All of the formulas are in the second link I posted. It’s an excerpt from an SAE paper. It literally explains everything. Maybe too much, I got a little lost in the weeds. For the math, it doesn’t matter.
In my haste speed reading the SAE paper I missed how the calculation is simplified for use with the mechanical gauges. With the bubble gauges, the scale on the gauge for reading caster is predetermined by the turn angle. The calculation can be done for any turn angle, but with the mechanical gauges the calculation has been simplified to the point that a predetermined angle is used to set the scale on the bubble gauges so you don’t have to do the math.
So, you have to use the angle specified for your gauge. Most of the older gauges use 20*, the fastrax gauge uses 15*.
The digital gauges can use any angle they’ve been programmed for. And the math can be done for any angle you want, but then you’d have to suffer through doing the calculation.