I have several shapes which are roughly elliptical. I know the major and minor axes and the true circumference, so I store them like this:
$$a={\text{axis}}_{\text{major}}\\ b={\text{axis}}_{\text{minor}}\\ c={{\text{circumference}}_{\text{measured}} \over {\text{circ}}(a,b)}$$
(where ${\text{circ}}(a,b)$ calculates a circumference from the axes using an approximation formula)
Next, I linearly interpolate the shapes. This means that $a$ and $b$ are linearly interpolated, and it turns out that $c$ is also roughly linear. I want to get an approximation of the new circumference without actually measuring it. I do this:
$$a'=v a_1 + (1-v) a_2\\ b'=v b_1 + (1-v) b_2\\ c'=v c_1 + (1-v) c_2\\ {\text{circumference}}'_{\text{guess}}={\text{circ}}(a',b') \times c'$$
This all works fine, but I feel as though it should be possible with 2 variables, not 3, but I can't figure out how it would work. My reasoning is that the "fiddle factor" ($c$) can be merged with the eccentricity, leaving only the scale factor. Could this be done (while maintaining linearity) or am I mistaken? How would this change if I added the fiddle factor instead of multiplying?