Family of Plane Curves in the Extended Gauss Plane Generated by One Function
This Demonstration shows four families of curves, each member of which satisfies an equation of the form , where, at any point on the parametrized curve, represents the arc length to that point and represents the spherical curvature at that point when the curve is projected onto the sphere via stereographic projection. If a curve on the sphere can be obtained from another via rotation of the sphere, then the two plane curves that are their stereographic projections are in the same family. The polar coordinates of an arbitrary point on the sphere define an axis of rotation; use that and an angle of rotation to get different curves in a family.
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 .