John Donnelly,
A Model of Continuous Plane Geometry that is Nowhere Geodesic,
Forum Geometricorum, 18 (2018) 255-273.
Abstract. We construct a model M_1 of plane geometry that satisfies
all of Hilbert's axioms for the euclidean plane (with the exception of Sided-Angle-Side), yet in which the
geodesic line segment connecting any two points A and B is never the
shortest path from A to B.
Moreover, the model M_1 is
continuous in the sense that it satisfies both the Ruler Postulate and
Protractor Postulate from Birkhoff's set of axioms
for the euclidean plane.
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