Hint: What is the formula for the arc-length of the portion of the circle that goes from start point to point N?
A geometric "construction"? chiro told you how to find the distance from the current point of contact to the point where N contacts the surface. That was what your orginal post asked. If you want a construction to give that point, put one point of compasses at the current point of contact, the other at N and strike an arc to cross the line.