Trisecting angles in an alternate topology

[Loren Booda]

Is there any topology where it is possible to trisect an angle using only straight lines and circles?

 

[HallsofIvy]

In other topologies, what do you mean by "straight lines" and "circles"? And how do measure angles?

 

[Loren Booda]

Lines are curves such that between every pair of points on the line, the segment there is a minimal geodesic; circles are a set of points all equidistant on lines from a center point; angles are measured regarding the curvature from local triangles.

Changing the problem slightly, all of these constructs are assumed to be on an arbitrary two-dimensional manifold. Is there any such topology where it is possible to trisect an angle using only lines and circles?