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 L1 (the vertical blue axis) or L3 (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: