The Binet ellipsoid occurs in the description of the free rotation of an asymetrical rigid body. The curves trace the path of the angular momentum vector as seen from a frame of reference fixed in the body:

The curves are given by the intersection of the two surfaces

\[ \frac{L_1^2}{2I_1} + \frac{L_2^2}{2I_2} + \frac{L_3^2}{2I_3} = T \hspace{5em} L_1^2 + L_2^2 + L_3^2 = L^2 \]with the conventions and restrictions

\[ I_1 < I_2 < I_3 \hspace{5em} 2TI_1 < L^2 < 2TI_3 \]The curves are thus the intersection of an angular momentum sphere with an energy ellipsoid. The equation for the sphere is used to eliminate either *L*_{1} (the vertical blue axis) or *L*_{3} (the left red axis) from the equation of the ellipsoid. The resulting ellipse is then parametrized by modifying a circle with appropriate scalings.

Complete code for this example: