In mathematics, swirl functions are special functions defined as follows：
S
(
k
,
n
,
r
,
θ
)
=
sin
(
k
cos
(
r
)
−
n
θ
)
where k, n are integers.
n is the number of blades, k is related to the shape of each blade.
The function S(k,n,r,θ) satisfies 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
(
k
,
n
,
r
,
θ
+
2
π
n
)
=
S
(
k
,
n
,
r
,
θ
)
