# By Chris Wilson And Geoff Zelder

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By Chris Wilson And Geoff Zelder
The Spherical Spiral By Chris Wilson And 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

Loxodrome Spiral M C Esher ( ), known for his art in optical illusions drew the Bolspiralen spiral, which is the best representation of Nunes’ theory Bolspiralen spiral 1958

Mercator’s Projection
Gerardus Mercator ( ), used Nunes’ loxodrome spiral which revolutionized the making of world maps Map 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

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.

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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