# 极坐标系

## 点的表示

### 极坐标系与平面直角坐标系之间的变换

${\displaystyle r={\sqrt {y^{2}+x^{2}}}\quad }$（参阅勾股定理
${\displaystyle \theta =\operatorname {atan2} (y,x)\quad }$atan2是已将象限纳入考量的反正切函数）

${\displaystyle \theta ={\begin{cases}\arctan({\frac {y}{x}})&{\text{if }}x>0\\\arctan({\frac {y}{x}})+\pi &{\text{if }}x<0{\text{ and }}y\geq 0\\\arctan({\frac {y}{x}})-\pi &{\text{if }}x<0{\text{ and }}y<0\\{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y>0\\-{\frac {\pi }{2}}&{\text{if }}x=0{\text{ and }}y<0\\0&{\text{if }}x=0{\text{ and }}y=0\end{cases}}}$

${\displaystyle x=r\cos \theta }$
${\displaystyle y=r\sin \theta }$

## 极坐标系方程

• 函数：用极坐标系描述的曲线方程称作极坐标方程，通常表示为r为自变量θ的函数
• 对称：极坐标方程经常会表现出不同的对称形式，如果r(−θ) = r（θ），则曲线关于极点（0°/180°）对称，如果r(π−θ) = r(θ)，则曲线关于极点（90°/270°）对称，如果r(θ−α) = r(θ)，则曲线相当于从极点逆时针方向旋转α°。

### 圆

${\displaystyle r^{2}-2rr_{0}\cos(\theta -\varphi )+r_{0}^{2}=a^{2}}$

${\displaystyle r(\theta )=a}$

#### 推导

${\displaystyle (x-p_{0}\cos \alpha )^{2}+(y-p_{0}\sin \alpha )^{2}=r^{2}}$

${\displaystyle x=p\cos \beta ,\qquad y=p\sin \beta }$

${\displaystyle p^{2}-2pp_{0}(\sin \beta \sin \alpha +\cos \beta \cos \alpha )+p_{0}^{2}=r^{2}}$

${\displaystyle p^{2}+p_{0}^{2}-2pp_{0}\cos(\beta -\alpha )=r^{2}}$

### 直线

${\displaystyle \theta =\varphi }$,

${\displaystyle r(\theta )={r_{0}}\sec(\theta -\varphi )}$.

### 玫瑰线

${\displaystyle r(\theta )=a\cos k\theta }$或者
${\displaystyle r(\theta )=a\sin k\theta }$

### 阿基米德螺线

${\displaystyle r(\theta )=a+b\theta }$

### 圆锥曲线

${\displaystyle r={\ell \over (1-e\cos \theta )}}$

${\displaystyle r={ep \over (1-e\cos \theta )}}$

## 复数

1. ${\displaystyle r(\cos \theta +i\sin \theta )}$，简写为${\displaystyle r\operatorname {cis} \theta }$
2. ${\displaystyle re^{i\theta }}$

${\displaystyle a=r\cos \theta }$
${\displaystyle b=r\sin \theta }$

• 乘法：${\displaystyle (r\operatorname {cis} \theta )*(R\operatorname {cis} \varphi )=rR\operatorname {cis} (\theta +\varphi )}$
• 除法：${\displaystyle {\frac {r\operatorname {cis} \theta }{R\operatorname {cis} \varphi }}={\frac {r}{R}}\operatorname {cis} (\theta -\varphi )}$
• 指数（棣莫弗定理，De Moivre's formula）: ${\displaystyle (r\operatorname {cis} \theta )^{n}=r^{n}\operatorname {cis} (n\theta )}$

## 微积分

### 微分

{\displaystyle {\begin{aligned}r{\frac {\partial u}{\partial r}}&=r{\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial r}}+r{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial r}},\\[2pt]{\frac {\partial u}{\partial \theta }}&={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial \theta }}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial \theta }},\end{aligned}}}

{\displaystyle {\begin{aligned}r{\frac {\partial u}{\partial r}}&=r{\frac {\partial u}{\partial x}}\cos \theta +r{\frac {\partial u}{\partial y}}\sin \theta =x{\frac {\partial u}{\partial x}}+y{\frac {\partial u}{\partial y}},\\[2pt]{\frac {\partial u}{\partial \theta }}&=-{\frac {\partial u}{\partial x}}r\sin \theta +{\frac {\partial u}{\partial y}}r\cos \theta =-y{\frac {\partial u}{\partial x}}+x{\frac {\partial u}{\partial y}}.\end{aligned}}}

{\displaystyle {\begin{aligned}r{\frac {\partial }{\partial r}}&=x{\frac {\partial }{\partial x}}+y{\frac {\partial }{\partial y}}\\[2pt]{\frac {\partial }{\partial \theta }}&=-y{\frac {\partial }{\partial x}}+x{\frac {\partial }{\partial y}}.\end{aligned}}}

{\displaystyle {\begin{aligned}{\frac {\partial u}{\partial x}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial x}}+{\frac {\partial u}{\partial \theta }}{\frac {\partial \theta }{\partial x}},\\[2pt]{\frac {\partial u}{\partial y}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial y}}+{\frac {\partial u}{\partial \theta }}{\frac {\partial \theta }{\partial y}},\end{aligned}}}

{\displaystyle {\begin{aligned}{\frac {\partial u}{\partial x}}&={\frac {\partial u}{\partial r}}{\frac {x}{\sqrt {x^{2}+y^{2}}}}-{\frac {\partial u}{\partial \theta }}{\frac {y}{x^{2}+y^{2}}}\\[2pt]&=\cos \theta {\frac {\partial u}{\partial r}}-{\frac {1}{r}}\sin \theta {\frac {\partial u}{\partial \theta }},\\[2pt]{\frac {\partial u}{\partial y}}&={\frac {\partial u}{\partial r}}{\frac {y}{\sqrt {x^{2}+y^{2}}}}+{\frac {\partial u}{\partial \theta }}{\frac {x}{x^{2}+y^{2}}}\\[2pt]&=\sin \theta {\frac {\partial u}{\partial r}}+{\frac {1}{r}}\cos \theta {\frac {\partial u}{\partial \theta }}.\end{aligned}}}

{\displaystyle {\begin{aligned}{\frac {\partial }{\partial x}}&=\cos \theta {\frac {\partial }{\partial r}}-{\frac {1}{r}}\sin \theta {\frac {\partial }{\partial \theta }}\\[2pt]{\frac {\partial }{\partial y}}&=\sin \theta {\frac {\partial }{\partial r}}+{\frac {1}{r}}\cos \theta {\frac {\partial }{\partial \theta }}.\end{aligned}}}

{\displaystyle {\begin{aligned}x&=r(\theta )\cos \theta \\y&=r(\theta )\sin \theta \end{aligned}}}

{\displaystyle {\begin{aligned}{\frac {dx}{d\theta }}&=r'(\theta )\cos \theta -r(\theta )\sin \theta \\[2pt]{\frac {dy}{d\theta }}&=r'(\theta )\sin \theta +r(\theta )\cos \theta .\end{aligned}}}

${\displaystyle {\frac {dy}{dx}}={\frac {r'(\theta )\sin \theta +r(\theta )\cos \theta }{r'(\theta )\cos \theta -r(\theta )\sin \theta }}.}$

### 向量微积分

${\displaystyle {\frac {d\mathbf {r} }{dt}}={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\theta }}{\hat {\boldsymbol {\theta }}},}$
${\displaystyle {\frac {d^{2}\mathbf {r} }{dt^{2}}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}.}$

${\displaystyle \mathbf {A} }$为被一条连接焦点与曲线上一点的线所划分出的区域，则${\displaystyle d\mathbf {A} }$就是由${\displaystyle \mathbf {r} }$${\displaystyle d\mathbf {r} }$所构平行四边形区域的一半。

${\displaystyle dA={\begin{matrix}{\frac {1}{2}}\end{matrix}}|\mathbf {r} \times d\mathbf {r} |}$,

## 极坐标与球坐标和圆柱坐标的联系

### 圆柱坐标系

${\displaystyle {x}={\rho }\,\cos \varphi }$
${\displaystyle {y}={\rho }\,\sin \varphi }$
${\displaystyle {z}={z}}$

### 球坐标系

${\displaystyle x=r\,\sin \theta \,\cos \varphi }$
${\displaystyle y=r\,\sin \theta \,\sin \varphi }$
${\displaystyle z=r\,\cos \theta }$

## 参考资料

• Adams, Robert; Christopher Essex. Calculus: a complete course Eighth. Pearson Canada Inc. 2013. ISBN 978-0-321-78107-9.
• Anton, Howard; Irl Bivens, Stephen Davis. Calculus Seventh. Anton Textbooks, Inc. 2002. ISBN 0-471-38157-8.
• Finney, Ross; George Thomas, Franklin Demana, Bert Waits. Calculus: Graphical, Numerical, Algebraic Single Variable Version. Addison-Wesley Publishing Co. June 1994. ISBN 0-201-55478-X.

1. ^ The MacTutor History of Mathematics archive: Coolidge's Origin of Polar Coordinates
2. ^ Coolidge, Julian. The Origin of Polar Coordinates. American Mathematical Monthly. 1952, 59: 78–85.
3. ^ Klaasen, Daniel. Historical Topics for the Mathematical Classroom.
4. ^ Miller, Jeff. Earliest Known Uses of Some of the Words of Mathematics. [2006-09-10]. （原始内容存档于1999-10-03）.
5. ^ Smith, David Eugene. History of Mathematics, Vol II. Boston: Ginn and Co. 1925: 324.
6. ^ Brown, Richard G. Andrew M. Gleason , 编. Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis. Evanston, Illinois: McDougal Littell Inc. 1997. ISBN 978-0-395-77114-3.
7. ^ Polar Coordinates and Graphing (PDF). [2006-09-22]. （原始内容 (PDF)存档于2012-02-15）.
8. ^ Principles of Physics. Brooks/Cole—Thomson Learning. 2005. ISBN 978-0-534-49143-7. 使用|coauthors=需要含有|author= (帮助)
9. ^ Torrence, Bruce Follett; Eve Torrence. The Student's Introduction to Mathematica. Cambridge University Press. 1999. ISBN 0-521-59461-8.
10. ^ Polar coordinates. [2006-05-25]. （原始内容存档于2006-04-27）.
11. ^ Ward, Robert L. Analytic Geometry: Polar Coordinates. [2006-05-25].
12. ^ Smith, Julius O. Euler's Identity. Mathematics of the Discrete Fourier Transform (DFT). W3K Publishing. 2003 [2006-09-22]. ISBN 978-0-9745607-0-0. （原始内容存档于2006-09-15）.
13. ^ Husch, Lawrence S. Areas Bounded by Polar Curves. [2006-11-25].
14. ^ Lawrence S. Husch. Tangent Lines to Polar Graphs. [2006-11-25].
15. ^ Wattenberg, Frank. Spherical Coordinates. 1997 [2006-09-16]. （原始内容存档于2012-02-15）.
16. ^ Santhi, Sumrit. Aircraft Navigation System. [2006-11-26].
17. ^ Emergency Procedures (PDF). [2007-01-15].