How to represent a roto-translating frame in a 3D space?

Good mornign everyone,

I need to represent in my code a rotating and translating frame {C} with respect to a fixed frame {W}.

The whole movement take place inside a 3D room with 4 walls.

I don't understand if I need to compute the homogeneus matrix that links {C} with {W} with variables parameters and, if so, how to do it.

Hoping that my question is not too general, I thank for the attention reserved me.