Swirl function

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

S(k,n,r,θ) = sin (kcos(r)−nθ)

where k, n are integers.

n is the number of blades, k is related to the shape of each blade.

Symmmetry

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

mirror symmetry

• f(−k,n,r,θ) =  − f(k,n,r,−θ)
• f(−k,n,r,θ) =  − f(k,−n,r,θ)
• f(−k,−n,r,θ) =  − f(k,n,r,θ)
• f(−k,n,r,−θ) =  − f(k,n,r,θ)
• f(−k,n,r,θ) =  − f(k,n,−r,−θ)
• f(−k,n,−r,−θ) =  − f(k,n,r,θ)
• f(−k,−n,−r,θ) =  − f(k,n,r,θ)
• f(−k,n,−r,−θ) =  − f(k,n,r,θ)

full symmetry

• f(k,−n,r,θ) = f(k,n,r,−θ)
• f(k,−n,r,−θ) = f(k,n,r,θ)
• f(k,n,−r,θ) = f(k,n,r,θ)
• f(k,n,−r,θ) = f(k,n,r,θ)
• f(k,n,−r,θ) = f(k,−n,r,−θ)
• f(k,−n,−r,−θ) = f(k,n,r,θ)
• f(k,n,−r,θ) − f(k,n,r,θ)

rotation symmetry

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