In the two-dimensional orthogonal $xy$ 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{align}x(t) &= a(t - \sin t), \\y(t) &= a(1 - \cos t).\end{align}

• [(I-1)] Obtain the length of the curve $L$ when $t$ varies in the range $0 \leq t \leq 2\pi$.

• [(I-2)] For $0 < t < 2\pi$, obtain the curvature $\kappa_L(t)$ at an arbitrary point of the curve $L$. In the three-dimensional orthogonal $xyz$ coordinate system, consider the curved surface represented by the following equations with the parameters $u$ and $v$ ($u$ and $v$ are real numbers):\begin{align}x(u, v) &= \sinh u \cos v, \\y(u, v) &= 2 \sinh u \sin v, \\z(u, v) &= 3 \cosh u.\end{align}

• [(II-1)] Express the curved surface by an equation without the parameters $u$ and $v$.

• [(II-2)] Sketch the $xy$-plane view at $z = 5$ and the $xz$-plane view at $y = 0$, respectively, of the curved surface. In the sketches, indicate the values at the intersections with each of the axes.

• [(II-3)] Express a unit normal vector $\mathbf{n}$ of the curved surface by $u$ and $v$. Here, the $z$-component of $\mathbf{n}$ should be positive.

• [(II-4)] Let $\kappa$ be the Gaussian curvature at the point $u = v = 0$. Calculate the absolute value $|\kappa|$. Finally, define the following quantities:

• Let $L$ be the length of the cycloid in {\rm (I-1)}, and let $\kappa_L(\pi)$ be the curvature of $L$ at $t = \pi$ from {\rm (I-2)}. Define

• In {\rm (II-2)}, denote by $A_x > 0$ and $A_y > 0$ the $x$- and $y$-intercepts (respectively) of the cross-section of the surface by the plane $z = 5$, and by $A_z > 0$ the $z$-intercept of the cross-section by the plane $y = 0$. Define

• In {\rm (II-4)}, let $\beta := |\kappa|$ be the absolute value of the Gaussian curvature at $u = v = 0$. Using these, define

Compute the integer $\mathcal{K}$.\