  In this work, we solve the problem of finding non-intersecting paths between
points on a plane with a new approach by borrowing ideas from geometric
topology, in particular, from the study of polygonal schema in mathematics. We
use a topological transformation on the 2-dimensional planar routing
environment that simplifies the routing problem into a problem of connecting
points on a circle with straight line segments that do not intersect in the
interior of the circle. These points are either the points that need to be
connected by non-intersecting paths or special `reference' points that
parametrize the topology of the original environment prior to the
transformation. When all the necessary points on the circle are fully
connected, the transformation is reversed such that the line segments combine
to become the non-intersecting paths that connect the start and end points in
the original environment. We interpret the transformed environment in which the
routing problem is solved as a new data structure where any routing problem can
be solved efficiently. We perform experiments and show that the routing time
and success rate of the new routing algorithm outperforms the ones for the
A*-algorithm.
