  The existence of kinematic formulas for area measures with respect to any
connected, closed subgroup of the orthogonal group acting transitively on the
unit sphere is established. In particular, the kinematic operator for area
measures is shown to have the structure of a co-product. In the case of the
unitary group the algebra associated to this co-product is described explicitly
in terms of generators and relations. As a consequence, a simple algorithm that
yields explicit kinematic formulas for unitary area measures is obtained.
