螺旋函数 (Swirl function)是一个以三角函数 定义的特殊函数 [1]
swirl function Maple plot
S
(
k
,
n
,
r
,
θ
)
=
s
i
n
(
k
∗
c
o
s
(
r
)
−
n
∗
θ
)
{\displaystyle S(k,n,r,\theta )=sin(k*cos(r)-n*\theta )}
其中k,n均为整数。k与螺旋叶的长度与形状有关，n为螺旋的叶片数。
镜像对称
S
(
k
,
n
,
r
,
θ
)
{\displaystyle S(k,n,r,\theta )}
S
(
k
,
−
n
,
r
,
θ
)
{\displaystyle S(k,-n,r,\theta )}
f
(
−
k
,
n
,
r
,
θ
)
=
−
f
(
k
,
n
,
r
,
−
θ
)
{\displaystyle f(-k,n,r,\theta )=-f(k,n,r,-\theta )}
f
(
−
k
,
n
,
r
,
θ
)
=
−
f
(
k
,
−
n
,
r
,
θ
)
{\displaystyle f(-k,n,r,\theta )=-f(k,-n,r,\theta )}
f
(
−
k
,
−
n
,
r
,
θ
)
=
−
f
(
k
,
n
,
r
,
θ
)
{\displaystyle f(-k,-n,r,\theta )=-f(k,n,r,\theta )}
f
(
−
k
,
n
,
r
,
−
θ
)
=
−
f
(
k
,
n
,
r
,
θ
)
{\displaystyle f(-k,n,r,-\theta )=-f(k,n,r,\theta )}
f
(
−
k
,
n
,
r
,
θ
)
=
−
f
(
k
,
n
,
−
r
,
−
θ
)
{\displaystyle f(-k,n,r,\theta )=-f(k,n,-r,-\theta )}
f
(
−
k
,
n
,
−
r
,
−
θ
)
=
−
f
(
k
,
n
,
r
,
θ
)
{\displaystyle f(-k,n,-r,-\theta )=-f(k,n,r,\theta )}
f
(
−
k
,
−
n
,
−
r
,
θ
)
=
−
f
(
k
,
n
,
r
,
θ
)
{\displaystyle f(-k,-n,-r,\theta )=-f(k,n,r,\theta )}
f
(
−
k
,
n
,
−
r
,
−
θ
)
=
−
f
(
k
,
n
,
r
,
θ
)
{\displaystyle f(-k,n,-r,-\theta )=-f(k,n,r,\theta )}
全对称
f
(
k
,
−
n
,
r
,
θ
)
=
f
(
k
,
n
,
r
,
−
θ
)
{\displaystyle f(k,-n,r,\theta )=f(k,n,r,-\theta )}
f
(
k
,
−
n
,
r
,
−
θ
)
=
f
(
k
,
n
,
r
,
θ
)
{\displaystyle f(k,-n,r,-\theta )=f(k,n,r,\theta )}
f
(
k
,
n
,
−
r
,
θ
)
=
f
(
k
,
n
,
r
,
θ
)
{\displaystyle f(k,n,-r,\theta )=f(k,n,r,\theta )}
f
(
k
,
n
,
−
r
,
θ
)
=
f
(
k
,
n
,
r
,
θ
)
{\displaystyle f(k,n,-r,\theta )=f(k,n,r,\theta )}
f
(
k
,
n
,
−
r
,
θ
)
=
f
(
k
,
−
n
,
r
,
−
θ
)
{\displaystyle f(k,n,-r,\theta )=f(k,-n,r,-\theta )}
f
(
k
,
−
n
,
−
r
,
−
θ
)
=
f
(
k
,
n
,
r
,
θ
)
{\displaystyle f(k,-n,-r,-\theta )=f(k,n,r,\theta )}
f
(
k
,
n
,
−
r
,
θ
)
−
f
(
k
,
n
,
r
,
θ
)
{\displaystyle f(k,n,-r,\theta )-f(k,n,r,\theta )}
旋转对称
S
(
k
,
n
,
r
,
θ
+
2
π
n
)
=
S
(
k
,
n
,
r
,
θ
)
{\displaystyle S(k,n,r,\theta +{\frac {2\pi }{n}})=S(k,n,r,\theta )}
级数展开
编辑
S
(
k
,
n
,
r
,
θ
)
≈
s
i
n
(
k
−
n
∗
θ
)
−
(
1
/
2
)
∗
c
o
s
(
k
−
n
∗
θ
)
∗
k
∗
r
2
+
(
−
(
1
/
8
)
∗
s
i
n
(
k
−
n
∗
θ
)
∗
k
2
+
(
1
/
24
)
∗
c
o
s
(
k
−
n
∗
θ
)
∗
k
)
∗
r
4
+
(
(
1
/
48
)
∗
s
i
n
(
k
−
n
∗
θ
)
∗
k
2
+
c
o
s
(
k
−
n
∗
θ
)
∗
(
−
(
1
/
720
)
∗
k
+
(
1
/
48
)
∗
k
3
)
)
∗
r
6
+
O
(
r
8
)
{\displaystyle S(k,n,r,\theta )\approx {sin(k-n*\theta )-(1/2)*cos(k-n*\theta )*k*r^{2}+(-(1/8)*sin(k-n*\theta )*k^{2}+(1/24)*cos(k-n*\theta )*k)*r^{4}+((1/48)*sin(k-n*\theta )*k^{2}+cos(k-n*\theta )*(-(1/720)*k+(1/48)*k^{3}))*r^{6}+O(r^{8})}}
S
(
k
,
n
,
r
,
θ
)
≈
s
i
n
(
k
∗
c
o
s
(
r
)
)
−
c
o
s
(
k
∗
c
o
s
(
r
)
)
∗
n
∗
θ
−
(
1
/
2
)
∗
s
i
n
(
k
∗
c
o
s
(
r
)
)
∗
n
2
∗
θ
2
+
(
1
/
6
)
∗
c
o
s
(
k
∗
c
o
s
(
r
)
)
∗
n
3
∗
θ
3
+
(
1
/
24
)
∗
s
i
n
(
k
∗
c
o
s
(
r
)
)
∗
n
4
∗
θ
4
−
(
1
/
120
)
∗
c
o
s
(
k
∗
c
o
s
(
r
)
)
∗
n
5
∗
θ
5
−
(
1
/
720
)
∗
s
i
n
(
k
∗
c
o
s
(
r
)
)
∗
n
6
∗
θ
6
+
(
1
/
5040
)
∗
c
o
s
(
k
∗
c
o
s
(
r
)
)
∗
n
7
∗
θ
7
+
(
1
/
40320
)
∗
s
i
n
(
k
∗
c
o
s
(
r
)
)
∗
n
8
∗
θ
8
+
O
(
θ
9
)
{\displaystyle S(k,n,r,\theta )\approx {sin(k*cos(r))-cos(k*cos(r))*n*\theta -(1/2)*sin(k*cos(r))*n^{2}*\theta ^{2}+(1/6)*cos(k*cos(r))*n^{3}*\theta ^{3}+(1/24)*sin(k*cos(r))*n^{4}*\theta ^{4}-(1/120)*cos(k*cos(r))*n^{5}*\theta ^{5}-(1/720)*sin(k*cos(r))*n^{6}*\theta ^{6}+(1/5040)*cos(k*cos(r))*n^{7}*\theta ^{7}+(1/40320)*sin(k*cos(r))*n^{8}*\theta ^{8}+O(\theta ^{9})}}
与其他特殊函数关系
编辑
S
(
k
,
n
,
r
,
θ
)
=
(
n
x
arccos
(
x
)
+
1
/
2
π
)
K
u
m
m
e
r
M
(
1
,
2
,
i
(
2
n
x
arccos
(
x
)
+
π
)
)
e
1
/
2
i
(
2
n
x
arccos
(
x
)
+
π
)
{\displaystyle S(k,n,r,\theta )={\frac {\left(nx\arccos \left(x\right)+1/2\,\pi \right){{\rm {KummerM}}\left(1,\,2,\,i\left(2\,nx\arccos \left(x\right)+\pi \right)\right)}}{{\rm {e}}^{1/2\,i\left(2\,nx\arccos \left(x\right)+\pi \right)}}}}
S
(
k
,
n
,
r
,
θ
)
=
−
i
(
2
n
x
arccos
(
x
)
+
π
)
W
h
i
t
t
a
k
e
r
M
(
0
,
1
/
2
,
i
(
2
n
x
arccos
(
x
)
+
π
)
)
4
n
x
arccos
(
x
)
+
2
π
{\displaystyle S(k,n,r,\theta )={\frac {-i\left(2\,nx\arccos \left(x\right)+\pi \right){{\rm {WhittakerM}}\left(0,\,1/2,\,i\left(2\,nx\arccos \left(x\right)+\pi \right)\right)}}{4\,nx\arccos \left(x\right)+2\,\pi }}}
S
(
k
,
n
,
r
,
θ
)
=
−
1
/
2
i
(
−
1
+
e
i
(
2
n
x
arccos
(
x
)
+
π
)
)
e
1
/
2
i
(
2
n
x
arccos
(
x
)
+
π
)
{\displaystyle S(k,n,r,\theta )={\frac {-1/2\,i\left(-1+{{\rm {e}}^{i\left(2\,nx\arccos \left(x\right)+\pi \right)}}\right)}{{\rm {e}}^{1/2\,i\left(2\,nx\arccos \left(x\right)+\pi \right)}}}}
S
(
k
,
n
,
r
,
θ
)
=
−
n
x
2
H
e
u
n
B
(
2
,
0
,
0
,
0
,
2
1
/
2
i
(
2
n
x
(
1
/
2
π
−
x
H
e
u
n
C
(
0
,
1
/
2
,
0
,
0
,
1
/
4
,
x
2
x
2
−
1
)
1
1
−
x
2
)
+
π
)
)
H
e
u
n
C
(
0
,
1
/
2
,
0
,
0
,
1
/
4
,
x
2
x
2
−
1
)
1
1
−
x
2
(
e
−
1
/
2
i
(
−
n
x
π
1
−
x
2
+
2
n
x
2
H
e
u
n
C
(
0
,
1
/
2
,
0
,
0
,
1
/
4
,
x
2
x
2
−
1
)
−
π
1
−
x
2
)
1
1
−
x
2
)
−
1
+
1
/
2
π
(
n
x
+
1
)
H
e
u
n
B
(
2
,
0
,
0
,
0
,
2
1
/
2
i
(
2
n
x
(
1
/
2
π
−
x
H
e
u
n
C
(
0
,
1
/
2
,
0
,
0
,
1
/
4
,
x
2
x
2
−
1
)
1
1
−
x
2
)
+
π
)
)
(
e
−
1
/
2
i
(
−
n
x
π
1
−
x
2
+
2
n
x
2
H
e
u
n
C
(
0
,
1
/
2
,
0
,
0
,
1
/
4
,
x
2
x
2
−
1
)
−
π
1
−
x
2
)
1
1
−
x
2
)
−
1
{\displaystyle S(k,n,r,\theta )=-n{x}^{2}{\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\left(2\,nx\left(1/2\,\pi -x{\it {HeunC}}\left(0,1/2,0,0,1/4,{\frac {{x}^{2}}{{x}^{2}-1}}\right){\frac {1}{\sqrt {1-{x}^{2}}}}\right)+\pi \right)}}\right){\it {HeunC}}\left(0,1/2,0,0,1/4,{\frac {{x}^{2}}{{x}^{2}-1}}\right){\frac {1}{\sqrt {1-{x}^{2}}}}\left({{\rm {e}}^{-1/2\,i\left(-nx\pi \,{\sqrt {1-{x}^{2}}}+2\,n{x}^{2}{\it {HeunC}}\left(0,1/2,0,0,1/4,{\frac {{x}^{2}}{{x}^{2}-1}}\right)-\pi \,{\sqrt {1-{x}^{2}}}\right){\frac {1}{\sqrt {1-{x}^{2}}}}}}\right)^{-1}+1/2\,\pi \,\left(nx+1\right){\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\left(2\,nx\left(1/2\,\pi -x{\it {HeunC}}\left(0,1/2,0,0,1/4,{\frac {{x}^{2}}{{x}^{2}-1}}\right){\frac {1}{\sqrt {1-{x}^{2}}}}\right)+\pi \right)}}\right)\left({{\rm {e}}^{-1/2\,i\left(-nx\pi \,{\sqrt {1-{x}^{2}}}+2\,n{x}^{2}{\it {HeunC}}\left(0,1/2,0,0,1/4,{\frac {{x}^{2}}{{x}^{2}-1}}\right)-\pi \,{\sqrt {1-{x}^{2}}}\right){\frac {1}{\sqrt {1-{x}^{2}}}}}}\right)^{-1}}
螺旋叶数与镜像对称
7,-2
7,2
7,-4
7,4
7,-6
7,6
7,-8
7,8
7,-10
7,10
7,-12
7,12
螺旋叶形
0,4
1,4
2,4
7,4
-5,4
-9,4
30,4
参考文献
编辑
^ 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.
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
