The Polar Coordinate System

Introduction to the Polar Coordinate System

The polar coordinate system is an alternate coordinate system where the two variables are [latex]r[/latex] and [latex]\theta[/latex], instead of [latex]x[/latex] and [latex]y[/latex].

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 [latex](x,y)[/latex] 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 [latex](r,θ)[/latex] 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 [latex]r[/latex] or [latex]ρ[/latex], and the angular coordinate by [latex]ϕ[/latex], [latex]θ[/latex], or [latex]t[/latex].

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 [latex]0[/latex] and polar axis [latex]L[/latex]. In green, the point with radial coordinate [latex]3[/latex] and angular coordinate [latex]60[/latex] degrees or [latex](3,60^{\circ})[/latex]. In blue, the point [latex](4,210^{\circ})[/latex].

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 ([latex]2\pi [/latex] rad being equal to [latex]360^{\circ}[/latex]). 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 [latex]ϕ[/latex] 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 [latex]x[/latex]axis now viewed as the polar axis and the origin as the pole. The first coordinate [latex]r[/latex] is the radius or length of the directed line segment from the pole. The angle [latex]θ[/latex], measured in radians, indicates the direction of [latex]r[/latex]. We move counterclockwise from the polar axis by an angle of [latex]θ[/latex],and measure a directed line segment the length of [latex]r[/latex] in the direction of [latex]θ[/latex]. Even though we measure [latex]θ[/latex] first and then [latex]r[/latex], the polar point is written with the [latex]r[/latex]-coordinate first. For example, to plot the point [latex](2,\frac{\pi }{4})[/latex],we would move [latex]\frac{\pi }{4}[/latex] units in the counterclockwise direction and then a length of [latex]2[/latex] 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 [latex](2,\frac{\pi }{4})[/latex],by moving [latex]\frac{\pi }{4}[/latex] units in the counterclockwise direction and then a length of [latex]2[/latex] from the pole.

Uniqueness of polar coordinates

Adding any number of full turns ([latex]360^{\circ} [/latex] or [latex]2\pi[/latex] 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([latex]r, \phi \pm n\cdot 360°[/latex]) or ([latex]−r, \phi \pm (2n + 1)\cdot 180°[/latex]), where [latex]n[/latex] is any integer. Moreover, the pole itself can be expressed as ([latex]0, ϕ[/latex]) for any angle [latex]ϕ[/latex].

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):
  • [latex]\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}[/latex]

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  [latex]x[/latex], [latex]y[/latex], [latex]r[/latex], and [latex]θ[/latex], from the definitions of [latex]\cos \theta[/latex] and [latex]\sin \theta[/latex].  Solving for the variables [latex]x[/latex] and [latex]y[/latex] yields the following formulas:

[latex]\displaystyle \cos \theta =\frac{x}{r}\quad\Rightarrow\quad x=r\cos \theta [/latex]

[latex]\displaystyle \sin \theta =\frac{y}{r}\quad\Rightarrow\quad y=r\sin \theta [/latex]

An easy way to remember the equations above is to think of [latex]\cos\theta[/latex] as the adjacent side over the hypotenuse and [latex]\sin\theta[/latex] as the opposite side over the hypotenuse.  Dropping a perpendicular from the point in the plane to the [latex]x[/latex]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 [latex](r,θ)[/latex] to rectangular coordinates [latex](x,y)[/latex] follow these steps:

1) Write [latex]\cos \theta =\frac{x}{r}\Rightarrow x=r\cos \theta [/latex] and [latex]\sin \theta =\frac{y}{r}\Rightarrow y=r\sin \theta [/latex].

2) Evaluate [latex]\cos\theta[/latex] and [latex]\sin\theta[/latex].

3) Multiply [latex]\cos\theta[/latex] by [latex]r[/latex] to find the [latex]x[/latex]-coordinate of the rectangular form.

4) Multiply [latex]\sin\theta[/latex] by [latex]r[/latex] to find the [latex]y[/latex]-coordinate of the rectangular form.

Example:   Write the polar coordinates [latex](3,\frac {\pi}{2})[/latex] as rectangular coordinates.

[latex]\displaystyle \begin{align} x &= r\cos \theta \\ &= 3cos \frac{\pi}{2}\\ &= 0 \end{align} [/latex]

[latex]\displaystyle \begin{align} y&=r\sin \theta\\&=3\sin\frac{\pi}{2}\\&=3 \end{align}[/latex]

The rectangular coordinates are [latex](0,3)[/latex].

Both points are 3 units up the y-axis.

Polar and Coordinate Grid of Equivalent Points: The rectangular coordinate [latex](0,3)[/latex] is the same as the polar coordinate [latex](3,\frac {\pi}{2})[/latex] 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:

[latex]\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}[/latex]

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 [latex](3,3)[/latex] to polar coordinates.

We are given the values of [latex]x[/latex] and [latex]y[/latex] and need to solve for [latex]\theta[/latex] and [latex]r[/latex].  Start by solving for [latex]\theta[/latex] using the [latex]\tan[/latex] function:

[latex]\displaystyle \begin{align} \tan \theta&=\frac{y}{x}\\&=\frac{3}{3}\\&=1\\ \end{align}[/latex]

So:

[latex]\displaystyle \begin{align} \theta &= \tan^{-1}\left( 1 \right)\\ &=\frac{\pi}{4} \end{align} [/latex]

Next substitute the values of [latex]x[/latex]
and [latex]y[/latex] into the formula [latex]r^2=x^2+y^2[/latex] and solve for [latex]r[/latex].

[latex]\displaystyle \begin{align} r^2&=x^2+y^2\\ &=3^2+3^2\\ &=18\\ \end{align}[/latex]

So:

[latex]\displaystyle \begin{align} r&=\sqrt{18}\\ &=3\sqrt{2} \end{align} [/latex]

The polar coordinates are [latex](3\sqrt2,\frac{\pi}{4})[/latex].

Note that [latex]r^2 = 18[/latex] implies [latex]r=\pm\sqrt{18}[/latex]. We chose to ignore the negative [latex]r[/latex] value. Also note that [latex]\tan^{-1}\left( 1 \right)[/latex] 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
[latex](-3\sqrt2,\frac{5\pi}{4})[/latex]
and [latex](3\sqrt2,-\frac{7\pi}{2})[/latex] will coincide with the original solution of [latex](3\sqrt2,\frac{\pi}{4})[/latex].

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 [latex]x=2+y^2[/latex]. 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 [latex]P(r,θ)[/latex] at the pole, and a line, the directrix, which is perpendicular to the polar axis.

For a conic with eccentricity [latex]e[/latex],

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

With this definition, we may now define a conic in terms of the directrix: [latex]x=±p[/latex], the eccentricity [latex]e[/latex], and the angle [latex]\theta[/latex]. Thus, each conic may be written as a polar equation in terms of [latex]r[/latex] and [latex]\theta[/latex].

For a conic with a focus at the origin, if the directrix is [latex]x=±p[/latex], where [latex]p[/latex] is a positive real number, and the eccentricity is a positive real number [latex]e[/latex], the conic has a polar equation:

[latex]\displaystyle r=\frac{e\cdot p}{1\: \pm\: e\cdot\cos\theta}[/latex]

For a conic with a focus at the origin, if the directrix is [latex]y=±p[/latex], where [latex]p[/latex] is a positive real number, and the eccentricity is a positive real number [latex]e[/latex], the conic has a polar equation:

[latex]\displaystyle r=\frac{e\cdot p}{1\: \pm\: e\cdot\sin\theta}[/latex]

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: [latex]r=a\:\cos n\theta[/latex]  and [latex]r=a\:\sin n\theta[/latex]  where [latex]a \ne 0[/latex].  If  [latex]n[/latex] is even, the curve has [latex]2n[/latex] petals. If  [latex]n[/latex] is odd, the curve has [latex]n[/latex] petals.
  • The formula that generates the graph of the Archimedes’ spiral is given by: [latex]r=θ[/latex] for  [latex]θ≥0[/latex].  As  [latex]\theta[/latex] increases,  [latex]r[/latex] 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 [latex]r=a + b\theta[/latex]
  • rose curve: A curve given by an equation of the form [latex]r = a\cos n\theta[/latex] or [latex]r=a\sin n\theta[/latex]

To graph in the rectangular coordinate system we construct a table of [latex]x[/latex] and [latex]y[/latex]  values. To graph in the polar coordinate system we construct a table of [latex]r[/latex] and [latex]\theta[/latex] values. We enter values of [latex]\theta[/latex] into a polar equation  and calculate [latex]r[/latex]. However, using the properties of symmetry and finding key values of [latex]\theta[/latex] and [latex]r[/latex] 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:

[latex]\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[/latex]

If [latex]n[/latex] is even, the curve has [latex]2n[/latex] petals. If [latex]n[/latex] is odd, the curve has [latex]n[/latex] 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:[latex]r=a\:\cos n\theta[/latex] and [latex]r=a\:\sin n\theta[/latex] where [latex]a≠0[/latex].

Investigating the Archimedes’ Spiral

Archimedes’ spiral is named for its discoverer, the Greek mathematician Archimedes ([latex]c. 287 BCE - c. 212 BCE[/latex]), 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:

[latex]\displaystyle r=a + b\theta \qquad \text{for} \qquad \theta\geq 0[/latex]

As [latex]\theta[/latex] increases, [latex]r[/latex] 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 [latex]r=θ[/latex] for [latex]θ≥0[/latex].