The Polar Coordinate System

Introduction to the Polar Coordinate System

The polar coordinate system is an alternate coordinate system where the two variables are
rr
 and
θ\theta
, instead of
xx
 and
yy
.

Learning Objectives

Discuss the characteristics of the polar coordinate system

Key Takeaways

Key Points

  • A polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.


Key Terms

  • radius: A distance measured from the pole.
  • angular coordinate: An angle measured from the polar axis, usually counter-clockwise.
  • pole: The reference point of the polar graph.
  • polar axis: A ray from the pole in the reference direction.


Introduction of Polar Coordinates

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.

When we think about plotting points in the plane, we usually think of rectangular coordinates 
(x,y)(x,y)
 in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. Polar coordinates are points labeled
(r,θ)(r,θ)
 and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.

The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth. The radial coordinate is often denoted by
rr
or
ρρ
, and the angular coordinate by
ϕϕ
,
θθ
, or
tt
.

The polar axis L extends horizontally to the right in this image. The point (3, 60 degrees) is 3 units out from the origin and 60 degrees up from the polar axis. The point (4, 210 degrees) is 4 units out from the origin and 210 degrees around from the polar axis. Examples of Polar Coordinates: Points in the polar coordinate system with pole
00
 and polar axis
LL
. In green, the point with radial coordinate
33
and angular coordinate
6060
degrees or
(3,60)(3,60^{\circ})
. In blue, the point
(4,210)(4,210^{\circ})
.


A polar coordinate system can be drawn with the four Cartesian quadrants for reference. It has rings of concentric circles at 1, 2, 3, etc units out, and slanted lines denoting the angles. Angles 0 and 180 are on the x-axis and 90 and 270 on the y axis. 30 and 60 degrees are in quadrant 1, 120 and 150 degrees are in quadrant 2, 210 and 240 are in quadrant 3, and 300 and 330 are in quadrant 4. Polar Graph Paper: A polar grid with several angles labeled in degrees


Angles in polar notation are generally expressed in either degrees or radians (
2π2\pi
rad being equal to
360360^{\circ}
). Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics.  In many contexts, a positive angular coordinate means that the angle
ϕϕ
is measured counterclockwise from the axis.  In mathematical literature, the polar axis is often drawn horizontal and pointing to the right.

Plotting Points Using Polar Coordinates

The polar grid is scaled as the unit circle with the positive
xx
-axis now viewed as the polar axis and the origin as the pole. The first coordinate 
rr
 is the radius or length of the directed line segment from the pole. The angle 
θθ
, measured in radians, indicates the direction of
rr
. We move counterclockwise from the polar axis by an angle of 
θθ
,and measure a directed line segment the length of 
rr
 in the direction of
θθ
. Even though we measure
θθ
 first and then 
rr
, the polar point is written with the
rr
-coordinate first. For example, to plot the point 
(2,π4)(2,\frac{\pi }{4})
,we would move 
π4\frac{\pi }{4}
 units in the counterclockwise direction and then a length of
22
from the pole. This point is plotted on the grid in Figure.

The point is in the first quadrant. Plotting a point on a Polar Grid: Plot of the point 
(2,π4)(2,\frac{\pi }{4})
,by moving 
π4\frac{\pi }{4}
 units in the counterclockwise direction and then a length of
22
 from the pole.


Uniqueness of polar coordinates

Adding any number of full turns (
360360^{\circ}
 or
2π2\pi
 radians) 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±n360°r, \phi \pm n\cdot 360°
) or (
r,ϕ±(2n+1)180°-r, \phi \pm (2n + 1)\cdot 180°
), where
nn
is any integer. Moreover, the pole itself can be expressed as (
0,ϕ0, ϕ
) for any angle
ϕϕ
.

Converting Between Polar and Cartesian Coordinates

Polar and Cartesian coordinates can be interconverted using the Pythagorean Theorem and trigonometry.

Learning Objectives

Derive and use the formulae for converting between Polar and Cartesian coordinates

Key Takeaways

Key Points

  • To convert from polar to rectangular (Cartesian) coordinates use the following formulas (derived from their trigonometric function definitions):
  • cosθ=xrx=rcosθsinθ=yry=rsinθr2=x2+y2tanθ=yx\cos \theta =\frac{x}{r}\rightarrow x=r\cos \theta\\\sin \theta =\frac{y}{r}\rightarrow y=r\sin \theta \\r^2=x^2+y^2\\\tan\theta=\frac{y}{x}


Polar Coordinates to Rectangular (Cartesian) Coordinates

When given a set of polar coordinates, we may need to convert them to rectangular coordinates. To do so, we can recall the relationships that exist among the variables  
xx
, 
yy
, 
rr
, and 
θθ
, from the definitions of
cosθ\cos \theta
 and
sinθ\sin \theta
.  Solving for the variables
xx
 and
yy
 yields the following formulas:

cosθ=xrx=rcosθ\displaystyle \cos \theta =\frac{x}{r}\quad\Rightarrow\quad x=r\cos \theta


sinθ=yry=rsinθ\displaystyle \sin \theta =\frac{y}{r}\quad\Rightarrow\quad y=r\sin \theta


An easy way to remember the equations above is to think of 
cosθ\cos\theta
 as the adjacent side over the hypotenuse and 
sinθ\sin\theta
 as the opposite side over the hypotenuse.  Dropping a perpendicular from the point in the plane to the
xx
-axis forms a right triangle, as illustrated in Figure below.

The point (x, y) or (r, theta) in the first quadrant has a line from the origin to it, r. This is the hypotenuse of a right triangle whose legs are the coordinates x and y. Trigonometry Right Triangle: A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.


To convert polar coordinates 
(r,θ)(r,θ)
 to rectangular coordinates
(x,y)(x,y)
 follow these steps:

1) Write
cosθ=xrx=rcosθ\cos \theta =\frac{x}{r}\Rightarrow x=r\cos \theta
 and
sinθ=yry=rsinθ\sin \theta =\frac{y}{r}\Rightarrow y=r\sin \theta
.

2) Evaluate 
cosθ\cos\theta
 and 
sinθ\sin\theta
.

3) Multiply
cosθ\cos\theta
 by
rr
 to find the 
xx
-coordinate of the rectangular form.

4) Multiply 
sinθ\sin\theta
 by 
rr
 to find the 
yy
-coordinate of the rectangular form.

Example:   Write the polar coordinates 
(3,π2)(3,\frac {\pi}{2})
 as rectangular coordinates.

x=rcosθ=3cosπ2=0\displaystyle \begin{align} x &= r\cos \theta \\ &= 3cos \frac{\pi}{2}\\ &= 0 \end{align}


y=rsinθ=3sinπ2=3\displaystyle \begin{align} y&=r\sin \theta\\&=3\sin\frac{\pi}{2}\\&=3 \end{align}


The rectangular coordinates are
(0,3)(0,3)
.

Both points are 3 units up the y-axis. Polar and Coordinate Grid of Equivalent Points: The rectangular coordinate
(0,3)(0,3)
 is the same as the polar coordinate
(3,π2)(3,\frac {\pi}{2})
 as plotted on the two grids above.


Rectangular (Cartesian) Coordinates to Polar Coordinates

To convert rectangular coordinates to polar coordinates, we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.

Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated below. Recall:

cosθ=xrx=rcosθsinθ=yry=rsinθr2=x2+y2tanθ=yx\displaystyle \begin{align} \cos \theta &=\frac{x}{r}\quad\Rightarrow\quad x=r\cos \theta \\\sin \theta &=\frac{y}{r}\quad\Rightarrow\quad y=r\sin \theta \\ r^2&=x^2+y^2\\\tan\theta&=\frac{y}{x} \end{align}


The point (x, y) or (r, theta) in the first quadrant has a line from the origin to it, r. This is the hypotenuse of a right triangle whose legs are the coordinates x and y. Trigonometry Right Triangle: A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.


Example:

Convert the rectangular coordinates 
(3,3)(3,3)
 to polar coordinates.

We are given the values of
xx
 and
yy
 and need to solve for
θ\theta
 and
rr
.  Start by solving for
θ\theta
 using the
tan\tan
 function:

tanθ=yx=33=1\displaystyle \begin{align} \tan \theta&=\frac{y}{x}\\&=\frac{3}{3}\\&=1\\ \end{align}


So:

θ=tan1(1)=π4\displaystyle \begin{align} \theta &= \tan^{-1}\left( 1 \right)\\ &=\frac{\pi}{4} \end{align}


Next substitute the values of
xx


and
yy
 into the formula
r2=x2+y2r^2=x^2+y^2
 and solve for
rr
.

r2=x2+y2=32+32=18\displaystyle \begin{align} r^2&=x^2+y^2\\ &=3^2+3^2\\ &=18\\ \end{align}


So:

r=18=32\displaystyle \begin{align} r&=\sqrt{18}\\ &=3\sqrt{2} \end{align}


The polar coordinates are
(32,π4)(3\sqrt2,\frac{\pi}{4})
.

Note that
r2=18r^2 = 18
 implies
r=±18r=\pm\sqrt{18}
. We chose to ignore the negative
rr
 value. Also note that
tan1(1)\tan^{-1}\left( 1 \right)
 has many answers. This corresponds to the non-uniqueness of polar coordinates. Multiple sets of polar coordinates can have the same location as our first solution. For example, the points

(32,5π4)(-3\sqrt2,\frac{5\pi}{4})


and 
(32,7π2)(3\sqrt2,-\frac{7\pi}{2})
 will coincide with the original solution of 
(32,π4)(3\sqrt2,\frac{\pi}{4})
.

Conics in Polar Coordinates

Polar coordinates allow conic sections to be expressed in an elegant way.

Learning Objectives

Describe the equations for different conic sections in polar coordinates

Key Takeaways

Key Points

  • Conic sections have several key features which define their polar equation; foci, eccentricity, and a directrix.
  • All conic sections have the same basic equation in polar coordinates, which demonstrates a connection between all of them.


Key Terms

  • eccentricity: A measure of deviation from a prescribed curve.
  • directrix: A fixed line used to described a curve.


Defining a Conic

Previously, we learned how a parabola is defined by the focus (a fixed point ) and the directrix (a fixed line).

The parabola opens to the right, with its line of symmetry the polar axis. Inside the parabola on the polar axis is the focus, and the directrix is to the left of the parabola perpendicular to the polar axis. Parts of a Parabola: Consider the parabola
x=2+y2x=2+y^2
. Any conic may be determined by three characteristics: a single focus, a fixed line called the directrix, and the ratio of the distances of each to a point on the graph.
We can define any conic in the polar coordinate system in terms of a fixed point, the focus
P(r,θ)P(r,θ)
 at the pole, and a line, the directrix, which is perpendicular to the polar axis.

For a conic with eccentricity 
ee
,

  1. If 
    0e<10≤e<1
    , the conic is an ellipse.
  2. If 
    e=1e=1
    , the conic is a parabola.
  3. If 
    e>1e>1
    , the conic is an hyperbola .


With this definition, we may now define a conic in terms of the directrix: 
x=±px=±p
, the eccentricity 
ee
, and the angle 
θ\theta
. Thus, each conic may be written as a polar equation in terms of 
rr
 and 
θ\theta
.

For a conic with a focus at the origin, if the directrix is 
x=±px=±p
, where 
pp
 is a positive real number, and the eccentricity is a positive real number 
ee
, the conic has a polar equation:

r=ep1±ecosθ\displaystyle r=\frac{e\cdot p}{1\: \pm\: e\cdot\cos\theta}


For a conic with a focus at the origin, if the directrix is 
y=±py=±p
, where 
pp
 is a positive real number, and the eccentricity is a positive real number 
ee
, the conic has a polar equation:

r=ep1±esinθ\displaystyle r=\frac{e\cdot p}{1\: \pm\: e\cdot\sin\theta}


Other Curves in Polar Coordinates

Some curves have a simple expression in polar coordinates, whereas they would be very complex to represent in Cartesian coordinates.

Learning Objectives

Describe the equations for spirals and roses in polar coordinates

Key Takeaways

Key Points

  • The formulas that generate the graph of a rose curve are given by:
    r=acosnθr=a\:\cos n\theta
      and 
    r=asinnθr=a\:\sin n\theta
      where 
    a0a \ne 0
    .  If  
    nn
     is even, the curve has 
    2n2n
     petals. If 
    nn
     is odd, the curve has 
    nn
     petals.
  • The formula that generates the graph of the Archimedes’ spiral is given by:
    r=θr=θ
     for  
    θ0θ≥0
    .  As  
    θ\theta
     increases, 
    rr
     increases at a constant rate in an ever-widening, never-ending, spiraling path.


Key Terms

  • Archimedes’ spiral: A curve given by an equation of the form
    r=a+bθr=a + b\theta
  • rose curve: A curve given by an equation of the form
    r=acosnθr = a\cos n\theta
    or
    r=asinnθr=a\sin n\theta


To graph in the rectangular coordinate system we construct a table of 
xx
 and 
yy
  values. To graph in the polar coordinate system we construct a table of 
rr
 and
θ\theta
 values. We enter values of 
θ\theta
 into a polar equation  and calculate 
rr
. However, using the properties of symmetry and finding key values of 
θ\theta
 and 
rr
 means fewer calculations will be needed.

Investigating Rose Curves

Polar equations can be used to generate unique graphs. The following type of polar equation produces a petal-like shape called a rose curve. Although the graphs look complex, a simple polar equation generates the pattern. The formulas that generate the graph of a rose curve are given by:

r=acos(nθ)andr=asin(nθ)wherea0\displaystyle r=a\cdot\cos \left( n\theta \right) \qquad \text{and} \qquad r=a\cdot\sin \left( n\theta \right) \qquad \text{where} \qquad a\ne 0


If 
nn
 is even, the curve has 
2n2n
 petals. If 
nn
 is odd, the curve has 
nn
 petals.

For the cosine curve, n is even, and the curve has 8 petals (four centered on each of the axes, and four in between them). The petals all meet at the origin. For the sine curve, n is odd, and the curve has 3 petals; one centered on the negative y axis, and one each in the second and first quadrants. The petals all meet at the origin. Rose Curves: Complex graphs generated by the simple polar formulas that generate rose curves:
r=acosnθr=a\:\cos n\theta
 and
r=asinnθr=a\:\sin n\theta
 where
a0a≠0
.


Investigating the Archimedes' Spiral

Archimedes’ spiral is named for its discoverer, the Greek mathematician Archimedes (
c.287BCEc.212BCEc. 287 BCE - c. 212 BCE
), who is credited with numerous discoveries in the fields of geometry and mechanics.

The formula that generates the graph of the Archimedes’ spiral is given by:

r=a+bθforθ0\displaystyle r=a + b\theta \qquad \text{for} \qquad \theta\geq 0


As 
θ\theta
 increases, 
rr
 increases at a constant rate in an ever-widening, never-ending, spiraling path.

Spirals going out counterclockwise from the origin. From theta = 0 to 2pi the spiral makes one rotation, and then another rotation if theta runs from 0 to 4pi. Archimedes' Spiral: The formula that generates the graph of a spiral is
r=θr=θ
 for
θ0θ≥0
.


Licenses and Attributions

More Study Resources for You

Show More