Presentation on theme: "By Chris Wilson And Geoff Zelder"— Presentation transcript:
1By Chris Wilson And Geoff Zelder The Spherical SpiralByChris WilsonAndGeoff Zelder
2HistoryPedro 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
3Loxodrome SpiralM C Esher ( ), known for his art in optical illusions drew the Bolspiralen spiral, which is the best representation of Nunes’ theoryBolspiralen spiral1958
4Mercator’s Projection Gerardus Mercator ( ), 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
5Plotting the spiralIn 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.
7A few ApplicationsA 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.
8Another 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.
9Some Fun with the Equation Here we let = 1, and We let. We end up with a sort of 3D Clothiod type figure.
10Here we let , and letWe let We end up with a cylindrical helix.
11Here we let , and letWe let We end up with this.