\section*{Problem 4}
I. In the two-dimensional orthogonal $x y$ coordinate system, consider the curve $L$ represented by the following equations with the parameter $t ( 0 \leq t \leq 2 \pi )$. Here, $a$ is a positive real constant.

$$\begin{aligned}
& x ( t ) = a ( t - \sin t ) \\
& y ( t ) = a ( 1 - \cos t )
\end{aligned}$$

\begin{enumerate}
  \item Obtain the length of the curve $L$ when $t$ varies in the range of $0 \leq t \leq 2 \pi$.
  \item Obtain the curvature at an arbitrary point of the curve $L$.
\end{enumerate}

Here, $t = 0$ and $t = 2 \pi$ are excluded.

II. In the three-dimensional orthogonal $x y z$ coordinate system, consider the curved surface represented by the following equations with the parameters $u$ and $v$ ( $u$ and $v$ are real numbers).

$$\begin{aligned}
& x ( u , v ) = \sinh u \cos v \\
& y ( u , v ) = 2 \sinh u \sin v \\
& z ( u , v ) = 3 \cosh u
\end{aligned}$$

\begin{enumerate}
  \item Express the curved surface by an equation without the parameters.
  \item Sketch the $x y$-plane view at $z = 5$ and the $x z$-plane view at $y = 0$, respectively, of the curved surface. In the sketches, indicate the values at the intersection with each of the axes.
  \item Express the unit normal vector $\boldsymbol { n }$ of the curved surface by $u$ and $v$. Here, the $z$-component of $\boldsymbol { n }$ should be positive.
  \item Let $\kappa$ be the Gaussian curvature at the point $u = v = 0$. Calculate the absolute value of $\kappa$.
\end{enumerate}