how to compute the angle of tilt of ellipse or ellipsoid

Question

i need you help in solving the following problem:

given the "center form" of ellipsoid:

• A : matrix of d*d where d is the dimension
• c : center of the ellipsoid hold in a vector

how can i compute the angle of tilt of the ellipsoid from these two inputs in Matlab? And how can i compute the angle between 2 axes i.e. the axes of the ellipsoid and the axes of the coordinate system?

Thanks in advance.

Answer 1

Following this post of mine, the eigenvectors of your matrix $A$ will form a rotation matrix which turns a rotated ellipse into its axis-aligned (i.e. diagonal matrix) analogon, and vice versa. This generalizes to higher dimensions.

To read the angle from that matrix, follow this Wikipedia article on how to turn a rotation matrix into an axis and angle representation. Unfortunately, the formula you obtain from this doesn't generalize to other dimensions.