Move a point(world coordinate) into a new coordinatesystem?

I'm experimenting with making a vonoroi diagram according  to a line's intersection of a surface. The vonoroi diagram is applyed to the surface.

When get the points of the intersection of the surfaces, I get the world coordinates. I want the coordinated to be defined according to each surface's new coordinate system.

I have played aroound with the plane CP component, but can't really get ot wotk. The points values are wrong, and the UV values seem to be correct but the vonoroi asks for coordinate points.

Any surgestions how I solve this problem.