The spherical spiral
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The Spherical Spiral. By Chris W ilson 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.

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The Spherical Spiral

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The spherical spiral

The Spherical Spiral

By

Chris Wilson

And

Geoff Zelder


History

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

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 spiral

1958


Mercator s projection

Mercator’s Projection

Gerardus Mercator (1512-1594), 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


The spherical spiral

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

The Spiral


A few applications

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.


The spherical spiral

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

Some Fun with the Equation

  • Here we let = 1, and . We let

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


The spherical spiral

  • Here we let , and let .

  • We let . We end up with a cylindrical helix.


The spherical spiral

  • Here we let , and let

    We let . We end up with this.


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