Loxodrome script for Sphere

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 From:  bemfarmer
8281.1 
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

EDITED: 26 Jan by BEMFARMER

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 From:  bemfarmer
8281.2 In reply to 8281.1 
For matching parameter values, where Loxodrome01 "Shift" corresponds to LogSpiral4 "Scale,"
the Loxodrome spiral is the inverse stereographic projection of the LogSpiral4 curve.
For Scale > 1, the planar log spiral expands, and the upper loxodrome shrinks toward the north pole.
Not sure how the bottom curve relates to the planar log spiral.

Loxodrome Sweep of a circle:

- Brian

EDITED: 28 Jan by BEMFARMER


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 From:  Michael Gibson
8281.3 In reply to 8281.2 
Reminds me of a bighorn sheep!

- Michael
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 From:  gunter511
8281.4 
Guys, I installed and tried the Loxodrom script and managed to get the top port of the spiral to connect to the vertical line but can't get the bottom part to do the same. I've played around with the parameters like whorls, etc. but nothing moves the bottom part. Any suggestions?





Thanks again!
Gunter

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 From:  bemfarmer
8281.5 In reply to 8281.4 
Hi Gunter,

My previous reply got zonked by windows internet connection and work distraction...

I do not think that I understand your question.

The loxodrome curve on the sphere is asymptotic with respect to the North and South poles.
It would take an infinite number of whorls to reach the poles.

The _loxodrome01 script can be run a second time with Top unchecked, to generate the bottom half, Southern Hemisphere curve.
The bottom half could be scaled in the top view, to meet the South pole, with the bottom curve still tangent to the top curve (per test),
but the bottom curve would no longer be a loxodrome.

- Brian
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 From:  Michael Gibson
8281.6 In reply to 8281.4 
Hi Gunter, another way to make a spherical spiral is to take an arc and run the Transform > Deform > Twist command on it, see here:
http://moi3d.com/forum/index.php?webtag=MOI&msg=8144.1

- Michael
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 From:  gunter511
8281.7 In reply to 8281.6 
Hi Michael,

That worked, yay! And super easy.

I think the problem was I wasn't twisting enough and so It looked like the spiral was on one side but with a much greater angle of twist it's in the middle.

Many thanks as always!
Gunter
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