#loxodrome

[2257ad]

[shonk]

Going Nowhere

(source code)

[tresbizarre]

Splice the mainbrace! Tap the Admiral! 

[ronbeckdesigns]

LOXODROME Sculpture by Naum Gabo, 1916

[curiosamathematica]

When you perfectly peel an orange like this, the orange peel approximates a rhumb line or loxodrome. This curve on a sphere originates at the north pole, spirals outwards intersecting the meridians at equal angles, back into the south pole. In navigation, the loxodrome shows up as the path of a ship maintaining a constant compass direction.

[rntnls]

Experiments in Processing: loxo0.gif & loxo1.gif

A loxodrome on the sphere and its stereographic projection, a logarithmic spiral.

[visualizingmath]

Loxodromes (Rhumb Lines)

In navigation, a rhumb line (or loxodrome) is a line crossing all meridians of longitude at the same angle, i.e. a path derived from a defined initial bearing. That is, upon taking an initial bearing, one proceeds along the same bearing, without changing the direction as measured relative to true north. 

A rhumb line can be contrasted with a great circle, which is the path of shortest distance between two points on the surface of a sphere, but whose bearing is non-constant. If you were to drive a car along a great circle you would hold the steering wheel fixed, but to follow a rhumb line you would have to turn the wheel, turning it more sharply as the poles are approached.

All loxodromes spiral from one pole to the other. Near the poles, they are close to being logarithmic spirals (on a stereographic projection they are exactly) so they wind round each pole an infinite number of times but reach the pole in a finite distance. The pole-to-pole length of a loxodrome is (assuming a perfect sphere) the length of the meridian divided by the cosine of the bearing away from true north. Loxodromes are not defined at the poles.

Above: Loxodrome Sconces by (the amazing) Paul Nylander.

[futro]