Refined algebraic domains with finite sets in the boundaries respecting differential geometry

We are interested in shapes of real algebraic curves in the plane and regions surrounded by them: they are named refined algebraic domains by the author. As characteristic finite sets, we consider points contained in two curves and the sets of singular points of the restrictions of the projections to the lines to the curves. As a new case, we respect differential geometry and consider inflection points and points of some double tangent lines of a single connected curve. We prove fundamental properties and investigate some examples. We have also previously considered the cases where the curves are straight lines, circles, or boundaries of ellipsoids for example. Such simple cases are trivial in our new consideration.