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
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 spiral
1958
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
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
1
z-axis
0.5
0
-0.5
-1
1
0.5
1
0.5
0
0
-0.5
y-axis
-0.5
-1
-1
x-axis
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
1
z-axis
0.5
0
-0.5
-1
2
1
1
0
0
-1
-1
y-axis
• Here we let
•
type figure.
-2
-2
x-axis
= 1, and
. We let
. We end up with a sort of 3D Clothiod
-3
x 10
8
z-axis
6
4
2
0
1
0.5
1
0.5
0
0
-0.5
y-axis
• Here we let
• We let
cylindrical helix.
-0.5
-1
-1
x-axis
, and let
.
. We end up with a
1
z-axis
0.5
0
-0.5
-1
1
0.5
2
1.5
0
1
0.5
-0.5
y-axis
• Here we let
We let
-1
0
-0.5
x-axis
, and let
. We end up with this.