This Demonstration shows a superposition of two sets of mesh functions inside a three-dimensional parametric plot of a sphere. Möbius transformations are interesting because they let us perform conformal mappings with dilation, rotation, reflection and inversion [1, 2]. You can also use Möbius transformations to visualize non-Euclidian axioms [3] and generate loxodromic spirals [3, 4]. The sphere in this Demonstration is parameterized using latitude and longitude ( and , respectively). The dashed purple and cyan lines are contours, and the solid magenta and blue lines are contours of the Möbius transformation of with variable offsets. The mesh offset values control the relative dominance of circles, lobes and spirals.

A general Möbius transformation of and using constants and has the form:

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Because this Demonstration uses the identity function , the Möbius transformation simplifies to:

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For small, nonzero offsets and , the resulting pole-to-pole spiral shape on this sphere can also be called a rumb line. These spirals have been known for centuries and arise in Mercator projections and maritime navigation.

References

[1] T. J. Osler and S. P. Waterpeace, Conformal Mapping and Its Applications, independently published, 2019.

[2] E. Kreyszig, Advanced Engineering Mathematics, 5th ed., New York: Wiley, 1983.

[3] M. Harvey, Geometry Illuminated: An Illustrated Introduction to Euclidean and Hyperbolic Plane Geometry, Washington, DC: Mathematical Association of America, 2015.

[4] D. Mumford, C. Series and D. Wright, Indra's Pearls: The Vision of Felix Klein, New York: Cambridge University Press, 2002.