螺旋函数

螺旋函数(Swirl function)是一个以三角函数定义的特殊函数： 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 )} 与 S ( k , − n , r , θ ) {\displaystyle S(k,-n,r,\theta )} 互为镜像对称. 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 )} 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 )} 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})}} 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 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.