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

.