Modern physical theories tell us that space is not euclidean. Yet, to most people, euclidean geometry seems most natural. Furthermore, most mathematical theories of non-euclidean geometry refer back to euclidean geometry in one way or another. For example, in differential geometry, the differential areas are almost always approximately Euclidean. Why is this? I want to suggest here a reason roo...

In this paper, vision theory for Euclidean, spherical and hyperbolic spaces is studied in a uniform framework using diierential geometry in spaces of constant curvature. It is shown that the epipolar geometry for Euclidean space can be naturally generalized to the spaces of constant curvature. In particular, it is shown that, in the general case, the bilinear epipolar constraint is exactly the ...

After the introduction of coordinates, it became possible to treat figures in plane and space by analytical methods, and calculus has been the main means for the study of curved figures. For example, one attaches the tangent line to a curve at each point. One sees how tangent lines change with points of the curve and gets an invariant called the curvature. C. F. Gauss, with whom differential ge...

It is well established from both colour difference and colour order perpectives that the colour space cannot be Euclidean. In spite of this, most colour spaces still in use today are Euclidean, and the best Euclidean colour metrics are performing comparably to state-of-the-art non-Euclidean metrics. In this paper, it is shown that a transformation from Euclidean to hyperbolic geometry (i.e., co...