Had a look at the 1st quadrant (0 .. 1r2p1 rad).
Worked with a vector "sweep" (origin to perimeter) for these graphs:
- unit circle
- square circumscribed unit circle
- quadric plane function x^4 + y^4= 1 ('real squircle')
- 'fake squircle'
The 'fake squircle' was constructed by taking the mean of vector
length (at the same angle) for square and unit circle.
The figure obtained by this procedure is a bulging square, situated
between circle and circumscribed square, with its four corners
sitting on the four bisectors of the four quadrants.
That's the region where the 'real squircle' lives; but in contrast,
that is a smooth curve.
I tried a couple of different means, notably harmonic, geometric and
arithmetic mean.
Using geometric mean, I noticed that the four corners of the 'fake
squircle' almost conincide (error below 2%) with the 'real' one
(while most of its graph lies *inside* of the 'real squircle'). Maybe
these corner positions gave rise to the mention of mean in this context.
I hesitate to include the awkward code I used, but as a starting
point I took your circle function (cos j. sin); my mean is different
from yours as I stayed on the same 'sweep vector' when comparing
different graph's vector lengths.
As RM would probably have written at this point: I hope this makes some sense.
Thanks
-M
At 2022-08-16 18:08, you wrote:
RD> I want to construct and plot a Squircle in J. [...]
RD>
RD> Has anyone a good idea for performing that calculation? Could the J
RD> function plot then draw the Squircle?
It is easy to "plot" parametric functions in the complex plane.
Here are three parametric functions, to be computed over parameter
inputs for a full circle, say, -pi to pi in 200 tiny steps:
circle =. cos j. sin
square =. cos j.&* sin
squircle =. (square + circle) % 2: NB. your "mean" of the functions.
NB. Show time!
plot (circle , square ,: squircle) i: 1p1 j. 200
Martin Neitzel
PS:
That * in cos j.&* sin is Sign, bulging out the circle to the full
1 _1 (square) borders.
Have some thoughts on what actually happens when the Sign * acts on and
returns a 0, and how that shows up.
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