Drawing a super-ellipse with a turtle - netlogo

Question

Obviously, any shape drawable by other means can be drawn by a turtle. Circles and squares are easy
rt 1 fd .0
and
if ticks mod 100 = 0 [rt 90]
fd 1
Super-ellipses not so much. (regular ellipses are not trivial either.)
The Wikipedia article on super-ellipses if you need to be refreshed on the topic.
Any input is appreciated.
Using a pendown turtle is there way to make a super-ellipse that emerges from turtle movement?

Answer 1

I have 1/4 of it, I suppose you could piece-wise put the other three together. Other values of n are not tested here. (using the Wiki notation, plus phi as an angle of rotating the whole thing.) And the placement of reset-ticks, pen-down, is sloppy, I know.
to go2
clear-all
reset-ticks
let a 6
let b 5
let phi 0
let n 3.5
create-turtles 1 [
let iNdx 1
repeat 90 [
show iNdx
show cos(iNdx)
if cos(iNdx) > 0 and sin(iNdx) > 0 [
let tx (a * (cos(iNdx) ^ (2 / n)))
let ty (b * (sin(iNdx) ^ (2 / n)))
let tx2 tx * cos(phi) - ty * sin(phi)
let ty2 tx * sin(phi) + ty * cos(phi)
setxy tx2 ty2
]
pen-down
set iNdx iNdx + 1
]
]
end
The ellipse looks simpler, but you be the judge
to go
clear-all
reset-ticks
let a 6
let b 5
let phi 45
create-turtles 1 [
let iNdx 1
repeat 360 [
let tx (a * cos(iNdx))
let ty (b * sin(iNdx))
let tx2 tx * cos(phi) - ty * sin(phi)
let ty2 tx * sin(phi) + ty * cos(phi)
setxy tx2 ty2
pen-down
set iNdx iNdx + 1
]
]
end
a generalization and simplification as a procedure.
to Super-ellipse [x y a b m n]
create-turtles 1 [
let iNdx 1
repeat 360 [
setxy (x + (abs cos iNdx)^(2 / m) * a * (sgn cos iNdx))
(y + (abs sin iNdx)^(2 / n) * b * (sgn sin iNdx))
pendown
set iNdx iNdx + 1]
]
end

Answer 2

The generalized form of another answer seems to produce the sort of thing I was thinking of. the closer the pen starts to one of the foci the closer the drawing is to a square. the n<1 super-ellipses are not achieved.
globals[c]
breed [pens pen]
breed [foci focus]
foci-own [dist distx disty]
to setup
ca
create-pens 1 [set heading 45 fd 10 pendown set C self]
;create-foci 1 [setxy (random-xcor / 2) (random-ycor / 2)]
create-foci 1 [setxy 10 10]
create-foci 1 [setxy 10 -10]
create-foci 1 [setxy -10 -10]
create-foci 1 [setxy -10 10]
end
to go
repeat 5100
[
ask foci [
set dist distance c
set distx xcor - [xcor] of c
set disty ycor - [ycor] of c
]
ask c
[
set heading 90 + atan ( sum [distx / dist] of foci / sum [dist] of foci)
( sum [disty / dist] of foci / sum [dist] of foci)
FD .0125
]
]
end