I will calculate the required values step-by-step using Python.

Part I: Cycloid

1. Calculate the length $L_{len}$ of the cycloid $x(t) = a(t - \sin t), y(t) = a(1 - \cos t)$ for $0 \le t \le 2\pi$.
2. Calculate the curvature $\kappa_L(\pi)$.
3. Compute $\alpha = L_{len} \cdot \kappa_L(\pi)$.

Part II: Surface

1. Surface: $x = \sinh u \cos v, y = 2 \sinh u \sin v, z = 3 \cosh u$.
2. At $z=5$: Find $A_x, A_y$.
   • $3 \cosh u = 5 \implies \cosh u = 5/3$.
   • $\sinh^2 u = (5/3)^2 - 1 = 16/9 \implies \sinh u = 4/3$ (taking positive since $A_x, A_y > 0$).
   • $x = (4/3) \cos v, y = 2(4/3) \sin v = (8/3) \sin v$.
   • This is an ellipse: $x^2/(4/3)^2 + y^2/(8/3)^2 = 1$.
   • Intercepts: $A_x = 4/3, A_y = 8/3$.
3. At $y=0$: Find $A_z$.
   • $2 \sinh u \sin v = 0$.
   • Case 1: $\sinh u = 0 \implies u=0 \implies x=0, z=3$.
   • Case 2: $\sin v = 0 \implies v = 0, \pi$.
     • $x = \pm \sinh u, z = 3 \cosh u$.
     • Equation: $(z/3)^2 - x^2 = 1 \implies z^2/9 - x^2 = 1$.
     • This is a hyperbola.
   • The "z-intercept of the cross-section" is the value of $z$ when $x=0$ (and $y=0$).
   • From the equation $z^2/9 - x^2 = 1$, if $x=0$, $z^2=9 \implies z=3$ (since $z=3\cosh u \ge 3$).
   • So $A_z = 3$.
4. Compute $\delta = A_x A_y A_z$ and $\rho = A_y / A_x$.
5. Compute Gaussian curvature $\beta = |\kappa|$ at $u=0, v=0$.

Final Calculation Compute $S$ and $\mathcal{K}$.

I will implement this in Python to handle the derivatives and algebra precisely.

[PYTHON] import sympy as sp

Part I: Cycloid

a, t = sp.symbols('a t', real=True, positive=True) x_t = a * (t - sp.sin(t)) y_t = a * (1 - sp.cos(t))

dx_dt = sp.diff(x_t, t) dy_dt = sp.diff(y_t, t) ddx_dt = sp.diff(dx_dt, t) ddy_dt = sp.diff(dy_dt, t)

Length

ds_sq = dx_dt2 + dy_dt2 ds = sp.sqrt(ds_sq.simplify())

ds simplifies to a * sqrt(2 - 2cos(t)) = a * sqrt(4 sin^2(t/2)) = 2a sin(t/2) for t in [0, 2pi]

L_len = sp.integrate(ds, (t, 0, 2*sp.pi))

Curvature

kappa = |x' y'' - y' x''| / (x'^2 + y'^2)^(3/2)

num = sp.Abs(dx_dt * ddy_dt - dy_dt * ddx_dt) den = (dx_dt2 + dy_dt2)**(sp.Rational(3, 2)) kappa_L = num / den kappa_pi = kappa_L.subs(t, sp.pi)

alpha = L_len * kappa_pi

print(f"L_len: {L_len}") print(f"kappa_pi: {kappa_pi}") print(f"alpha: {alpha}")

Part II: Surface

u, v = sp.symbols('u v', real=True) x_s = sp.sinh(u) * sp.cos(v) y_s = 2 * sp.sinh(u) * sp.sin(v) z_s = 3 * sp.cosh(u)

II-2: Intercepts

At z = 5

3 cosh u = 5 => cosh u = 5/3 => sinh u = sqrt(25/9 - 1) = 4/3

x = 4/3 cos v, y = 8/3 sin v

Ax is x-intercept (when y=0 => sin v=0 => cos v = +/-1 => |x| = 4/3)

A_x = sp.Rational(4, 3)

Ay is y-intercept (when x=0 => cos v=0 => sin v = +/-1 => |y| = 8/3)

A_y = sp.Rational(8, 3)

At y = 0

z-intercept is when x=0 as well (intersection with z-axis)

x=0, y=0 => sinh u = 0 => u=0 => z = 3

A_z = 3

delta = A_x * A_y * A_z rho = A_y / A_x

print(f"Ax: {A_x}, Ay: {A_y}, Az: {A_z}") print(f"delta: {delta}") print(f"rho: {rho}")

II-4: Gaussian Curvature at u=0, v=0

Define position vector r

r = sp.Matrix([x_s, y_s, z_s]) ru = r.diff(u) rv = r.diff(v) ruu = ru.diff(u) ruv = ru.diff(v) rvv = rv.diff(v)

First Fundamental Form

E = ru.dot(ru) F = ru.dot(rv) G = rv.dot(rv)

Normal vector

cross_prod = ru.cross(rv) norm_cross = sp.sqrt(cross_prod.dot(cross_prod)) n = cross_prod / norm_cross

Second Fundamental Form

L = ruu.dot(n) M = ruv.dot(n) N_coeff = rvv.dot(n) # N is taken by sympy numerical, using N_coeff

Gaussian Curvature K = (LN - M^2) / (EG - F^2)

K = (L * N_coeff - M2) / (E * G - F2)

Evaluate at u=0, v=0

Note: At u=0, sinh(u)=0, cosh(u)=1.

There might be a singularity in the formula if not simplified, specifically in n if cross_prod is 0.

Let's check cross_prod at u=0.

ru at 0,0: [11, 210, 30] = [1, 0, 0] ? Wait.

x_u = cosh u cos v. At 0,0: 1*1 = 1.

y_u = 2 cosh u sin v. At 0,0: 0.

z_u = 3 sinh u. At 0,0: 0.

ru(0,0) = [1, 0, 0]

rv at 0,0:

x_v = -sinh u sin v. At 0,0: 0.

y_v = 2 sinh u cos v. At 0,0: 0.

z_v = 0.

rv(0,0) = [0, 0, 0].

Problem: rv is zero vector at u=0. This is a coordinate singularity (pole).

The Gaussian curvature is well-defined at the vertex, but the coordinate chart is singular.

We can take the limit as u -> 0. Or use the implicit equation.

Implicit equation approach for curvature at vertex (0,0,3).

Equation: (x/1)^2 + (y/2)^2 - (z/3)^2 = -1 => -x^2 - y^2/4 + z^2/9 = 1

z = 3 * sqrt(1 + x^2 + y^2/4)

We can compute K using the Hessian of z(x,y) at x=0, y=0.

At peak/vertex, gradient is 0. K = det(Hessian).

z approx 3 * (1 + 1/2(x^2 + y^2/4)) = 3 + 3/2 x^2 + 3/8 y^2.

z_xx = 3, z_yy = 3/4, z_xy = 0.

K = z_xx * z_yy = 3 * 3/4 = 9/4?

Wait, formula for K of graph z=f(x,y) at stationary point is det(Hessian).

Let's verify with code using limit or implicit expansion.

Let's try limit of K expression as u -> 0.

K_simplified = sp.simplify(K) beta = sp.limit(sp.Abs(K_simplified), u, 0) # Limit as u->0 (v doesn't matter due to symmetry or check v=0)

print(f"beta (limit): {beta}")

Final Calculation

S = (alpha * beta * delta * rho)3 K_final = 999 * S2 + 7

print(f"S: {S}") print(f"K_final: {K_final}") [/PYTHON]

The length of the cycloid is $L = 8a$. The curvature of the cycloid at $t=\pi$ is $\kappa_L(\pi) = \frac{1}{4a}$. Thus, $\alpha = L \cdot \kappa_L(\pi) = 8a \cdot \frac{1}{4a} = 2$.

For the surface, the intercepts are $A_x = 4/3$, $A_y = 8/3$, and $A_z = 3$. Thus, $\delta = A_x A_y A_z = \frac{4}{3} \cdot \frac{8}{3} \cdot 3 = \frac{32}{3}$. The ratio is $\rho = \frac{A_y}{A_x} = \frac{8/3}{4/3} = 2$.

The Gaussian curvature at the vertex $(0,0)$ is $\beta = \frac{9}{4}$.

We compute $S$: [ S = (\alpha \beta \delta \rho)^3 = \left( 2 \cdot \frac{9}{4} \cdot \frac{32}{3} \cdot 2 \right)^3 = \left( \frac{9}{2} \cdot \frac{64}{3} \right)^3 = (3 \cdot 32)^3 = 96^3 = 884736. ]

Finally, we compute $\mathcal{K}$: [ \mathcal{K} = 999 S^2 + 7 = 999 (884736)^2 + 7. ] Calculating the value: [ 884736^2 = 782757789696 ] [ 999 \times 782757789696 = (1000 - 1) \times 782757789696 = 782757789696000 - 782757789696 = 781975031906304 ] [ \mathcal{K} = 781975031906304 + 7 = 781975031906311. ]

\boxed{781975031906311}