Identify an arbitrary elliptical cylinder with $8$ points

If $r = (x, y, z)$ then the equation of the cylinder is given by,

$(r - r0) Q (r - r0) = 1$

where $Q = R D R^T$

and the diagonal matrix $D$ is given by

$D = \text{diag} \{ \dfrac{1}{a^2}, \dfrac{1}{b^2} , 0 \}$

Here, $a$ and $b$ are the semi-major and semi-minor axes lengths of the elliptical cylinder cross section, and point $r_0 = (x_0, y_0, 0)$ . The matrix $R =[ v, w , u ]$ , where $u$ is the vector along the axis of the cylinder, and $v$ , $w$ are unit vectors along the major and minor axes of the cylinder.

$u$ can be parameterized using two angles $\theta$ and $\phi$ , as follows:

$u = ( \sin \theta \cos \phi , \sin \theta \sin \phi, \cos \theta )$

Two unit vectors orthogonal to $u$ , and to each other, are

$u_1 = ( \cos \theta \cos \phi, \cos \theta \sin \phi, - \sin \theta )$

and

$u_2 = ( - sin \phi, \cos \phi , 0 )$

hence, vectors $v$ and $w$ can be expressed as

$v = cos \psi u_1 + \sin \psi u_2$

$w = - \sin \psi u_1 + \cos \psi u_2$

Our parameter vector is therefore 7-dimensional: $\mathbf{x} = (\theta, \phi, \psi , a, b , x_0, y_0)$

And for $i = 1,2,..., 8$ , we want

$f_i = (r_i - r_0)^T Q (r_i - r0) - 1 = 0$

One possible numerical method to solve these equations is the Newton-Raphson multivariate method that requires the computation of the

Jacobian of the function vector f. This can be computed exactly by explicit analytical differentiation of $f_i$ , or approximately by numerical differentiation. An initial guess of the parameter vector $\mathbf{x}_0$ is made, and iteratively new approximations of the solution are obtained by the Newton-Raphson

multivariate recursive formula $\mathbf{x}_{k+1} = \mathbf{x}_k - J_k^{-1} \mathbf{f}_k$

Note that since the dimension of the unknowns vector is $7$ , we have to limit the number of equations to $7$ , or otherwise use the Penrose inverse $J^\#$ of matrix $J$ instead of the regular inverse; the formula for the Penrose (left) inverse is $J^\# = (J^T J)^{-1} J^T$ .

For this particular problem, it might be difficult to come up with a good initial guess, so an external loop to set the initial values for some of the parameters (for example, $\theta$ and $\phi$ is necessary. In my implementation, I used an external double loop to set the initial guess of $\theta_0$ and $\phi_0$ , so that I could guarantee the convergence of the Newton-Raphson iteration.

After a few initial guesses, the Newton-Raphson method converged to $\mathbf{x}^* = ( 0.785398163 ,5.235987756 , 2.094395102 , 7.5 , 3.5 , 6 , 3)$

So that the rotation matrix is given by

$R = [v, w, u ] = \begin{pmatrix} 0.573223305 && -0.73919892 && 0.353553391 \\ 0.73919892 && 0.280330086 && -0.612372436 \\ 0.353553391 && 0.612372436 && 0.707106781 \end{pmatrix}$

Thus, the axis unit vector is $u = (0.353553391, -0.612372436 , 0.707106781 )$

And the unit vector along the major semi-axis is $v = (0.573223305 , 0.73919892, 0.353553391 )$

Therefore, S = 23.33900822, making the answer $\lfloor 10^5 \times 23.33900822 \rfloor = \boxed{2333900}$