Spherical Spiral
(3 · cos(u/2 − π/2) · cos(8u + t), 3 · sin(u/2 − π/2), 3 · cos(u/2 − π/2) · sin(8u + t))
Open in the app The dials and keys named below are the app's.
What it draws
Call u/2 − π/2 the latitude. The second expression, 3sin(u/2 − π/2), is the height, and the distance from the vertical axis is 3cos(u/2 − π/2). That distance is split into x and z by the angle 8u + t, the longitude. Squared and added, the three coordinates come to 9 at every u: the curve lies exactly on a sphere of radius 3.
At Turns of u 1 the latitude runs from −π/2 to π/2, pole to pole once, while the longitude turns eight times. That is eight loops, from the bottom of the sphere to the top. They crowd near the poles, because a circle of latitude is only 2π·3cos of the latitude around: 18.85 at the equator, nothing at a pole. The clock adds to the longitude and to nothing else. So the whole spiral turns rigidly about the vertical axis, a full turn every 2π ≈ 6.28 seconds at Speed 1.
Longitude against latitude
Since the latitude is u/2 − π/2, the longitude 8u + t is 16 times the latitude plus a constant. Longitude and latitude therefore advance in fixed proportion. A curve on a sphere with that property is a Clelia curve, and one whose ratio is 2 or more is called a spherical spiral. It is not the rhumb line, or loxodrome, which crosses every meridian at the same angle. A rhumb line winds around a pole infinitely often and never arrives, while this curve reaches both poles after its eight loops.
History
The Italian mathematician Guido Grandi named these curves clelie in a book of 1728. He dedicated it to Clelia Grillo Borromeo, a Genoese noblewoman and mathematician who had founded her own academy nine years before.
Try
- Turns of u to 2: the latitude carries on past the north pole and back down, so a second set of eight loops returns to the south pole and the curve closes on itself.
- Open the winding out. Change both 8u to 4u: four loops instead of eight, and the sphere shows through.
- Top: from above, the distance from the axis swells from nothing at the south pole to 3 at the equator and shrinks back, so the loops read as a flat spiral out and in again.
- Turn on trace: the pen climbs the whole way from the south pole to the north, one loop at a time.