Contents 1 History 2 Conventions 2.1 Uniqueness of polar coordinates 3 Converting between polar and Cartesian coordinates 4 Polar equation of a curve 4.1 Circle 4.2 Line 4.3 Polar rose 4.4 Archimedean spiral 4.5 Conic sections 5 Intersection of two polar curves 6 Complex numbers 7 Calculus 7.1 Differential calculus 7.2 Integral calculus (arc length) 7.3 Integral calculus (area) 7.3.1 Generalization 7.4 Vector calculus 7.4.1 Centrifugal and Coriolis terms 7.4.1.1 Co-rotating frame 8 Connection to spherical and cylindrical coordinates 9 Applications 9.1 Position and navigation 9.2 Modeling 10 See also 11 References 12 External links

History See also: History of trigonometric functions Hipparchus The concepts of angle and radius were already used by ancient peoples of the first millennium BC. The Greek astronomer and astrologer Hipparchus (190–120 BC) created a table of chord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.[2] In On Spirals, Archimedes describes the Archimedean spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system. From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Mecca (qibla)—and its distance—from any location on the Earth.[3] From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. The calculation is essentially the conversion of the equatorial polar coordinates of Mecca (i.e. its longitude and latitude) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the great circle through the given location and the Earth's poles, and whose polar axis is the line through the location and its antipodal point.[4] There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidge's Origin of Polar Coordinates.[5] Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs. In Method of Fluxions (written 1671, published 1736), Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems.[6] In the journal Acta Eruditorum (1691), Jacob Bernoulli used a system with a point on a line, called the pole and polar axis respectively. Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates. The actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in George Peacock's 1816 translation of Lacroix's Differential and Integral Calculus.[7][8] Alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them.[5]

Conventions A polar grid with several angles labeled in degrees The radial coordinate is often denoted by r or ρ, and the angular coordinate by ϕ, θ, or t. The angular coordinate is specified as ϕ by ISO standard 31-11. Angles in polar notation are generally expressed in either degrees or radians (2π rad being equal to 360°). Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics.[9] In many contexts, a positive angular coordinate means that the angle ϕ is measured counterclockwise from the axis. In mathematical literature, the polar axis is often drawn horizontal and pointing to the right. Uniqueness of polar coordinates Adding any number of full turns (360°) to the angular coordinate does not change the corresponding direction. Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the same point can be expressed with an infinite number of different polar coordinates (r, ϕ ± n×360°) or (−r, ϕ ± (2n + 1)180°), where n is any integer.[10] Moreover, the pole itself can be expressed as (0, ϕ) for any angle ϕ.[11] Where a unique representation is needed for any point, it is usual to limit r to non-negative numbers (r ≥ 0) and ϕ to the interval [0, 360°) or (−180°, 180°] (in radians, [0, 2π) or (−π, π]).[12] One must also choose a unique azimuth for the pole, e.g., ϕ = 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 ⁡ ( 6 x ) + 2 {\displaystyle y=\sin(6x)+2} is mapped onto r = sin ⁡ ( 6 φ ) + 2 {\displaystyle 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 ⁡ φ {\displaystyle x=r\cos \varphi } y = r sin ⁡ φ {\displaystyle 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:[13] r = x 2 + y 2 {\displaystyle r={\sqrt {x^{2}+y^{2}}}\quad } (as in the Pythagorean theorem or the Euclidean norm), and φ = atan2 ⁡ ( y , x ) {\displaystyle \varphi =\operatorname {atan2} (y,x)\quad } , where atan2 is a common variation on the arctangent function defined as atan2 ⁡ ( y , x ) = { arctan ⁡ ( y x ) if  x > 0 arctan ⁡ ( y x ) + π if  x < 0  and  y ≥ 0 arctan ⁡ ( y x ) − π if  x < 0  and  y < 0 π 2 if  x = 0  and  y > 0 − π 2 if  x = 0  and  y < 0 undefined if  x = 0  and  y = 0 {\displaystyle \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.

Intersection of two polar curves The graphs of two polar functions r = f ( θ ) {\displaystyle r=f(\theta )} and r = g ( θ ) {\displaystyle r=g(\theta )} have possible intersections of three types: In the origin if the equations f ( θ ) = 0 {\displaystyle f(\theta )=0} and g ( θ ) = 0 {\displaystyle g(\theta )=0} have at least one solution each. All the points [ g ( θ i ) , θ i ] {\displaystyle [g(\theta _{i}),\theta _{i}]} where θ i {\displaystyle \theta _{i}} are the solutions to the equation f ( θ ) = g ( θ ) {\displaystyle f(\theta )=g(\theta )} . All the points [ g ( θ i ) , θ i ] {\displaystyle [g(\theta _{i}),\theta _{i}]} where θ i {\displaystyle \theta _{i}} are the solutions to the equation f ( θ + ( 2 k + 1 ) π ) = − g ( θ ) {\displaystyle f(\theta +(2k+1)\pi )=-g(\theta )} where k {\displaystyle 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 + i y {\displaystyle 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 ⋅ ( cos ⁡ φ + i sin ⁡ φ ) {\displaystyle z=r\cdot (\cos \varphi +i\sin \varphi )} and from there as z = r e i φ {\displaystyle z=re^{i\varphi }} where e is Euler's number, which are equivalent as shown by Euler's formula.[15] (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 φ 0 ⋅ r 1 e i φ 1 = r 0 r 1 e i ( φ 0 + φ 1 ) {\displaystyle r_{0}e^{i\varphi _{0}}\cdot r_{1}e^{i\varphi _{1}}=r_{0}r_{1}e^{i(\varphi _{0}+\varphi _{1})}} Division: r 0 e i φ 0 r 1 e i φ 1 = r 0 r 1 e i ( φ 0 − φ 1 ) {\displaystyle {\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): ( r e i φ ) n = r n e i n φ {\displaystyle (re^{i\varphi })^{n}=r^{n}e^{in\varphi }}

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.

See also Mathematics portal Curvilinear coordinates List of canonical coordinate transformations Log-polar coordinates Polar decomposition

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