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Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,60°). In blue, the point (4,210°).

গনিতে পোলার স্থানাংক ব্যবস্থা এক ধরনের দ্বিমাত্রিক স্থানাংক ব্যাবস্থা যেখানে সমতলের প্র্রতিটি বিন্দুর অবস্থান একটি বিন্দু ও একটি অক্ষরেখার সাপেক্ষে নির্নয় করা হয়।

মুল বিন্দুকে পোলার স্থানাংক ব্যাবস্থায় সাপেক্ষ বিন্দু এবং ধনাত্ব x অক্ষকে পোলার স্থানাংক ব্যবস্থায় রেফারেন্স অক্ষ হিসাবে ধরা হয়। মুল বিন্দু থেকে বিন্দুর দুরত্বকে ব্যাসার্ধ ভেক্টর বা ব্যাসার্ধ বলা হয় যাকে 'r' দ্বারা প্রকাশ করা হয় এবং r এর সাথে ধনাত্বক x অক্ষের উৎপন্ন ধনাত্বক কোণকে 'θ' দ্বার প্রকাশ করা হয়। একটি বিন্দকে পোলার স্থানাংক ব্যাবস্থাকে (r,θ) দ্বারা প্রকাশ করা হয়।

প্রকাশের উপায়

একটি পোলার স্থানাংক ব্যবস্থা বিভিন্ন কোণের মাধ্যমে ষাট মুলক পদ্ধতিতে

ব্যাসার্ধ ভেক্টরকে r উৎপন্ন কোনকে θ দ্বারা প্রকাশ করা হয়েছে।

পোলার স্থানাংক ব্যাবস্থায় কোনকে ষাট মুলক এককে ডিগ্রিতে এবং বৃত্তিয় এককে রেডিয়ানে প্রকাশ করা হয়। ষাট মুলক পদ্ধতি সাধারনত নেভিগেশন, সারভে সহ অনেক ফলিত মাধ্যমে প্রকাশ করা হয় যেখানে বৃত্তিয় পদ্ধতি গনিত এবং গাণিতিক পদার্থ বিজ্ঞানে ব্যবহৃত হয়।

অনেকের মতে একটি ধনাত্বক কোণ উৎপন্ন হয় যখন কোন টি স্থানাংকের অক্ষের সাথে ঘড়ির কাটার বিপরীতে নির্নয় করা হয়। গানিতিক সংস্কৃতিতে পোলার স্থানাংকের অক্ষ অনুভুমিক ভাবে আকা হয় এবং তা ডান দিকে অসিম পর্যন্ত আকা হয়

প্রতিটি বিন্দুর সতন্ত্র পোলার স্থানাংক ব্যবস্থা

কোণের প্রতিটি মানের সাথে প্রতিবার 360 ডিগ্রি বা 2(pi) কোণ যুক্ত করলে যে কোণ বিন্দুর দিক অপরিবর্তিত থাকে। আবার ব্যাসার্ধ ভেক্টরের মান ঋনাত্বক হলে তা বিপরীত দিকে দুরত্ব বুঝায়। সে ক্ষেত্রে একই বিন্দুকে অসীম সংখ্যাক মান দ্বারা প্রকাশ করা যাবে যেমন: (r, θ ± n×360°) অথবা (−r, θ ± (2n + 1)180°) যেখানে n যে কোন পূর্ন সংখ্যা। সেক্ষেত্রে পোলার স্থানাংকের সতন্ত্রতা প্রশ্নবাদ্ধ হয়। এই ত্রুটি নিরসনের জন্য ব্যসার্ধ ভেক্টরের মান সিমাবদ্ধ এবং তা ঋনাত্বক বাস্তব সংখ্যা গুলোর মাঝে আবদ্ধ (r ≥ 0)। এবং কৌনিক ব্যবধানের মান ষাট মুলক পদ্ধতিতে [0, 360°) এবং বৃত্তিয় পদ্ধতিতে [0, 2π) এর মধ্যে সিমাবদ্ধ রাখা হয়। স্থানাংক নির্নয়ের জন্য সতন্ত্র অক্ষ রেখাও প্রয়োজন। সতন্ত্র অক্ষ রেখাটিই মূল অক্ষ বি সাপেক্ষ অক্ষ যেখানে θ = 0.

Converting between polar and Cartesian coordinates

A diagram illustrating the relationship between polar and Cartesian coordinates.

A curve on the Cartesian plane can be mapped into polar coordinates. In this animation, $y=\sin(6x)+2$ is mapped onto $r=\sin(6\varphi )+2$. Click on image for details.

The polar coordinates r and ϕ can be converted to the Cartesian coordinates x and y by using the trigonometric functions sine and cosine:

$x=r\cos \varphi \,$

$y=r\sin \varphi \,$

The Cartesian coordinates x and y can be converted to polar coordinates r and ϕ with r ≥ 0 and ϕ in the interval (−π, π] by:

$r={\sqrt {x^{2}+y^{2}}}\quad$ (as in the Pythagorean theorem or the Euclidean norm), and

$\varphi =\operatorname {atan2} (y,x)\quad$,

where atan2 is a common variation on the arctangent function defined as

$\operatorname {atan2} (y,x)={\begin{cases}\arctan({\frac {y}{x}})&{\mbox{if }}x>0\\\arctan({\frac {y}{x}})+\pi &{\mbox{if }}x<0{\mbox{ and }}y\geq 0\\\arctan({\frac {y}{x}})-\pi &{\mbox{if }}x<0{\mbox{ and }}y<0\\{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y>0\\-{\frac {\pi }{2}}&{\mbox{if }}x=0{\mbox{ and }}y<0\\{\text{undefined}}&{\mbox{if }}x=0{\mbox{ and }}y=0\end{cases}}$

The value of ϕ above is the principal value of the complex number function arg applied to x+iy. An angle in the range [0, 2π) may be obtained by adding 2π to the value in case it is negative.

Polar equation of a curve

The equation defining an algebraic curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of ϕ. The resulting curve then consists of points of the form (r(ϕ), ϕ) and can be regarded as the graph of the polar function r.

Different forms of symmetry can be deduced from the equation of a polar function r. If r(−ϕ) = r(ϕ) the curve will be symmetrical about the horizontal (0°/180°) ray, if r(π − ϕ) = r(ϕ) it will be symmetric about the vertical (90°/270°) ray, and if r(ϕ − α) = r(ϕ) it will be rotationally symmetric by α counterclockwise about the pole.

Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, and cardioid.

For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.

Circle

A circle with equation r(ϕ) = 1

The general equation for a circle with a center at (r₀, $\gamma$) and radius a is

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

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

$r(\varphi )=a\,$

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

When r₀ = a, or when the origin lies on the circle, the equation becomes

$r=2a\cos(\varphi -\gamma )$.

In the general case, the equation can be solved for r, giving

$r=r_{0}\cos(\varphi -\gamma )+{\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\varphi -\gamma )}}$,

the solution with a minus sign in front of the square root gives the same curve.

Line

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

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

$\varphi =\gamma \,$,

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

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

Otherwise stated (r₀, ɣ) is the point in which the tangent intersects the imaginary circle of radius r₀.

Polar rose

A polar rose is a famous mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,

$r(\varphi )=a\cos(k\varphi +\gamma _{0})\,$

for any constant ɣ₀ (including 0). If k is an integer, these equations will produce a k-petaled rose if k is odd, or a 2k-petaled rose if k is even. If k is rational but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The variable a represents the length of the petals of the rose.

Archimedean spiral

One arm of an Archimedean spiral with equation r(ϕ) = ϕ / 2π for 0 < ϕ < 6π

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

$r(\varphi )=a+b\varphi .\,$

Changing the parameter a will turn the spiral, while b controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one for ϕ > 0 and one for ϕ < 0. The two arms are smoothly connected at the pole. Taking the mirror image of one arm across the 90°/270° line will yield the other arm. This curve is notable as one of the first curves, after the conic sections, to be described in a mathematical treatise, and as being a prime example of a curve that is best defined by a polar equation.

Conic sections

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:

$r={\ell \over {1-e\cos \varphi }}$

where e is the eccentricity and $\ell$ is the semi-latus rectum (the perpendicular distance at a focus from the major axis to the curve). If e > 1, this equation defines a hyperbola; if e = 1, it defines a parabola; and if e < 1, it defines an ellipse. The special case e = 0 of the latter results in a circle of radius $\ell$.

Intersection of two polar curves

The graphs of two polar functions $r=f(\theta )$ and $r=g(\theta )$ have possible intersections in 3 cases:

1. In the origin if the equations $f(\theta )=0$ and $g(\theta )=0$ have at least one solution each.
2. All the points $[g(\theta _{i}),\theta _{i}]$ where $\theta _{i}$ are the solutions to the equation $f(\theta )=g(\theta )$.
3. All the points $[g(\theta _{i}),\theta _{i}]$ where $\theta _{i}$ are the solutions to the equation $f(\theta +(2k+1)\pi )=-g(\theta )$ where $k$ is an integer.

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

Every complex number can be represented as a point in the complex plane, and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form). The complex number z can be represented in rectangular form as

$z=x+iy\,$

where i is the imaginary unit, or can alternatively be written in polar form (via the conversion formulae given above) as

$z=r\cdot (\cos \varphi +i\sin \varphi )$

and from there as

$z=re^{i\varphi }\,$

where e is Euler's number, which are equivalent as shown by Euler's formula. (Note that this formula, like all those involving exponentials of angles, assumes that the angle ϕ is expressed in radians.) To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used.

For the operations of multiplication, division, and exponentiation of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form. From the laws of exponentiation:

• Multiplication:

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

• Division:

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

• Exponentiation (De Moivre's formula):

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

Calculus

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.

Differential calculus

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)

$r{\frac {\partial u}{\partial r}}=r{\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial r}}+r{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial r}},$

${\frac {\partial u}{\partial \varphi }}={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial \varphi }}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial \varphi }},$

or

$r{\frac {\partial u}{\partial r}}=r{\frac {\partial u}{\partial x}}\cos \varphi +r{\frac {\partial u}{\partial y}}\sin \varphi =x{\frac {\partial u}{\partial x}}+y{\frac {\partial u}{\partial y}},$

${\frac {\partial u}{\partial \varphi }}=-{\frac {\partial u}{\partial x}}r\sin \varphi +{\frac {\partial u}{\partial y}}r\cos \varphi =-y{\frac {\partial u}{\partial x}}+x{\frac {\partial u}{\partial y}}.$

Hence, we have the following formulae:

$r{\frac {\partial }{\partial r}}=x{\frac {\partial }{\partial x}}+y{\frac {\partial }{\partial y}}\,$

${\frac {\partial }{\partial \varphi }}=-y{\frac {\partial }{\partial x}}+x{\frac {\partial }{\partial y}}.$

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

${\frac {\partial u}{\partial x}}={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial x}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial x}},$

${\frac {\partial u}{\partial y}}={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial y}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial y}},$

or

${\frac {\partial u}{\partial x}}={\frac {\partial u}{\partial r}}{\frac {x}{\sqrt {x^{2}+y^{2}}}}-{\frac {\partial u}{\partial \varphi }}{\frac {y}{x^{2}+y^{2}}}=\cos \varphi {\frac {\partial u}{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial u}{\partial \varphi }},$

${\frac {\partial u}{\partial y}}={\frac {\partial u}{\partial r}}{\frac {y}{\sqrt {x^{2}+y^{2}}}}+{\frac {\partial u}{\partial \varphi }}{\frac {x}{x^{2}+y^{2}}}=\sin \varphi {\frac {\partial u}{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial u}{\partial \varphi }}.$

Hence, we have the following formulae:

${\frac {\partial }{\partial x}}=\cos \varphi {\frac {\partial }{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial }{\partial \varphi }}\,$

${\frac {\partial }{\partial y}}=\sin \varphi {\frac {\partial }{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial }{\partial \varphi }}.$

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.

$x=r(\varphi )\cos \varphi \,$

$y=r(\varphi )\sin \varphi \,$

Differentiating both equations with respect to ϕ yields

${\frac {dx}{d\varphi }}=r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi \,$

${\frac {dy}{d\varphi }}=r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi .\,$

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

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

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

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

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 an arbitrary positive integer. Thus Δϕ, the length of each subinterval, is equal to b − a (the total length 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

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

Hence, the total area of all of the sectors is

$\sum _{i=1}^{n}{\tfrac {1}{2}}r(\varphi _{i})^{2}\,\Delta \varphi .$

As the number of subintervals n is increased, the approximation of the area continues to improve. In the limit as 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:

$J=\det {\frac {\partial (x,y)}{\partial (r,\varphi )}}={\begin{vmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \varphi }}\\[8pt]{\frac {\partial y}{\partial r}}&{\frac {\partial y}{\partial \varphi }}\end{vmatrix}}={\begin{vmatrix}\cos \varphi &-r\sin \varphi \\\sin \varphi &r\cos \varphi \end{vmatrix}}=r\cos ^{2}\varphi +r\sin ^{2}\varphi =r.$

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:

$\iint _{R}f(x,y)\,dA=\int _{a}^{b}\int _{0}^{r(\varphi )}f(r,\varphi )\,r\,dr\,d\varphi .$

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 mentioned above is retrieved by taking f identically equal to 1. A more surprising application of this result yields the Gaussian integral

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

Vector calculus

Vector calculus can also be applied to polar coordinates. For a planar motion, let $\mathbf {r}$ be the position vector (rcos(ϕ), rsin(ϕ)), with r and ϕ depending on time t.

We define the unit vectors

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

in the direction of r and

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

in the plane of the motion perpendicular to the radial direction, where ${\hat {\mathbf {k} }}$ is a unit vector normal to the plane of the motion.

Then

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

${\dot {\mathbf {r} }}=({\dot {x}},\ {\dot {y}})={\dot {r}}(\cos \varphi ,\ \sin \varphi )+r{\dot {\varphi }}(-\sin \varphi ,\ \cos \varphi )={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\varphi }}{\hat {\boldsymbol {\varphi }}}\ ,$

${\ddot {\mathbf {r} }}=({\ddot {x}},\ {\ddot {y}})={\ddot {r}}(\cos \varphi ,\ \sin \varphi )+2{\dot {r}}{\dot {\varphi }}(-\sin \varphi ,\ \cos \varphi )+r{\ddot {\varphi }}(-\sin \varphi ,\ \cos \varphi )-r{\dot {\varphi }}^{2}(\cos \varphi ,\ \sin \varphi )\ =$

$\left({\ddot {r}}-r{\dot {\varphi }}^{2}\right){\hat {\mathbf {r} }}+\left(r{\ddot {\varphi }}+2{\dot {r}}{\dot {\varphi }}\right){\hat {\boldsymbol {\varphi }}}\ =({\ddot {r}}-r{\dot {\varphi }}^{2}){\hat {\mathbf {r} }}+{\frac {1}{r}}\;{\frac {d}{dt}}\left(r^{2}{\dot {\varphi }}\right){\hat {\boldsymbol {\varphi }}}$

Centrifugal and Coriolis terms

See also: Mechanics of planar particle motion and Centrifugal force (rotating reference frame)

The term $r{\dot {\varphi }}^{2}$ is sometimes referred to as the centrifugal term, and the term $2{\dot {r}}{\dot {\varphi }}$ as the Coriolis term. For example, see Shankar. Although these equations bear some resemblance in form to the centrifugal and Coriolis effects found in rotating reference frames, nonetheless these are not the same things. For example, the physical centrifugal and Coriolis forces appear only in non-inertial frames of reference. In contrast, these terms, that appear when acceleration is expressed in polar coordinates, are a mathematical consequence of differentiation; these terms appear wherever polar coordinates are used. In particular, these terms appear even when polar coordinates are used in inertial frames of reference, where the physical centrifugal and Coriolis forces never appear.

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:

${\boldsymbol {F}}+{\boldsymbol {F_{cf}}}+{\boldsymbol {F_{Cor}}}=m{\ddot {\boldsymbol {r}}}\ ,$

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

$F_{r}+mr\Omega ^{2}=m{\ddot {r}}$

$F_{\varphi }-2m{\dot {r}}\Omega =mr{\ddot {\varphi }}\ ,$

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

$F_{r}=m{\ddot {r}}-mr{\dot {\varphi }}^{2}\$

$F_{\varphi }=mr{\ddot {\varphi }}+2m{\dot {r}}{\dot {\varphi }}\ .$

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.

Connection to spherical and cylindrical coordinates

The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate system.

Applications

Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. They are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. For instance, the examples above show how elementary polar equations suffice to define curves—such as the Archimedean spiral—whose equation in the Cartesian coordinate system would be much more intricate. Moreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates. The initial motivation for the introduction of the polar system was the study of circular and orbital motion.

Position and navigation

Polar coordinates are used often in navigation as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance, aircraft use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a clockwise direction, rather than counterclockwise, as in the mathematical system. Heading 360 corresponds to magnetic north, while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively. Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read zero-niner-zero by air traffic control).

Modeling

Systems displaying radial symmetry provide natural settings for the polar coordinate system, with the central point acting as the pole. A prime example of this usage is the groundwater flow equation when applied to radially symmetric wells. Systems with a radial force are also good candidates for the use of the polar coordinate system. These systems include gravitational fields, which obey the inverse-square law, as well as systems with point sources, such as radio antennas.

Radially asymmetric systems may also be modeled with polar coordinates. For example, a microphone's pickup pattern illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves. The curve for a standard cardioid microphone, the most common unidirectional microphone, can be represented as r = 0.5 + 0.5sin(ϕ) at its target design frequency. The pattern shifts toward omnidirectionality at lower frequencies.

See also

• Curvilinear coordinates
• List of canonical coordinate transformations
• Log-polar coordinates
• Polar decomposition

References

General

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

Specific

External links

• "Polar coordinates", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Coordinate Converter — converts between polar, Cartesian and spherical coordinates
• Polar Coordinate System Dynamic Demo

Category:Coordinate systems