螺旋函数

镜像对称

• ${\displaystyle S(k,n,r,\theta )}$ 与${\displaystyle S(k,-n,r,\theta )}$  互为镜像对称.
• ${\displaystyle f(-k,n,r,\theta )=-f(k,n,r,-\theta )}$ 
• ${\displaystyle f(-k,n,r,\theta )=-f(k,-n,r,\theta )}$ 
• ${\displaystyle f(-k,-n,r,\theta )=-f(k,n,r,\theta )}$ 
• ${\displaystyle f(-k,n,r,-\theta )=-f(k,n,r,\theta )}$ 
• ${\displaystyle f(-k,n,r,\theta )=-f(k,n,-r,-\theta )}$ 

• ${\displaystyle f(-k,n,-r,-\theta )=-f(k,n,r,\theta )}$ 

• ${\displaystyle f(-k,-n,-r,\theta )=-f(k,n,r,\theta )}$ 

• ${\displaystyle f(-k,n,-r,-\theta )=-f(k,n,r,\theta )}$ 

全对称

• ${\displaystyle f(k,-n,r,\theta )=f(k,n,r,-\theta )}$ 
• ${\displaystyle f(k,-n,r,-\theta )=f(k,n,r,\theta )}$ 

• ${\displaystyle f(k,n,-r,\theta )=f(k,n,r,\theta )}$ 

• ${\displaystyle f(k,n,-r,\theta )=f(k,n,r,\theta )}$ 

• ${\displaystyle f(k,n,-r,\theta )=f(k,-n,r,-\theta )}$ 

• ${\displaystyle f(k,-n,-r,-\theta )=f(k,n,r,\theta )}$ 
• ${\displaystyle f(k,n,-r,\theta )-f(k,n,r,\theta )}$ 

旋转对称

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

${\displaystyle S(k,n,r,\theta )\approx {sin(k-n*\theta )-(1/2)*cos(k-n*\theta )*k*r^{2}+(-(1/8)*sin(k-n*\theta )*k^{2}+(1/24)*cos(k-n*\theta )*k)*r^{4}+((1/48)*sin(k-n*\theta )*k^{2}+cos(k-n*\theta )*(-(1/720)*k+(1/48)*k^{3}))*r^{6}+O(r^{8})}}$ 

${\displaystyle S(k,n,r,\theta )\approx {sin(k*cos(r))-cos(k*cos(r))*n*\theta -(1/2)*sin(k*cos(r))*n^{2}*\theta ^{2}+(1/6)*cos(k*cos(r))*n^{3}*\theta ^{3}+(1/24)*sin(k*cos(r))*n^{4}*\theta ^{4}-(1/120)*cos(k*cos(r))*n^{5}*\theta ^{5}-(1/720)*sin(k*cos(r))*n^{6}*\theta ^{6}+(1/5040)*cos(k*cos(r))*n^{7}*\theta ^{7}+(1/40320)*sin(k*cos(r))*n^{8}*\theta ^{8}+O(\theta ^{9})}}$ 

• ${\displaystyle S(k,n,r,\theta )={\frac {\left(nx\arccos \left(x\right)+1/2\,\pi \right){{\rm {KummerM}}\left(1,\,2,\,i\left(2\,nx\arccos \left(x\right)+\pi \right)\right)}}{{\rm {e}}^{1/2\,i\left(2\,nx\arccos \left(x\right)+\pi \right)}}}}$ 

• ${\displaystyle S(k,n,r,\theta )={\frac {-i\left(2\,nx\arccos \left(x\right)+\pi \right){{\rm {WhittakerM}}\left(0,\,1/2,\,i\left(2\,nx\arccos \left(x\right)+\pi \right)\right)}}{4\,nx\arccos \left(x\right)+2\,\pi }}}$ 

• ${\displaystyle S(k,n,r,\theta )={\frac {-1/2\,i\left(-1+{{\rm {e}}^{i\left(2\,nx\arccos \left(x\right)+\pi \right)}}\right)}{{\rm {e}}^{1/2\,i\left(2\,nx\arccos \left(x\right)+\pi \right)}}}}$ 

• ${\displaystyle S(k,n,r,\theta )=-n{x}^{2}{\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\left(2\,nx\left(1/2\,\pi -x{\it {HeunC}}\left(0,1/2,0,0,1/4,{\frac {{x}^{2}}{{x}^{2}-1}}\right){\frac {1}{\sqrt {1-{x}^{2}}}}\right)+\pi \right)}}\right){\it {HeunC}}\left(0,1/2,0,0,1/4,{\frac {{x}^{2}}{{x}^{2}-1}}\right){\frac {1}{\sqrt {1-{x}^{2}}}}\left({{\rm {e}}^{-1/2\,i\left(-nx\pi \,{\sqrt {1-{x}^{2}}}+2\,n{x}^{2}{\it {HeunC}}\left(0,1/2,0,0,1/4,{\frac {{x}^{2}}{{x}^{2}-1}}\right)-\pi \,{\sqrt {1-{x}^{2}}}\right){\frac {1}{\sqrt {1-{x}^{2}}}}}}\right)^{-1}+1/2\,\pi \,\left(nx+1\right){\it {HeunB}}\left(2,0,0,0,{\sqrt {2}}{\sqrt {1/2\,i\left(2\,nx\left(1/2\,\pi -x{\it {HeunC}}\left(0,1/2,0,0,1/4,{\frac {{x}^{2}}{{x}^{2}-1}}\right){\frac {1}{\sqrt {1-{x}^{2}}}}\right)+\pi \right)}}\right)\left({{\rm {e}}^{-1/2\,i\left(-nx\pi \,{\sqrt {1-{x}^{2}}}+2\,n{x}^{2}{\it {HeunC}}\left(0,1/2,0,0,1/4,{\frac {{x}^{2}}{{x}^{2}-1}}\right)-\pi \,{\sqrt {1-{x}^{2}}}\right){\frac {1}{\sqrt {1-{x}^{2}}}}}}\right)^{-1}}$ 

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

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.