Back in 2001, I was looking for mathematical curves to fit the facing curve measurement shapes I was collecting.  I tested a Power Curve equation but it did not fit the shapes I wanted as well as the Elliptical Curve equation.  So I discarded it to my notebook.

Recently, a client wanted me to copy a facing schedule he gave me.  It had a more parabolic shape to it with more curvature coming off the table and becoming flatter towards the tip.  So I broke out the Power Curve and found that it fit his facing schedule well.  The form I used is as follows:

Y = S-C*(X*25.4*2)^(1/P)

X = feeler in inches
Y = glass gauge reading in mm*2
S = start point of the curve from the tip
C = a constant
P = the power of the curve.  2 would be for a square root.  You could use ^P instead of ^(1/P).  But I like 1/P since higher values of P have more curve to them.

I set it up a spreadsheet and used a least squares methods to solve for S, C, and P.  For my client, his curve was close to Sf, C5, P=3.  This was for a .124" tip opening on tenor.

I'm not a huge fan of this type of curve but it works decent.  It has more resistance in it than an elliptical curve unless you use a more extreme ellipse.  So you might need a .010" smaller tip opening or so than you would use with an elliptical or radial curve.   

With a soft enough reed to make the low notes come out, it had good high notes and altissimo.  I felt the right hand notes were more resistant and stuffy to me.

For those that were looking at using two radial curves to give more resistance at the break, you may find that the Power Curve works well.