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학습 가이드 > Boundless Algebra

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 360∘360^{\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 (360∘360^{\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±n⋅360°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):
  • cos⁡θ=xr→x=rcos⁡θsin⁡θ=yr→y=rsin⁡θr2=x2+y2tan⁡θ=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 cos⁡θ\cos \theta and sin⁡θ\sin \theta.  Solving for the variables xx and yy yields the following formulas: cos⁡θ=xr⇒x=rcos⁡θ\displaystyle \cos \theta =\frac{x}{r}\quad\Rightarrow\quad x=r\cos \theta sin⁡θ=yr⇒y=rsin⁡θ\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 cos⁡θ\cos\theta as the adjacent side over the hypotenuse and sin⁡θ\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 cos⁡θ=xr⇒x=rcos⁡θ\cos \theta =\frac{x}{r}\Rightarrow x=r\cos \theta  and sin⁡θ=yr⇒y=rsin⁡θ\sin \theta =\frac{y}{r}\Rightarrow y=r\sin \theta . 2) Evaluate cos⁡θ\cos\theta and sin⁡θ\sin\theta. 3) Multiply cos⁡θ\cos\theta by rr to find the xx-coordinate of the rectangular form. 4) Multiply sin⁡θ\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.

\displaystyle \begin{align} x &= r\cos \theta \\ &= 3cos \frac{\pi}{2}\\ &= 0 \end{align} \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: \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: \displaystyle \begin{align} \tan \theta&=\frac{y}{x}\\&=\frac{3}{3}\\&=1\\ \end{align} So: \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. \displaystyle \begin{align} r^2&=x^2+y^2\\ &=3^2+3^2\\ &=18\\ \end{align} So: \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 tan⁡−1(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 0≤e<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=e⋅p1 ± e⋅cos⁡θ\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=e⋅p1 ± e⋅sin⁡θ\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=a cos⁡nθr=a\:\cos n\theta  and r=a sin⁡nθr=a\:\sin n\theta  where a≠0a \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  θ≥0θ≥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=acos⁡nθr = a\cos n\theta or r=asin⁡nθr=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=a⋅cos⁡(nθ)andr=a⋅sin⁡(nθ)wherea≠0\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=a cos⁡nθr=a\:\cos n\theta and r=a sin⁡nθr=a\:\sin n\theta where a≠0a≠0.

Investigating the Archimedes' Spiral

Archimedes’ spiral is named for its discoverer, the Greek mathematician Archimedes (c.287BCE−c.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+bθforθ≥0\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 θ≥0θ≥0.

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