نظام إحداثيات قطبية

الإحداثيات الكروية أو القطبية، وهي نبين موقع نقطة P وإحداثياتها الثلاث ρ,θ and φ.

النظام الإحداثي القطبي (إنگليزية: polar coordinate system) في الرياضيات والفيزياء هو مجموعة متغيرات تمكننا من معرفة مكان نقطة ما في الفضاء الثلاثي الأبعاد. وعلى عكس الإحداثيات الديكارتية الذي يستعمل ثلاثة أبعاد (س، ص، ع) لتحديد مقوع نقطة في الفراغ، يستعمل نطام الإحداثي الكروي أو القطبي نصف القطر ρ وزاوية المسقط على الدائرة الإستوائية θ وزاوية المسقط على الدائرة القطبية φ .

 تحويل الإحداثيات الكروية إلى إحداثيات خطية ثلاثية

يمكن تحويل الإحداثيات الكروية إلى الإحداثيات الخطية الثلاثية بواسطة عمليات رياضية بسيطة. (أنظر تباين). بعض المسائل في الطبيعة تسهل حلها باستعمال الإحداثيات الخطية، وبعض المسائل يسهل حلها باستخدام الإحداثيات الكروية، مثل انتشار الأشعة حول مصباح أو انتشار الأشعه حول الشمس.

وتذكر الدوامات في المياه، فهذه حالة خاصة من الإحداثيات الكروية وتسمي الإحداثيات الدائرية ، وهي تعمل بمعرفة نصف القطر ρ وزاوية واحدة θ.

كما نستعمل في حياتنا اليومية لتحديد موقع مدينة على سطح الكرة الأرضية خط الطول وخط العرض. أي مقياسان اثنان يلزمنان لذلك، وهذا صحيح طالما كان نصف القطر للكرة الأرضية ثابت.

أردنا بذلك أن نوضح لماذا توجد أنظمة مختلفة، ويكمن السر في اختيار النطام المناسب ،وإلا تهنا في متاهات التحويلات الرياضية.

إذا أردت معرفة مدار المحطة الفضائية الدولية فلا شك أنك ستختار لذلك النظام الإحداثي القطبي .

 المعادلة القطبية لمنحنى

 الدائرة

A circle with equation r(θ) = 1

The general equation for a circle with a center at (r₀, ${\displaystyle{\displaystyle\varphi}}$) and radius a is

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

This can be simplified in various ways, to conform to more specific cases, such as the equation

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

for a circle with a center at the pole and radius a.

 الخط

A polar rose with equation r(θ) = 2 sin 4θ

Radial lines (those running through the pole) are represented by the equation

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

where φ is the angle of elevation of the line; that is, φ = arctan m where m is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line θ = φ perpendicularly at the point (r₀, φ) has the equation

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

 لولب أرخميدس

One arm of an Archimedean spiral with equation r(θ) = θ for 0 < θ < 6π

The Archimedean spiral is a famous spiral that was discovered by Archimedes, which also can be expressed as a simple polar equation. It is represented by the equation

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

 قطاعات قمعية

Ellipse, showing semi-latus rectum

A conic section with one focus on the pole and the other somewhere on the 0° ray (so that the conic's major axis lies along the polar axis) is given by:

${\displaystyle{\displaystyle r={\ell\over{1+e\cos\theta}}}}$

 Complex numbers

An illustration of a complex number z plotted on the complex plane

An illustration of a complex number plotted on the complex plane using Euler's formula

• الضرب:

${\displaystyle{\displaystyle r_{0}e^{i\theta_{0}}\cdot r_{1}e^{i\theta_{1}}=r_{0}r_{1}e^{i(\theta_{0}+\theta_{1})}\,}}$

• Division:

${\displaystyle{\displaystyle{\frac{r_{0}e^{i\theta_{0}}}{r_{1}e^{i\theta_{1}}}}={\frac{r_{0}}{r_{1}}}e^{i(\theta_{0}-\theta_{1})}\,}}$

• Exponentiation (De Moivre's formula):

${\displaystyle{\displaystyle(re^{i\theta})^{n}=r^{n}e^{in\theta}\,}}$

 الحسبان

Calculus can be applied to equations expressed in polar coordinates.

The angular coordinate φ is expressed in radians throughout this section, which is the conventional choice when doing calculus.

 الحسبان التفاضلي

Using x = r cos φ and y = r sin φ, one can derive a relationship between derivatives in Cartesian and polar coordinates. For a given function, u(x,y), it follows that (by computing its total derivatives) or

$${\displaystyle{\begin{aligned} \displaystyle r{\frac{du}{dr}}&\displaystyle=r{\frac{\partial u}{\partial x}}\cos\varphi+r{\frac{\partial u}{\partial y}}\sin\varphi=x{\frac{\partial u}{\partial x}}+y{\frac{\partial u}{\partial y}},\\ \displaystyle{\frac{du}{d\varphi}}&\displaystyle=-{\frac{\partial u}{\partial x}}r\sin\varphi+{\frac{\partial u}{\partial y}}r\cos\varphi=-y{\frac{\partial u}{\partial x}}+x{\frac{\partial u}{\partial y}}.\end{aligned}}}$$

Hence, we have the following formula:

$${\displaystyle{\begin{aligned} \displaystyle r{\frac{d}{dr}}&\displaystyle=x{\frac{\partial}{\partial x}}+y{\frac{\partial}{\partial y}}\\ \displaystyle{\frac{d}{d\varphi}}&\displaystyle=-y{\frac{\partial}{\partial x}}+x{\frac{\partial}{\partial y}}.\end{aligned}}}$$

Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function u(r,φ), it follows that

$${\displaystyle{\begin{aligned} \displaystyle{\frac{du}{dx}}&\displaystyle={\frac{\partial u}{\partial r}}{\frac{\partial r}{\partial x}}+{\frac{\partial u}{\partial\varphi}}{\frac{\partial\varphi}{\partial x}},\\ \displaystyle{\frac{du}{dy}}&\displaystyle={\frac{\partial u}{\partial r}}{\frac{\partial r}{\partial y}}+{\frac{\partial u}{\partial\varphi}}{\frac{\partial\varphi}{\partial y}},\end{aligned}}}$$

or

$${\displaystyle{\begin{aligned} \displaystyle{\frac{du}{dx}}&\displaystyle={\frac{\partial u}{\partial r}}{\frac{x}{\sqrt{x^{2}+y^{2}}}}-{\frac{\partial u}{\partial\varphi}}{\frac{y}{x^{2}+y^{2}}}\\ &\displaystyle=\cos\varphi{\frac{\partial u}{\partial r}}-{\frac{1}{r}}\sin\varphi{\frac{\partial u}{\partial\varphi}},\\ \displaystyle{\frac{du}{dy}}&\displaystyle={\frac{\partial u}{\partial r}}{\frac{y}{\sqrt{x^{2}+y^{2}}}}+{\frac{\partial u}{\partial\varphi}}{\frac{x}{x^{2}+y^{2}}}\\ &\displaystyle=\sin\varphi{\frac{\partial u}{\partial r}}+{\frac{1}{r}}\cos\varphi{\frac{\partial u}{\partial\varphi}}.\end{aligned}}}$$

Hence, we have the following formulae:

$${\displaystyle{\begin{aligned} \displaystyle{\frac{d}{dx}}&\displaystyle=\cos\varphi{\frac{\partial}{\partial r}}-{\frac{1}{r}}\sin\varphi{\frac{\partial}{\partial\varphi}}\\ \displaystyle{\frac{d}{dy}}&\displaystyle=\sin\varphi{\frac{\partial}{\partial r}}+{\frac{1}{r}}\cos\varphi{\frac{\partial}{\partial\varphi}}.\end{aligned}}}$$

To find the Cartesian slope of the tangent line to a polar curve r(φ) at any given point, the curve is first expressed as a system of parametric equations.

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

Differentiating both equations with respect to φ yields

$${\displaystyle{\begin{aligned} \displaystyle{\frac{dx}{d\varphi}}&\displaystyle=r^{\prime}(\varphi)\cos\varphi-r(\varphi)\sin\varphi\\ \displaystyle{\frac{dy}{d\varphi}}&\displaystyle=r^{\prime}(\varphi)\sin\varphi+r(\varphi)\cos\varphi.\end{aligned}}}$$

Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point (r(φ), φ):

$${\displaystyle{\frac{dy}{dx}}={\frac{r^{\prime}(\varphi)\sin\varphi+r(\varphi)\cos\varphi}{r^{\prime}(\varphi)\cos\varphi-r(\varphi)\sin\varphi}}.}$$

For other useful formulas including divergence, gradient, and Laplacian in polar coordinates, see curvilinear coordinates.

 Integral calculus (arc length)

The arc length (length of a line segment) defined by a polar function is found by the integration over the curve r(φ). Let L denote this length along the curve starting from points A through to point B, where these points correspond to φ = a and φ = b such that 0 < b − a < 2π. The length of L is given by the following integral

$$L=\int_{a}^{b}{\sqrt{\left[r(\varphi)\right]^{2}+\left[{\tfrac{dr(\varphi)}{d\varphi}}\right]^{2}}}d\varphi$$

 Integral calculus (area)

The integration region R is bounded by the curve r(φ) and the rays φ = a and φ = b.

Let R denote the region enclosed by a curve r(φ) and the rays φ = a and φ = b, where 0 < b − a ≤ 2π. Then, the area of R is

$${\displaystyle{\frac{1}{2}}\int_{a}^{b}\left[r(\varphi)\right]^{2}\,d\varphi.}$$

The region R is approximated by n sectors (here, n = 5).

A planimeter, which mechanically computes polar integrals

This result can be found as follows. First, the interval [a, b] is divided into n subintervals, where n is some positive integer. Thus Δφ, the angle measure of each subinterval, is equal to b − a (the total angle measure of the interval), divided by n, the number of subintervals. For each subinterval i = 1, 2, ..., n, let φ_i be the midpoint of the subinterval, and construct a sector with the center at the pole, radius r(φ_i), central angle Δφ and arc length r(φ_i)Δφ. The area of each constructed sector is therefore equal to

$${\displaystyle\left[r(\varphi_{i})\right]^{2}\pi\cdot{\frac{\Delta\varphi}{2\pi}}={\frac{1}{2}}\left[r(\varphi_{i})\right]^{2}\Delta\varphi.}$$

Hence, the total area of all of the sectors is

$${\displaystyle\sum_{i=1}^{n}{\tfrac{1}{2}}r(\varphi_{i})^{2}\,\Delta\varphi.}$$

As the number of subintervals n is increased, the approximation of the area improves. Taking n → ∞, the sum becomes the Riemann sum for the above integral.

A mechanical device that computes area integrals is the planimeter, which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element linkage effects Green's theorem, converting the quadratic polar integral to a linear integral.

 Generalization

Using Cartesian coordinates, an infinitesimal area element can be calculated as dA = dx dy. The substitution rule for multiple integrals states that, when using other coordinates, the Jacobian determinant of the coordinate conversion formula has to be considered:

$$J=\det{\frac{\partial(x,y)}{\partial(r,\varphi)}}={\begin{vmatrix}{\frac{\partial x}{\partial r}}&{\frac{\partial x}{\partial\varphi}}\\ {\frac{\partial y}{\partial r}}&{\frac{\partial y}{\partial\varphi}}\end{vmatrix}}={\begin{vmatrix}\cos\varphi&-r\sin\varphi\\ \sin\varphi&r\cos\varphi\end{vmatrix}}=r\cos^{2}\varphi+r\sin^{2}\varphi=r.$$

Hence, an area element in polar coordinates can be written as

$$dA=dx\,dy\ =J\,dr\,d\varphi=r\,dr\,d\varphi.$$

Now, a function, that is given in polar coordinates, can be integrated as follows:

$${\displaystyle\iint_{R}f(x,y)\,dA=\int_{a}^{b}\int_{0}^{r(\varphi)}f(r,\varphi)\,r\,dr\,d\varphi.}$$

Here, R is the same region as above, namely, the region enclosed by a curve r(φ) and the rays φ = a and φ = b. The formula for the area of R is retrieved by taking f identically equal to 1.

A graph of ${\displaystyle{\displaystyle f(x)=e^{-x^{2}}}}$ and the area between the function and the ${\displaystyle{\displaystyle x}}$-axis, which is equal to ${\displaystyle{\displaystyle{\sqrt{\pi}}}}$.

A more surprising application of this result yields the Gaussian integral:

$${\displaystyle\int_{-\infty}^{\infty}e^{-x^{2}}\,dx={\sqrt{\pi}}.}$$

 حسبان المتجهات

حسبان المتجهات can also be applied to polar coordinates. For a planar motion, let ${\displaystyle{\displaystyle\mathbf{r}}}$ be the position vector (r cos(φ), r sin(φ)), with r and φ depending on time t.

We define an orthonormal basis with three unit vectors: radial, transverse, and normal directions. The radial direction is defined by normalizing ${\displaystyle{\displaystyle\mathbf{r}}}$:

$${\displaystyle{\hat{\mathbf{r}}}=(\cos(\varphi),\sin(\varphi))}$$

Radial and velocity directions span the plane of the motion, whose normal direction is denoted

${\displaystyle{\displaystyle{\hat{\mathbf{k}}}}}$

:

$${\displaystyle{\hat{\mathbf{k}}}={\hat{\mathbf{v}}}\times{\hat{\mathbf{r}}}.}$$

The transverse direction is perpendicular to both radial and normal directions:

$${\displaystyle{\hat{\boldsymbol{\varphi}}}=(-\sin(\varphi),\cos(\varphi))={\hat{\mathbf{k}}}\times{\hat{\mathbf{r}}}\ ,}$$

Then

$${\displaystyle{\begin{aligned} \displaystyle\mathbf{r}&\displaystyle=(x,\ y)=r(\cos\varphi,\ \sin\varphi)=r{\hat{\mathbf{r}}}\ ,\\ \displaystyle{\dot{\mathbf{r}}}&\displaystyle=\left({\dot{x}},\ {\dot{y}}\right)={\dot{r}}(\cos\varphi,\ \sin\varphi)+r{\dot{\varphi}}(-\sin\varphi,\ \cos\varphi)={\dot{r}}{\hat{\mathbf{r}}}+r{\dot{\varphi}}{\hat{\boldsymbol{\varphi}}}\ ,\\ \displaystyle{\ddot{\mathbf{r}}}&\displaystyle=\left({\ddot{x}},\ {\ddot{y}}\right)\\ &\displaystyle={\ddot{r}}(\cos\varphi,\ \sin\varphi)+2{\dot{r}}{\dot{\varphi}}(-\sin\varphi,\ \cos\varphi)+r{\ddot{\varphi}}(-\sin\varphi,\ \cos\varphi)-r{\dot{\varphi}}^{2}(\cos\varphi,\ \sin\varphi)\\ &\displaystyle=\left({\ddot{r}}-r{\dot{\varphi}}^{2}\right){\hat{\mathbf{r}}}+\left(r{\ddot{\varphi}}+2{\dot{r}}{\dot{\varphi}}\right){\hat{\boldsymbol{\varphi}}}\\ &\displaystyle=\left({\ddot{r}}-r{\dot{\varphi}}^{2}\right){\hat{\mathbf{r}}}+{\frac{1}{r}}\;{\frac{d}{dt}}\left(r^{2}{\dot{\varphi}}\right){\hat{\boldsymbol{\varphi}}}.\end{aligned}}}$$

This equation can be obtained by taking derivative of the function and derivatives of the unit basis vectors.

For a curve in 2D where the parameter is ${\displaystyle{\displaystyle\theta}}$ the previous equations simplify to:

$${\displaystyle{\begin{aligned} \displaystyle\mathbf{r}&\displaystyle=r(\theta){\hat{\mathbf{e}}}_{r}\\ \displaystyle{\frac{d\mathbf{r}}{d\theta}}&\displaystyle={\frac{dr}{d\theta}}{\hat{\mathbf{e}}}_{r}+r{\hat{\mathbf{e}}}_{\theta}\\ \displaystyle{\frac{d^{2}\mathbf{r}}{d\theta^{2}}}&\displaystyle=\left({\frac{d^{2}r}{d\theta^{2}}}-r\right){\hat{\mathbf{e}}}_{r}+{\frac{dr}{d\theta}}{\hat{\mathbf{e}}}_{\theta}\end{aligned}}}$$

 Centrifugal and Coriolis terms

انظر أيضاً: Centrifugal force

Position vector r, always points radially from the origin.

Velocity vector v, always tangent to the path of motion.

Acceleration vector a, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.

Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2d space, but a plane in any higher dimension.

The term ${\displaystyle{\displaystyle r{\dot{\varphi}}^{2}}}$ is sometimes referred to as the centripetal acceleration, and the term ${\displaystyle{\displaystyle 2{\dot{r}}{\dot{\varphi}}}}$ as the Coriolis acceleration. For example, see Shankar.

Note: these terms, that appear when acceleration is expressed in polar coordinates, are a mathematical consequence of differentiation; they appear whenever polar coordinates are used. In planar particle dynamics these accelerations appear when setting up Newton's second law of motion in a rotating frame of reference. Here these extra terms are often called fictitious forces; fictitious because they are simply a result of a change in coordinate frame. That does not mean they do not exist, rather they exist only in the rotating frame.

Inertial frame of reference S and instantaneous non-inertial co-rotating frame of reference S′. The co-rotating frame rotates at angular rate Ω equal to the rate of rotation of the particle about the origin of S′ at the particular moment t. Particle is located at vector position r(t) and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angle ϕ normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distance r need not be related to the radius of curvature of the path.

 Co-rotating frame

For a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneous co-rotating frame of reference. To define a co-rotating frame, first an origin is selected from which the distance r(t) to the particle is defined. An axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin. Then, at the selected moment t, the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, dφ/dt. Next, the terms in the acceleration in the inertial frame are related to those in the co-rotating frame. Let the location of the particle in the inertial frame be (r(t), φ(t)), and in the co-rotating frame be (r′(t), φ′(t)). Because the co-rotating frame rotates at the same rate as the particle, dφ′/dt = 0. The fictitious centrifugal force in the co-rotating frame is mrΩ², radially outward. The velocity of the particle in the co-rotating frame also is radially outward, because dφ′/dt = 0. The fictitious Coriolis force therefore has a value −2m(dr/dt)Ω, pointed in the direction of increasing φ only. Thus, using these forces in Newton's second law we find:

$${\displaystyle\mathbf{F}+\mathbf{F}_{\text{cf}}+\mathbf{F}_{\text{Cor}}=m{\ddot{\mathbf{r}}}\,,}$$

where over dots represent derivatives with respect to time, and F is the net real force (as opposed to the fictitious forces). In terms of components, this vector equation becomes:

$${\displaystyle{\begin{aligned} \displaystyle F_{r}+mr\Omega^{2}&\displaystyle=m{\ddot{r}}\\ \displaystyle F_{\varphi}-2m{\dot{r}}\Omega&\displaystyle=mr{\ddot{\varphi}}\ ,\end{aligned}}}$$

which can be compared to the equations for the inertial frame:

$${\displaystyle{\begin{aligned} \displaystyle F_{r}&\displaystyle=m{\ddot{r}}-mr{\dot{\varphi}}^{2}\\ \displaystyle F_{\varphi}&\displaystyle=mr{\ddot{\varphi}}+2m{\dot{r}}{\dot{\varphi}}\ .\end{aligned}}}$$

This comparison, plus the recognition that by the definition of the co-rotating frame at time t it has a rate of rotation Ω = dφ/dt, shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame.

For general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous osculating circle of its motion, not to a fixed center of polar coordinates. For more detail, see centripetal force.

 انظر أيضا

• نظام إحداثي
• نظام إحداثي ديكارتي
• إحداثيات

 وصلات خارجية

• Hazewinkel, Michiel, ed. (2001), "Polar coordinates", Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Coordinate Converter — converts between polar, Cartesian and spherical coordinates
• Polar Coordinate System Dynamic Demo