Parametric equations with periodic functions can produce some beautiful results. The initial idea here was to see the effect of combining two polar roses, out of phase. In the end, I introduced an additional variable on top of this – a polar rose would have parametric coordinates

(cos(mt)cos(t), cos(mt)sin(t))

but I adjusted it to

(cos(mt)cos(t), cos(nt)sin(t))

to allow for more variety in the results.

All 9 gifs at the end of this post were made with the same parametric form:

The ‘acos(kt+b)’ part represents a polar rose whose size depends on a, number of petals depends on k and phase depends on b. By animating variable b, effectively we are rotating this rose over time.

Overall, we see the combined effect of the first curve plus the rotating rose. For example:

The original Desmos version that I used is here.

Hover/click to see the values of m, n, a and k for each one (b is the variable used for the animation).

n=4, m=2, a=0.3, k=8

n=4, m=2, a=1, k=12

n=4, m=2, a=3.5, k=2

n=4, m=4, a=1, k=2

n=5, m=3, a=3, k=3

n=10, m=8, a=3, k=2

n=10, m=10, a=3, k=2

n=4, m=3, a=1, k=0

n=8, m=6, a=2, k=0

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