Notes on Loxodrome script for a sphere
The loxodrome curve, on the surface of the earth, is known as a rhumb line, and such curves are still relevant to navigation by ship or airplane.
Note that Karsten has spoken of doing a loxodrome Node.
This spherical _Loxodrome script forms a curve on the surface of a sphere, by means of parametric stereographic mapping, as defined by Alfred Gray,
"Modern Differential Geometry of Curves and Surfaces with Mathematica," Third Edition. Chapter 8, Construction of Space Curves, AG8.pdf, page 250-251, q.v..
The general parametric stereographic map function is defined from the plane, Rsquared, onto the sphere, Ssquared. (Would a 2D slider be useful?)
(The mapping may be useable for other planar curves?)
"The spherical loxodrome is the image under a stereographic map of a logarithmic spiral." The parametric planar logarithmic spiral, as per the LogSpiral4 MoI script,
is "a * pow(b*t)", where "a" and "b" are parameters. The loxodrome parametric equation is on page 251.
Note that there are several other equations for loxodromes. This may be confusing, and the equations could use some reconciliation.
"Loxodromes exist for all surfaces of revolution."
For the same handedness (CW or CCW), run the script twice, unchecking Top to yield a Bottom curve, so as to get a pair of curves,
a "north" curve and a "south" curve, which can be joined.
Parameter "a" is called "Shift" in this script, and changes the position of the curve(s) with respect to the poles, and causes shrinkage or expansion
of the lengths of the curve(s). The default value of "a = 1.0" causes top and bottom curves to be of equal lenth, and terminate at the equator.
For other values of "a", top and bottom curves of the same handedness (CW or CCW), will be of different lengths, and joinable either north or south of the equator.
Parameter "b" is called "growth" in this script, and alters the rhumb line compass angle.
Parameter "Whorls" increases the number of turns, inwards toward the north, or south, poles, which are asymptotic points.
For artistic creations, such as trimming a sphere, connecting circular arrays of loxodromes near the poles may be needed,
such as with segments of a circle, or blends. For a large number of Whorls, the Point Count should be increased, to maintain proper curvature.
UnChecking the "Clockwise" checkbox results in a "left handed" spiral.
UnChecking the "Top" checkbox results in a curve in the Southern hemisphere.
(Scripting to get CW/CCW and Top/Bottom to work together, so that the top and bottom curves would be the same handedness, was done mostly by trial and error. )
- Brian
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