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The two-dimensional unit sphere , centered at the origin, is a model of the Gauss plane together with a point at infinity. This model is realized via a stereographic projection from the pole onto the plane through the equator. The group of rigid motions on coincides with the group of rotations that preserves . Any spherical curve on , parameterized by an arc length parameter , is defined up to a rigid motion on by a function called spherical curvature. The group of rigid motions on induces on the plane via the stereographic projection a subgroup of the Möbius group. The transformations of this subgroup are represented by the functions , , , where is the field of complex numbers. A relation between the spherical curvature of a curve on and the Euclidean curvature of its corresponding plane curve is an invariant under the group and determines any plane curve up to transformation from the group . For example, in snapshot 3, the function defines plane curves equivalent to the logarithmic spiral. Since the natural Cesàro equation for the curvature defines a Cornu's spiral in the Euclidean plane, the curve in snapshot 1 could be called a Cornu's spiral in the Möbius plane with a group of rigid motion .