating the intersection with a hemisphere and then treating the intersection as point in (hemi?) spherical space Example (Hemisphere as there would be two points for a full sphere)
This then turns into a (tessellating?) (hemi?) spherical geometry problem where going below the equator brings you back round to the opposite side Example
Does anyone know any better/simpler/more intuitive methods?
Its also of note that that I plan to apply this method in an arbitrary number number of dimensions (I need to be able to do this for a ratio with any number of parts)
Thanks and I apologise if any terminology is wrong