PolySkelion script

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 From:  bemfarmer
6936.1 
After a lot of modification to the math, and conversion to Moi type script, and adapting a .js version to the methods of Max Smirnov in his Gear script,
here is the PolySkelion script.
PolySkelions may have as few as 2 tentacles.
There are slender and chubby versions, depending upon the "relax" slider.
The spirals are Archimedean spirals, including Archimedes with exponent 1, and Fermat with exponent 0.5.
The negative exponents did not work, and so are not included.

A new PolySkelion may be evolving, with Galactic spiral arms, if the math works.

- Brian

EDITED: 24 Sep 2014 by BEMFARMER

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 From:  Mike (MGG942)
6936.2 In reply to 6936.1 
Cool!
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 From:  bemfarmer
6936.3 In reply to 6936.1 
Here is my FLAWED galactic skelion script.
Something is wrong with the math, or ?
The script can be used to make the spirals needed, but they must be manipulated a lot.

After forming the N spirals, discard all but the two at 1 o'clock and 3 o'clock.

Move the lower tail of 1 to touch the upper tail of 3. Draw 2 tangent lines at their intersection.
Rotate 1, centered on intersection using its tangent line as a "handle," to overlay the tangents.
This makes a small angle at the tentacle tip, which can be blended away later.
This makes spiral 1 and spiral 2 tangent at the intersection.
Run separate, and discard the upper tail of 1, and the lower tail of 3.
The red curve below is the result.
Create an N-gon, using side version, with the centers of spiral 1 and spiral 2, to yield its center.
The remaining two "half" spiral curves, from spiral center to spiral center may be joined, and circular arrayed by N,
using the center of the N-gon as the circular array center. The blue spirals are the result.
N is seven for the attached picture.
Note the points inside the teal heptagon are the intersection points of spirals 1 and 3, and they are not on the teal heptagon.

- Brian




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 From:  Max Smirnov (SMIRNOV)
6936.4 In reply to 6936.3 
Wow! Looks great :)

P.S.
code:
Line 38: lastN = N.value; lastS = S.value; lastWD = WD.value; lastW = W.value; lastP = P.value; lastX === X.value;
                                                                                                      ^^^
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 From:  bemfarmer
6936.5 In reply to 6936.4 
Thanks Max

H'mm, I'd blame === on the cat, but he does not type that well...

- Brian
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 From:  bemfarmer
6936.6 
A closer look at the join point of the ends of the spiral arms shows a slight tangent mismatch, often less than 2 degrees,
which makes sense, since the join point is not the midpoint between the centers.
You can see the slight irregularity just by looking at the curves.
A comparison between tangents for 100 points versus 1000 points shows an extremely slight difference made by point count.
The 100 point tangent difference was even very slightly better. So the difference is mostly due to the difference in the radii of the spirals.
The Galaxy spirals tangent mismatch just shows up more due to their greater "pitch" change.
The tangents can be manually matched as described above, with the compensation occurring in the center of the spiral.

Taking the derivative of the curves at the end of the spiral arms should enable the compensation to be done by script.
The degrees to rotate is the difference in the arctangents of the tangents.
The derivative of the Galaxy spiral has been calculated.

- Brian

EDITED: 27 Sep 2014 by BEMFARMER


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