Full Version: Doyle spirals, Doyle flower

Show messages:  1-19  20-24

Hi James,

Thank you for the nice style by circle size version.

When time permits, I'll add some style to the Doyle Spiral node, by arm, or position in arm.
Or gradients.
Adding an indexByArm output, and an indexByPositionInArm output might suffice?

There are two arm possibilities, "rows" or "columns" corresponding to p and q.

Maybe even rotation animation.

MoI scaling would move the two attractors apart.
Note that circle C of the Doyle cell is mapped and relocated from center (1,0) to center (0,0), with mapped A and mapped B circles still tangent.
For a different moebius transformation separation of the attractors, the formulas would have to be recalculated. I think this would be a scaling as well(?).

- Brian

I'll have to try sphere replacement. Cannot remember if such a script was written yet.

p.s. It seems that one circle near -1 does not get colored?

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The create rainbow function may have a "bug" which causes the White/Black 4th circle in to not have "blue" shade of style(???)

I modified the for loop to i <= 255, instead of i<255, and now the 4th circle has a "blue" style, BUT it is too intense.
Maybe the sigmoid equations need to slightly modified???
Due to my ignorance of the sigmoid equations, whether or not this is a bug is unclear.

Karsten's (?) create rainbow function is located in nodeeditor extension/libs folder, inside basicfunctions.js.
The function is called by setStyle node.

- Brian

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For the color adding nodes, Switching the constant 255,0 to 0,255 results in the irregular 4th circle to be orange, but too dark a shade, and the two large circles on the far right to be white/black. Adding the <= to the rainbow for loop colors the two large circles to blue shade.
So I conclude that there are two problems.
1. add the <= to fix blue colors.
2. something is wrong with the way the 4th circle is measured and/or turned into a style index. Maybe related to number of digits??

I wonder if the new circumference availability for MoI 4 could be used for indexing?
May try some of the recent Gradients code.

- Brian

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Deriving the moebius transformation that maps 3 points to 3 other points
is explained here(section 3.4.11 Building a mobius transformation):
https://mphitchman.com/geometry/section3-4.html
Using equated cross ratios. (handling infinity is also explained.)

For turning the moebius transformation formula into a formula involving x and y,
see the 4th answer here:
https://math.stackexchange.com/questions/36542/real-and-imaginary-parts-of-the-m%C3%B6bius-transformation
Uses a, b, c, d, (real numbers with imaginary parts a2= b2= c2 = d2 = 0) ,
and alpha, beta, gamma, and delta.

After about 8 pages of tedius hand calculation with many corrected mistakes,
the moebius transformation and the x and y formulas were calculated, for the current mapping of the attractor points to -1 and +1, and also to
-2 and +2 . The simple expected result is that this scaling by 2 is equivalent to multiplying the x and y formulas by 2. So a scaling input could be added to the double doyle. Or the scaling could be done after forming the double doyle, by MoI scaling.

The original moebius transformation is V(z) = (z-1)/(z+1).
The 2 times scale moebius transformation is V(z) = 2*(z-1)(z+1).

For the x and y formulas:
The new denominator is the same as used in double doyle node.
denom = (x+1)*(x+1) + y*y.
The numerator is 2 times the original numerator.
New numer = 2*((x*x - 1) + (y*y))

- Brian

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The DoyleSpiral node does not have a complete pointarray output.
The angles and x,y, lengths are not there. (See Spherepoints node).
Need to review pointarray docs, Max and/or Karsten's explanations...for a better understanding of pointarray format...
- Brian

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Show messages:  1-19  20-24