What path does a star travel across the night sky?

If one is not standing on the equator, one sees stars of a night time move from East to West, effectively rotating about either the North Celestial Pole or the South Celestial Pole.

So if you ponder what apparent path does a star move in, your first answer may be a circle - quickly updated to be an ellipse because things are tilted according to your lattitude.

However lets say we want to be really exact - like we are doing astrophotography and arc seconds matter. The ignoring turbulence (which can move a star back and forth a few arc seconds) the apparent altitude of a star and the observers elevation above sea level will change a key factor - refraction cause by the air itself.

The closer you are to sea level - the more air is above you and the more refraction kicks in. The closer you observe to the horizon the more air of changing density you are looking through and the more refraction kicks in.

Advanced telescopes model this refraction compensating for refraction using King rates of tracking I believe. So in fact the stars don't actually revolve around the North and South Celestial poles, technically they revolve around the refracted NCP and SCP respectively (which may be an arc minute or two above or below the pole).

So to my question - if refraction changes linearly with a star's elevation - is the best description of a stars movement across the night sky (ignoring turbulence) a hyperbola?

Many thanks to anyone who can definitely answer this!

Matthew