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**HallsofIvy** "Spherical geometry" is the geometry you get if you take "point" to be a point on the surface of a given sphere, "line" to be a great circle on that sphere. Spherical geometries have the property that all lines intersect- there are no "parallel" lines. "Differential geometry" can be the geometry on any surface- the important thing is that we be able to use Calculus to fid such things a tangent lines or planes to the surface. "Axiomatic Geometry" is any form of geometry in which we are given "axioms" that are used to prove theorems in the geometry.

I have no idea why you would think there would be a particular geometry for "crop circles"! Since crop circles occur on portions of the surface of the earth, small enough that the curvature of the earth, "Euclidean Geometry" would be appropriate. And I do not know what you mean by "artistic geometry".