The Spherical SpiralBy

Chris WilsonAnd

Geoff Zelder

History

Pedro Nunes, a sixteenth century Portuguese cosmographer discovered that the shortest distance from point A to point B on a sphere is not a straight line, but an arc known as the great circle route.

Nunes gave early navigators two possible routes across open seas. One being the shortest route and the other being a route following a constant direction, generally about a 60 degree angle, in relation to the cardinal points known as the rhumb line or the loxodrome spiral.

Pedro Nunes 1502-1579

Loxodrome Spiral

M C Esher (1898-1972), known for his art in optical illusions drew the Bolspiralen spiral, which is the best representation of Nunes’ theory

Bolspiralen spiral1958

Mercator’s Projection

Gerardus Mercator (1512-1594), used Nunes’ loxodrome spiral which revolutionized the making of world mapsMap makers have to distort the geometry of the globe in order to reproduce a spherical surface on a flat surface

Plotting the spiral

In this case we let run from 0 to k , so the larger k is the more times the spiral will circumnavigate the sphere. We let , where controls the spacing of the spirals, and controls the closing of the top and bottom of the spiral.

The Spiral

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A few Applications• A spherical spiral display which rotates about a

vertical axis was proposed in the 60’s as a 3-D radar display. A small high intensity light beam is shot into mirrors in the center which control the azimuth and elevation. A fixed shutter with slits in it would control the number of targets that could be displayed at one time.

Another use is a high definition 3-D projection technique to produce many 2-D images in different directions so the image could be viewed from any angle, this creates a sort of fishbowl effect.

Some Fun with the Equation

• Here we let = 1, and . We let

• . We end up with a sort of 3D Clothiod type figure.

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• Here we let , and let .• We let . We end up with a

cylindrical helix.

• Here we let , and let We let . We end up with this.

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