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Yes, as whuber says, the gnomonic projection is the way to go. GeographicLib (written by me) includes an ellipsoidal generalization of this projection in which geodesics are very nearly staight. For example, if you consider two points 4000 km apart, the straight line connecting them in the gnomonic projection deviates from the geodesic by less than 30 m. ...


Do the computations using the Cartesian formulas in a Gnomonic projection. These projections cover a single hemisphere and map all portions of great circles to line segments. Presumably, in the representation of your spherical polygons as sequences of (lon,lat) values it is implicit that each successive pair of vertices is to be traveled along a geodesic. ...

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