Nested Ellipses, and Ellipse Whirl may be constructed using the MoI centerEllipse factory, and some mathematics, in a node program.
https://benice-equation.blogspot.com/2019/01/nested-ellipses.html
The identity of benice shows up as C.J. Chen.
This person has multiple geometric art graphics posted, without showing the mathematics.
In certain orientations, the Ellipse Whirl generates the illusion of logarithmic spirals. There are similarities to the spiral arms of a galaxy. Clockwise or CounterClockwise rotation with respect to the milky way galaxy is possible.
For the mathematics:
Let bigEllipse be an ellipse centered at the origin, (0,0).
Initially, bigEllipse is non-rotating, and non-scaling, and has
it's first (semi-) axis, of length "a", on the x-axis.
It's second (semi-) axis, of length "b", is on the y-axis.
(It could also be considered to be in a rotating frame, in other art.)
nestedEllipse1 is the first ellipse nested inside bigEllipse, and is tangent to bigEllipse in two locations, (so long as it is smaller).
Its first semi-axis is of (variable) length "aPrime".
Its second semi-axis is of (variable) length "bPrime".
There can be many more (similar) ellipses inside. (recursive)
The ratio r = aPrime/a = bPrime/b.
The nested ellipses rotate counterclockwise, and scale, inside bigEllipse, maintaining two point tangency, so long as the nested ellipse is smaller than its circumscribed ellipse, (turn not equal to n*PI). The rotation angle of nestedEllipse1, with respect to bigEllipse, is called "tilt", or "t". When the "aPrime" axis overlays the "b" axis, "tilt" = PI/2 radians, and nestedEllipse1 is of minimum scale, and is "vertically" oriented. (tilt "t" acts as a parametric angle.)
The initial nest structure has nestedEllipse1 "vertical", and nestedEllipse2 "horizontal", etc.
nestedEllipse2 must rotate at twice the angular rate (omega) of nestedEllipse1, etc. (I think).
Link's to the mathematics:
https://math.stackexchange.com/questions/3773593/given-ellipse-of-axes-a-and-b-find-axes-of-tangential-and-concentric-ellips
https://math.stackexchange.com/questions/4059601/want-to-rotate-and-scale-an-ellipse-so-it-is-tangent-to-the-first-one
https://commons.wikimedia.org/wiki/File:Nested_Ellipses.png
https://www.mathisfunforum.com/viewtopic.php?id=24759
(I have not yet implemented the mathematics.)
- Brian
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Algorithm draft:
Use ellipse factory to draw "parent", bigEllipse, with "a" and "b".
Orient Line/Line, (make copies,) can easily create initial nest of multiple ellipses.
Let tilt angle "t" range from PI/2 to 3PI/2, based upon numpoints.
For each value of "t", calculate "delta" and "r".
For each "r" value, calculate aPrime and bPrime of child ellipse.
Using polar coordinates with angle "t", aPrime, and bPrime,
calculate the two axis points for current scaled nestedEllipse1, and
use the two points in the ellipse factory, to draw tilted, scaled nestedEllipse1.
Repeat for nestedEllipse2, using nestedEllipse1 as parent, instead of bigEllipse. Use 2*t and rotated start angle...
Planar color in bands?...
