Are Two Ellipsoids in Contact? Algebraic Separation Condition for Ellipsoids

An algebraic expression for characterizing the 3-D spatial configurations formed by two ellipsoids.

A proximity query that is expressed as an algebraic condition for realtime
continuous contact detection for ellipsoids moving under rigid body transformations.
The algebraic condition is a quartic polynomial equation, also named as separation condition
or characteristic equation, which relates the geometric parameters of shape, spatial orientation,
and position of two ellipsoids. Depending on the sign of all four roots, it is possible to
determine the contact status. The resolution of the characteristic equation is straightforward,
leading to a simple and yet efficient algorithm for contact detection of ellipsoidal bodies that
computes the exact time interval of contact.

References:
Wang, W., Wang, J., Kim, M.-S.
An algebraic condition for the separation of two ellipsoids.
Computer Aided Geometric Design,
18(6):531–539, 2001.

Jia, X., Choi, Y.-K., Mourrain, B., Wang, W.
An algebraic approach to continuous collision detection for ellipsoids.
Computer Aided Geometric Design,
28:164–176, 2011.