螺旋函數 (Swirl function)是一個以三角函數 定義的特殊函數 [1]
swirl function Maple plot
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{\displaystyle S(k,n,r,\theta )=sin(k*cos(r)-n*\theta )}
其中k,n均為整數。k與螺旋葉的長度與形狀有關，n為螺旋的葉片數。
鏡像對稱
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{\displaystyle S(k,n,r,\theta )}
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{\displaystyle S(k,-n,r,\theta )}
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{\displaystyle f(-k,n,r,\theta )=-f(k,n,r,-\theta )}
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{\displaystyle f(-k,n,r,\theta )=-f(k,-n,r,\theta )}
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{\displaystyle f(-k,-n,r,\theta )=-f(k,n,r,\theta )}
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{\displaystyle f(-k,n,r,-\theta )=-f(k,n,r,\theta )}
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{\displaystyle f(-k,n,r,\theta )=-f(k,n,-r,-\theta )}
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{\displaystyle f(-k,n,-r,-\theta )=-f(k,n,r,\theta )}
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{\displaystyle f(-k,-n,-r,\theta )=-f(k,n,r,\theta )}
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{\displaystyle f(-k,n,-r,-\theta )=-f(k,n,r,\theta )}
全對稱
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{\displaystyle f(k,-n,r,\theta )=f(k,n,r,-\theta )}
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{\displaystyle f(k,-n,r,-\theta )=f(k,n,r,\theta )}
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{\displaystyle f(k,n,-r,\theta )=f(k,n,r,\theta )}
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{\displaystyle f(k,n,-r,\theta )=f(k,n,r,\theta )}
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{\displaystyle f(k,n,-r,\theta )=f(k,-n,r,-\theta )}
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{\displaystyle f(k,-n,-r,-\theta )=f(k,n,r,\theta )}
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{\displaystyle f(k,n,-r,\theta )-f(k,n,r,\theta )}
旋轉對稱
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{\displaystyle S(k,n,r,\theta +{\frac {2\pi }{n}})=S(k,n,r,\theta )}
級數展開
編輯
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{\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})}}
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{\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})}}
與其他特殊函數關係
編輯
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{\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)}}}}
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{\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 }}}
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{\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)}}}}
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{\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}}
螺旋葉數與鏡像對稱
7,-2
7,2
7,-4
7,4
7,-6
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7,-8
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7,-10
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7,-12
7,12
螺旋葉形
0,4
1,4
2,4
7,4
-5,4
-9,4
30,4
參考文獻
編輯
^ 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
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30,4
