## Rolling Ellipse

### Problem 525

An ellipse `E`(`a`, `b`) is given at its initial position by equation:

$\frac {x^2} {a^2} + \frac {(y - b)^2} {b^2} = 1$

The ellipse rolls without slipping along the `x` axis for one complete turn. Interestingly, the length of the curve generated by a focus is independent from the size of the minor axis:

$F(a,b) = 2 \pi \text{ } max(a,b)$

This is not true for the curve generated by the ellipse center. Let `C`(`a`,`b`) be the length of the curve generated by the center of the ellipse as it rolls without slipping for one turn.

You are given `C`(2, 4) ~ 21.38816906.

Find `C`(1, 4) + `C`(3, 4). Give your answer rounded to 8 digits behind the decimal point in the form *ab.cdefghij*.