For the case of a 3D rigid body to which any transformation in may be applied, the same general principles apply. The quaternion parameterization once again becomes the right way to represent because using (4.20) avoids all trigonometric functions in the same way that (4.18) avoided them for . Unfortunately, (4.20) is not linear in the configuration variables, as it was for (4.18), but it is at least polynomial. This enables semi-algebraic models to be formed for . Type FV, VF, and EE contacts arise for the case. From all of the contact conditions, polynomials that correspond to each patch of can be made. These patches are polynomials in seven variables: , , , , , , and . Once again, a special primitive must be intersected with all others; here, it enforces the constraint that unit quaternions are used. This reduces the dimension from back down to . Also, constraints should be added to throw away half of , which is redundant because of the identification of antipodal points on .
Steven M LaValle 2012-04-20