Thoughts on a Stirling Engine

It has occurred to me that in the thermodynamic description of the stirling cycle that there is no mention of sin or cos functions. Why is it then that every stirling engine I have ever come across has a cyclicly connected motion ?

It is on the face of it a very convenient way of moving the pistons of the larger and smaller cylinders in relation to one another with a close adherence to the thermodynamic path of the cycle shown below.