Wave grooves on (ring) mesh

For straight grooves, I use the Bevel tool, and some simple extruding and scaling operations. However, I also want to place an oblique line on a ring (so seperate the ring into two piece for a 2 coloured ring). Furthermore, not only straight but also wave-shaped grooves as follows. (from the wedding ring front view)

wave

I want to ask for help on how to place an oblique line and wave-shaped grooves on ring.

Thank you in advance for your help!

can you show the whole ring !

and is this for printing ?
do you need a 3D mesh or can be UV map ?

happy bl

Yes, It will be a 3D mesh.

I work with several different cross-sections, which I use to form a ring using a circle, then I convert into a mesh.

I would be interested in exactly above subdivisions (in other view this two type), so that I will be able to texture the two parts differently.

WavingShape

I want to ask for help on how can I get the ring to be divided into 2 parts obliquely or wavy.

I attach the blender file.
https://drive.google.com/file/d/1G0PIXwi9c-vDuCI0jOOKCLs16CEkB75l/view?usp=sharing

cut as a plane cut or some weird shape ?

you can use the bisect tool to cut and have the cross section is filled up

happy bl