Loxodromic Möbius Mesh inside a Sphere

{view 1,-1}

{view 2,-1}

{view 3,0.165}

{t offset,0.004}

{s offset,0}

{plot points,59}

{mesh t,t-like,8}

{mesh s,s-like,8}

{t start,-2π}

{t opacity,1.}

{s opacity,1.}

{t-like opacity,1.}

{s-like opacity,1.}

{plot opacity,0.4}

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

{t,s}

contours, and the solid magenta and blue lines are contours of the Möbius transformation of

{t,s}

with variable offsets. The mesh offset values control the relative dominance of circles, lobes and spirals.

Details

A general Möbius transformation of

and

using constants

and

has the form:

f(t+si)-(a+bi)

1-f(t+si)×(a-bi)

Because this Demonstration uses the identity function

f(t+si)=t+si

, the Möbius transformation simplifies to:

(s+ti)-(a+bi)

1-(s+ti)×(a-bi)

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.

Permanent Citation

Alexandra L. Brosius

"Loxodromic Möbius Mesh inside a Sphere"
http://demonstrations.wolfram.com/LoxodromicMoebiusMeshInsideASphere/
Wolfram Demonstrations Project
Published: October 9, 2024