Swirl function

In mathematics, swirl functions are special functions defined as follows：

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

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

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

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

This article "Swirl function" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Swirl function. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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