---

Problem 525

---

Rolling Ellipse

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:
$F(a,b)=2\pi max(a,b)$

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.

---

翻滚的椭圆

椭圆E(a, b)的初始位置由以下方程给出：
$\frac{x^2}{a^2}+\frac{(y-b)^2}{b^2}=1$

椭圆沿着x轴无滑动地滚动一周。有趣的是，椭圆焦点的运动轨迹长度与椭圆短轴的长度无关：
$F(a,b)=2\pi max(a,b)$

对于椭圆中心而言则并非如此；记C(a,b)为椭圆无滑动地滚动一周时椭圆中心的运动轨迹长度。

已知C(2, 4) ~ 21.38816906。

求C(1, 4) + C(3, 4)。将你的答案四舍五入到小数点后8位小数，即格式为ab.cdefghij。

---