Consider the following interpretation of a geometry. Begin with a punctured sphere in Euclidean 3-space. This is a sphere with one point, P, removed, where everything else about the sphere looks normal. Let points be points in the normal sense on the surface of the punctured sphere. Let straight lines be defined as circles on the surface of the sphere that pass through the point P (note: these are the only infinite straight lines for this infinite geometric model). Is this infinite model an incidence geometry? If so, does Playfair's Axiom hold in this model? Why or why not. Is this model Isomorphic to any other geometric models we know? (hint: a punctured sphere is often transformed by stereographic projection into a familar shape that is easier to work with)
Consider the following interpretation of a geometry. Begin with a punctured sphere in Euclidean 3-space. This is a sphere with one point, P, removed, where everything else about the sphere looks normal. Let points be points in the normal sense on the surface of the punctured sphere. Let straight lines be defined as circles on the surface of the sphere that pass through the point P (note: these are the only infinite straight lines for this infinite geometric model). Is this infinite model an incidence geometry? If so, does Playfair's Axiom hold in this model? Why or why not. Is this model Isomorphic to any other geometric models we know? (hint: a punctured sphere is often transformed by stereographic projection into a familar shape that is easier to work with)
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