Swirl function

In mathematics, swirl functions are special functions defined as follows[1]：

S(k,n,r,\theta )=\sin(k\cos(r)-n\theta )

where k and n are integers, and r and θ are polar coordinates.

When these functions are graphed, they usually resemble a swirling fan blade, where n is the number of blades, k is related to the shape of each blade.

Symmetry

The function S(k,n,r,θ) satisfies the following relations:

mirror symmetry

$f(-k,n,r,\theta )=-f(k,n,r,-\theta )$
$f(-k,n,r,\theta )=-f(k,-n,r,\theta )$
$f(-k,-n,r,\theta )=-f(k,n,r,\theta )$
$f(-k,n,r,-\theta )=-f(k,n,r,\theta )$
$f(-k,n,r,\theta )=-f(k,n,-r,-\theta )$
$f(-k,n,-r,-\theta )=-f(k,n,r,\theta )$
$f(-k,-n,-r,\theta )=-f(k,n,r,\theta )$
$f(-k,n,-r,-\theta )=-f(k,n,r,\theta )$

full symmetry

$f(k,-n,r,\theta )=f(k,n,r,-\theta )$
$f(k,-n,r,-\theta )=f(k,n,r,\theta )$
$f(k,n,-r,\theta )=f(k,n,r,\theta )$
$f(k,n,-r,\theta )=f(k,n,r,\theta )$
$f(k,n,-r,\theta )=f(k,-n,r,-\theta )$
$f(k,-n,-r,-\theta )=f(k,n,r,\theta )$
$f(k,n,-r,\theta )-f(k,n,r,\theta )$

rotation symmetry

S\left(k,n,r,\theta +{\frac {2\pi }{n}}\right)=S(k,n,r,\theta )

examples

First number is n, second is k

7,-2
7,2
7,-4
7,4
7,-6
7,6
7,-8
7,8
7,-10
7,10
7,-12
7,12

0,4
1,4
2,4
7,4
-5,4
-9,4
30,4

References

Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 36–37 and 86, 1999.

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[1] Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 36–37 and 86, 1999.