螺旋函数

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螺旋函数(Swirl function)是一个以三角函数定义的特殊函数：

$$S(k,n,r,\theta)=sin(k*cos(r)-n*\theta)$$

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。

## 对称性

镜像对称

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

全对称

- $$f(k, -n, r, \theta)=f(k, n, r, -\theta)$$
- $$f(k, -n, r, -\theta)=f(k, n, r, \theta)$$
- $$f(k, n, -r, \theta)=f(k, n, r, \theta)$$
- $$f(k, n, -r, \theta)=f(k, n, r, \theta)$$
- $$f(k, n, -r, \theta)=f(k, -n, r, -\theta)$$
- $$f(k, -n, -r, -\theta)=f(k, n, r, \theta)$$
- $$f(k, n, -r, \theta)-f(k, n, r, \theta)$$

旋转对称

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

## 级数展开

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

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

## 与其他特殊函数关系

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

## 图例

螺旋叶数与镜像对称

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- 7,-2

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- 7,2

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- 7,-4

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- 7,4

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- 7,-6

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- 7,6

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- 7,-8

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- 7,8

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- 7,-10

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- 7,10

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- 7,-12

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- 7,12

螺旋叶形

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- 0,4

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- 1,4

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- 2,4

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- 7,4

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- -5,4

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- -9,4

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- 30,4