Mathematically predicting mag & bias values?

[R Reynolds]

I'm building a formed steel frame, a part of which is shown below, to fit inside an archway . 

I wanted the vertical column to intersect nicely with the curved section, so I built the curved beam first and made a copy of it as a background template.  After completing the orthogonal Tee section with all the corners and five pointers, the final step was splicing the Tee section into the curve and massaging it's cp's, mags and bias values so it's shape matches that of the original curve. 

My question is based on the fact that when you lathe a circle in A:M, no matter how many points make up the circle, the correct mag is always calculated to get the splines to lie on a circle. A four point circle gets mags of 167.39 while a twelve point circle uses mags of 105.78. So does a mathematical formula exist (at least on paper) that says if you want a spline that matches a mathematically defined shape, an ellipse for instance, input the coords. of a CP anywhere on the line and the output is the correct mag and bias to get the best match?

I'm guessing the answer is no but I thought it worth asking.

[robcat2075]

I know there must be a formula because Stitch with "Maintain curvature" works automatically.

Lathe out your perfect four-point circle, go to Add Mode, then hold down SHIFT while you Add a new CP anywhere on the circle.

The circle will remain a circle. The new CP has its biases automatically set and its neighboring CPS have had their biases adjusted for the new situation.

It is probably a pretty hairy formula since it has to correctly handle not just circles and ovals but ANY spline curvature.

This spline-inserting technique would eliminate manual bias fiddling for all of your new CPs except the one that are part of the sharp corners.

 

While there is this Maintain Curvature ability for adding CPs, I regret there is no similar power for deleting CPs.

[R Reynolds]

Dang! I've used that technique before and I still didn't think of it for this case. Thanks for the reminder.