The Thompson coupling is a relatively recent design of constant-velocity coupling, that is, principally based on the double Cardan mechanism. An extra mechanism comprising a spherical pantograph serves to align the intermediate shaft of this coupling and so maintains the constant velocity of the double Cardan mechanism, in a modular fashion. This technical note serves to introduce basic closed form expressions for the coupling’s geometry—which may then be used to derive linkage accelerations and dynamic forces. The expressions are derived using standard identities in spherical geometry. The resulting dynamic model then informs a basic conceptual design optimization, which object is intended to reduce induced driveline vibrations, when the coupling is articulated at nonzero angles of torque transmission.