A planar cut through an Elliptical Cylinder

The equation of an elliptical cylinder, can be written in vector-matrix form as

$r^T Q r = 1$

where $r = [x, y, z]^T$ , and

$Q = \begin{bmatrix} 1/a^2 && 0 && 0 \\ 0 && 1/b^2 && 0 \\ 0 && 0 && 0 \end{bmatrix}$

And the cutting plane vector equation is $r = r_0 + V u$ , where $V = [ v_1, v_2 ]$ is a 3 x 2 matrix, whose columns $v_1$ and $v_2$ are two unit vectors orthogonal to each other and to the normal to the plane.

Therefore,

$(r_0 + Vu)^T Q (r_0 + Vu ) = 1$

$u^T V^T Q V u + 2 r_0^T Q V u = 1 - r_0^T Q r_0$

$(u - u_0)^T V^T Q V (u - u_0) = c$

where $u_0 = - (V^TQ V)^{-1} V^T Q r_0$ and $c = 1 - r_0^T Q r_0 + u_0^T V^T Q V u_0$

Let $B = (1/c) V^T Q V$ , then

$(u - u_0)^T B (u - u_0) = 1$

Next we diagonalize matrix B, by finding a rotation matrix R , and a diagonal matrix D, such that

$R^T B R = D$

hence, if we make the change of coordinate, $u = R w + u_0$ , then

$w^T D w = 1$

which is the well-known equation of an ellipse, given that the diagonal entries of D are positive, which is the case.

Hence the ellipse in the w-plane, and consequently in the u-plane, has semi-major and semi-minor axes equal to the reciprocal of the square roots of the diagonal entries of D.

Performing the calculations, results in a semi-minor axis $m = 10.6066$ and a semi-major axis $M = 16.32993$ , making the answer $M + m =\boxed{ 26.937}$ .