Split surface

Is there any other way to split the surface by curve on it other than the way on the image?Capture

extrude that curve and use split with brep component to split the surface with the extrusion

alternatively if you want to fake the split, and if resolution (of your surface) isn’t compromised, then you may split the boundary (curve) of the surface with the splitting curve (on the surface), use the two halves of the surface boundary plus the splitting curve to get 4 edges for each half to construct two edge surfaces