Euclidean, Spherical, and Projective Geometry

Spherical geometry is very tangible because we live on a sphere. Projective geometry seems much more exotic, and yet they’re closely related. To show how they’re connected, let’s go back to the origins of non-Euclidean geometry. Euclidean geometry Here are Euclid’s five postulates, statements about what geometric objects can be constructred. To draw a straight line from any point to any point. To produce (extend) a finite straight line continuously in a straight line. To describe a circle with any centre and distance (radius). That all right angles are equal to one another. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. Clearly there’s something different about the fifth postulate, the parallel postulate. It’s much longer and more complicated than the other postulates. There was…

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