Family of Plane Curves in the Extended Gauss Plane Generated by One Function
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 =f(σ), 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.
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