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Transitional geometry

We develop a transitional geometry, that is, a family of geometries of constant curvatures which makes a continuous connec-tion between the hyperbolic, Euclidean and spherical geometries. In this transitional setting, several geometric entities like points, lines, dis-tances, triangles, angles, area, curvature, etc. as well as trigonometric formulae and other properties transit in a continuous manner from one geometry to another. AMS classification: 01-99 ; 53-02 ; 53-03 ; 53A35.