Doing a cut around a curved surface without taking off parts of the feature

I am currently trying to make a cut around a curved surface.  I want the generic shape shown in the first image to go all the way around without losing any of the material towards to the top of the part. I need the curve to start and be flush with the top of the part like the picture shown, but it always cuts off segments of the parts that need to be remained. Any advice? I do not want it to look like the picture with the black circle. I need the top of the cut to be with the top surface of the part the whole way around.

SolidworksParts And Features