R - Interpolate between lines that are not vertical or horizontal oriented

Question

I am looking for a way to interpolate between lines in a raster. I have a raster file with x and y in GK coordinates and a value for each point on the line: Plot with lines to interpolate between (I am not yet allowed to add pictures)

Is there a function or a way that allows to only interpolate between the lines? Using interpolate from the Field package does an interpolation everywhere in the raster space, which is not useful for me.

The aim is to have a filled contour between the lines. So something like this: Filled contour between two vertical lines. I am able to do that for a vertical or horizontal combination of lines. But not for any other angles.