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Polar coordinate system

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   A polar grid with several angles labeled.
   Enlarge
   A polar grid with several angles labeled.

   In mathematics, the polar coordinate system is a two-dimensional
   coordinate system in which points are given by an angle and a distance
   from a central point known as the pole (equivalent to the origin in the
   more familiar Cartesian coordinate system). The polar coordinate system
   is used in many fields, including mathematics, physics, engineering,
   navigation and robotics. It is especially useful in situations where
   the relationship between two points is most easily expressed in terms
   of angles and distance; in the Cartesian coordinate system, such a
   relationship can only be found through trigonometric formulae. For many
   types of curves, a polar equation is the simplest means of
   representation; for some others, it is the only such means.

History

   It is known that the Greeks used the concepts of angle and radius. The
   astronomer Hipparchus (190-120 BC) tabulated 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. In On
   Spirals, Archimedes describes his famous spiral, a function whose
   radius depends on the angle. The Greek work, however, did not extend to
   a full coordinate system.

   There are various accounts of who first introduced 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. Grégoire de Saint-Vincent and Bonaventura Cavalieri
   independently introduced the concepts at about the same time.
   Saint-Vincent wrote about them privately in 1625 and published in 1647,
   while Cavalieri published in 1635 with a corrected version appearing in
   1653. Cavalieri first utilized 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
   was the first to look upon polar coordinates as a method of locating
   any point in the plane. Newton examined the transformations between
   polar coordinates and nine other coordinate systems. In 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.

   Alexis Clairaut and Leonhard Euler are credited with extending the
   concept of polar coordinates to three dimensions.

Plotting points with polar coordinates

   The points (3,60°) and (4,210°)
   Enlarge
   The points (3,60°) and (4,210°)

   As with all two-dimensional coordinate systems, there are two polar
   coordinates: r (the radial coordinate) and θ (the angular coordinate,
   polar angle, or azimuth angle, sometimes represented as φ or t). The r
   coordinate represents the radial distance from the pole, and the θ
   coordinate represents the anticlockwise (counterclockwise) angle from
   the 0° ray (sometimes called the polar axis), known as the positive
   x-axis on the Cartesian coordinate plane.

   For example, the polar coordinates (3,60°) would be plotted as a point
   3 units from the pole on the 60° ray. The coordinates (−3,240°) would
   also be plotted at this point because a negative radial distance is
   measured as a positive distance on the opposite ray
   (240° − 180° = 60°).

   One important aspect of the polar coordinate system not present in the
   Cartesian coordinate system is the ability to express a single point
   with an infinite number of different coordinates. In general, the point
   (r, θ) can be represented as (r, θ ± n×360°) or (−r, θ ± (2n + 1)180°),
   where n is any integer. If the r coordinate of a point is 0, then
   regardless of the θ coordinate, the point will be located at the pole.

Use of radian measure

   Angles in polar notation are generally expressed in either degrees or
   radians, using the conversion 2π rad = 360°. The choice depends largely
   on the context. Navigation applications use degree measure, while some
   physics applications (specifically rotational mechanics) use radian
   measure, based on the ratio of the radius of the circle to its
   circumference.

Converting between polar and Cartesian coordinates

   The two polar coordinates r and θ can be converted to Cartesian
   coordinates by

          x = r \cos \theta \,
          y = r \sin \theta. \,

   From the above two formulas, r and θ can be defined in terms of x and
   y:

          r = \sqrt{x^2 + y^2} \,
          \theta = \begin{cases} \arctan(\frac{y}{x}) & \mbox{if }x>0
          \mbox{ and }y\ge0\\ \arctan(\frac{y}{x}) + 2\pi& \mbox{if }x>0
          \mbox{ and }y<0\\ \arctan(\frac{y}{x}) + \pi& \mbox{if }x<0\\
          \frac{\pi}{2} & \mbox{if }x=0 \mbox{ and }y>0\\ \frac{3\pi}{2} &
          \mbox{if }x=0 \mbox{ and }y<0 \end{cases}

   With this formula θ is obtained in [0, 2π), or [0°, 360°).

Polar equations

   The equation of a curve expressed in polar coordinates is known as a
   polar equation, and is usually written with r as a function of θ.

   Polar equations may exhibit different forms of symmetry. If
   r(−θ) = r(θ) then the curve will be symmetrical about the horizontal
   (0°/180°) ray; if r(π−θ) = r(θ) then it will be symmetric about the
   vertical (90°/270°); if r(θ−α) = r(θ) then it will be rotationally
   symmetric α° counterclockwise about the pole.

Circle

   A circle with equation r(θ) = 1.
   Enlarge
   A circle with equation r(θ) = 1.

   The general equation for any circle with a centre at (r[0], φ) and
   radius a is

          r^2 - 2 r r_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

          r(\theta)=a \,

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

Line

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

          \theta = \varphi \, ,

   where φ is the angle of elevation of the line; that is, φ = arctan m
   with m the slope of the line in the Cartesian coordinate system. Any
   line that does not run through the pole is perpendicular to some radial
   line. The line that crosses the line θ = φ perpendicularly at the point
   (r[0], φ) has equation

          r(\theta) = {r_0}\sec(\theta-\varphi) \, .

Polar rose

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

   A polar rose is a famous mathematical curve which looks like a petalled
   flower, and which can only be expressed as a polar equation. It is
   given by the equations

          r(\theta) = a \cos k\theta \, OR
          r(\theta) = a \sin k\theta \,

   If k is an integer, these equations will produce a k-petalled rose if k
   is odd, or a 2k-petalled rose if k is even. If k is not an integer, a
   disc is formed, as the number of petals is also not an integer. Note
   that with these equations it is impossible to make a rose with 2 more
   than a multiple of 4 (2, 6, 10, 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(θ) = θ for 0 < θ < 6π.
   Enlarge
   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 only as a polar equation. It is
   represented by the equation:

          r(\theta) = a+b\theta \, .

   Changing the parameter a will turn the spiral, while b controls the
   distance between the arms, which is always constant.. The Archimedean
   spiral has two arms, one for θ > 0 and one for θ < 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 known curves where the radius of the
   function is dependent on the angle, which is the basis of the polar
   coordinate system.

Conic sections

   Ellipse, showing semi-latus rectum
   Enlarge
   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 = {l\over (1 + e \cos \theta)}

   where e is the eccentricity and l 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 1.

Other curves

   Because of the circular nature of the polar coordinate system, it is
   much simpler to describe many curves with an equation in polar rather
   than Cartesian form. Among the best known of these curves are
   lemniscates, limaçons, and cardioids.

Complex numbers

   Complex numbers, written in rectangular form as a + bi, can also be
   expressed in polar form in two different ways:
    1. r(\cos\theta+i\sin\theta) \, , abbreviated r \mbox{ cis } \theta \,
    2. r e^{i\theta} \,

   which are equivalent as per Euler's formula. To convert between
   rectangular and polar complex numbers, the following conversion
   formulas are used:

          a = r \cos \theta \,
          b = r \sin \theta \,
          and therefore r = \sqrt{a^2 + b^2} \,

   For the operations of multiplication, division, exponentiation, and
   finding roots of complex numbers, it is much easier to use polar
   complex numbers than rectangular complex numbers. In abbreviated form:
     * Multiplication: (r \mbox{ cis } \theta)(R \mbox{ cis } \varphi) =
       rR \mbox{ cis } (\theta+\varphi) \,
     * Division: \frac{r \mbox{ cis } \theta}{R \mbox{ cis } \varphi} =
       \frac{r}{R} \mbox{ cis } (\theta-\varphi) \,
     * Exponentiation ( De Moivre's formula): (r \mbox{ cis } \theta)^n =
       r^n \mbox{ cis } (n\theta) \,

Calculus

   Calculus can be applied to equations expressed in polar coordinates.

Differential calculus

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

          x=r(\theta)\cos(\theta) \,
          y=r(\theta)\sin(\theta) \,

   Differentiating both equations with respect to θ yields

          \tfrac{dx}{d\theta}=r'(\theta)\cos(\theta)-r(\theta)\sin(\theta)
          \,
          \tfrac{dy}{d\theta}=r'(\theta)\sin(\theta)+r(\theta)\cos(\theta)
          \,

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

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

Integral calculus

   To find the area under a curve r(θ) on a closed interval [a, b], where
   0 < b − a < 2π, the curve is first expressed as a Riemann sum. First
   the interval [a, b] is divided into n subintervals, where n is an
   arbitrary positive integer. Thus Δθ, 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 θ[i]
   be the midpoint of the subinterval, and construct a sector with the
   centre at the pole, radius r(θ[i]), and central angle Δθ. The area of
   each constructed sector is therefore equal to
   \tfrac12r(\theta_i)^2\Delta\theta . Hence, the total area of all of the
   sectors is

          \sum_{i=1}^n \tfrac12r(\theta_i)^2\Delta\theta.

   As the number of subintervals n is increased, the approximation of the
   area continues to improve. Thus, the area under the curve r(θ) on
   [a, b] can be defined as

          \lim_{n \to \infty} \sum_{i=1}^n
          \tfrac12r(\theta_i)^2\Delta\theta,

   the Riemann sum for the integral

          \frac12\int_a^b r(\theta_i)^2 d\theta.

Vector calculus

   Vector calculus can also be applied to polar coordinates. Let
   \mathbf{r} be the position vector (r\cos(\theta),r\sin(\theta))\, ,
   with r and θ depending on time t, \hat{\mathbf{r}} be a unit vector in
   the direction \mathbf{r} and \hat{\boldsymbol\theta} be a unit vector
   at right angles to \mathbf{r} . The first and second derivatives of
   position are

          \frac{d\mathbf{r}}{dt} = \dot r\hat{\mathbf{r}} +
          r\dot\theta\hat{\boldsymbol\theta}, .
          \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}.

   Let \mathbf{A} be the area swept out by a line joining the focus to a
   point on the curve. In the limit d\mathbf{A} is half the area of the
   parallelogram formed by \mathbf{r} and d\mathbf{r} ,

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

   and the total area will be the integral of d\mathbf{A} with respect to
   time.

Three dimensions

   The polar coordinate system is extended into three dimensions with two
   different coordinate systems, the cylindrical and spherical coordinate
   systems, both of which include the two-dimensional polar coordinates as
   a subset.

Cylindrical coordinates

   2 points plotted with cylindrical coordinates
   Enlarge
   2 points plotted with cylindrical coordinates

   The cylindrical coordinate system is a coordinate system that
   essentially extends the two-dimensional polar coordinate system by
   adding a third coordinate measuring the height of a point above the
   plane, similar to the way in which the Cartesian coordinate system is
   extended into three dimensions. The third coordinate is usually denoted
   h, making the three cylindrical coordinates (r, θ, h).

   The three cylindrical coordinates can be converted to Cartesian
   coordinates by

          {x}={r} \,\cos\theta
          {y}={r} \, \sin\theta
          {z}={h} \,

Spherical coordinates

   A point plotted using spherical coordinates
   Enlarge
   A point plotted using spherical coordinates

   Polar coordinates can also be extended into three dimensions using the
   coordinates (ρ, φ, θ), where ρ is the distance from the pole, φ is the
   angle from the z-axis (called the colatitude or zenith and measured
   from 0 to 180°) and θ is the angle from the x-axis (as in the polar
   coordinates). This coordinate system, called the spherical coordinate
   system, is similar to the latitude and longitude system used for Earth,
   with the latitude being the complement of φ, determined by δ = 90° − φ,
   and the longitude being measured by l = θ − 180°.

   The three spherical coordinates are converted to Cartesian coordinates
   by

          x =\rho \, \sin\phi \, \cos\theta
          y =\rho \, \sin\phi \, \sin\theta
          z =\rho \, \cos\phi

Applications

Robotics

   Many robots capable of movement use the polar coordinate system (or a
   slightly modified version of it) for navigation. This is very
   convenient for artificial intelligence, as the centre of the coordinate
   system (the pole) is always placed at the robot's present location.
   Therefore, the robot need not calculate where it is within the
   coordinate system at any given time; all it needs to determine is in
   which direction and how far to move. If robots were to navigate using
   the Cartesian coordinate system, distance and angles required for
   movement would have to be calculated using algebra and trigonometry. In
   the polar coordinate system, a robot simply needs to be told how far to
   move, and in what direction (given as an angle), and the robot knows
   exactly where to go.

Aviation

   In the air, aircraft use a slightly modified version of the polar
   coordinate system for navigation. In this coordinate system, the 0° ray
   (generally called heading 360) is vertical, and the angles continue in
   a clockwise, rather than a counterclockwise, direction. 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 niner-zero by air traffic control).

Archimedean spiral

   The Archimedean spiral has many real-world applications. Scroll
   compressors, made from two interleaved Archimedean spirals of the same
   size, are used for compressing liquids and gases. The grooves of very
   early gramophone records form an Archimedean spiral, making the grooves
   evenly spaced and maximizing the amount of music that could be fit onto
   the record (although this was later changed to allow better sound
   quality). Archimedean spirals are also used in DLP projection systems
   to minimize the "Rainbow Effect", making it look as if multiple colors
   are displayed at the same time, when in reality red, green, and blue
   are being cycled extremely fast.

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