Polar Equations

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A polar rose (Rhodonea Curve)
This polar rose is created with the polar equation:  r = cos(\pi\theta) .


Basic Description

Polar equations are used to create interesting curves, and in most cases they are periodic like sine waves. Other types of curves can also be created using polar equations besides roses, such as Archimedean spirals and limaçons.

A More Mathematical Explanation

Note: understanding of this explanation requires: *calculus, trigonometry


The general polar equations form to create a rose is UNIQ35a640552e89765-math-00000001-QI [...]


The general polar equations form to create a rose is r = a \sin(n \theta) or r = a \cos(n \theta). Note that the difference between sine and cosine is \sin(\theta) = \cos(\theta-\frac{\pi}{2}), so choosing between sine and cosine affects where the curve starts and ends. a represents the maxium value r can be, i.e. the maximum radius of the rose. n affects the number of petals on the graph:

  • If n is an odd integer, then there would be n petals, and the curve repeats itself every \pi.
  • If n is an even integer, then there would be 2n petals, and the curve repeats itself every 2 \pi.
  • If n is a rational fraction (p/q where p and q are integers), then the curve repeats at the \theta = \pi q k , where k = 1 if pq is odd, and k = 2 if pq is even.
  • If n is irrational, then there are an infinite number of petals.

Below is an applet to graph polar roses:

If you can see this message, you do not have the Java software required to view the applet.

Other Polar Curves

Archimedean Spirals

Archimedes' Spiral
 r = a\theta
Fermat's Spiral
 r = \pm a\sqrt\theta
Hyperbolic spiral
 r = \frac{a}{\theta}

It begins at an infinite distance from the pole, and
winds faster as it approaches closer to the pole.
 r^2 \theta = a^2

The word "limaçon" derives from the Latin word "limax," meaning snail. The general equation for a limaçon is r = b + a\cos(\theta).

  • If b \geq 2a, then the curve is convex.
  • If  2a > b > a , then it is dimpled.
  • If  b = a, then it becomes a cardioid.
  • If  b = a/2, then it is a trisectrix.
 r = \cos(\theta)
 r = 0.5 + \cos(\theta)
  r = 1 + \cos(\theta)
  r = 2 + \cos(\theta)

Finding Derivatives[2]

Consider the polar curve r = f(\theta).
If we turn it into parametric equations, we would get:

  • x = r \cos(\theta) = f(\theta) \cos(\theta)
  • y = r \sin(\theta) = f(\theta) \sin(\theta)

Using the method of finding the derivative of parametric equations and the product rule, we would get:
\frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)}

Finding Areas and Arc Lengths[2]

To find the area of a sector of a circle, where  r is the radius, you would use  A = \frac{1}{2} r^2 \theta .
Therefore, for  r = f(\theta), the formula for the area of a polar region is:

 A = \int_a^b\! \frac{1}{2} r^2 d\theta

The formula to find the arc length for r = f(\theta) and assuming r is continuous is:

 L = \int_a^b\! \sqrt{r^2 + {\bigg(\frac{dr}{d\theta}\bigg)} ^2} d\theta

Why It's Interesting

Polar coordinates are often used in navigation, such as aircrafts. They are also used to plot gravitational fields and point sources. Furthermore, polar patterns are seen in the directionality of microphones, which is the direction at which the microphone picks up sound. A well-known pattern is a cardioid.

Possible Future work

  • More details can be written about the different curves, maybe they can get their own pages.
  • Applets can be made to draw these different curves, like the one on the page for roses.

Teaching Materials

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About the Creator of this Image

The images on this page were created using C++ with OpenGL.

Related Links

Additional Resources

Polar Coordinates


Wolfram MathWorld: Rose, Limacon, Archimedean Spiral
Wikipedia: Polar Coordinate System
  1. Weisstein, Eric W. (2011). http://mathworld.wolfram.com/Limacon.html. Wolfram:MathWorld.
  2. 2.0 2.1 Stewert, James. (2009). Calculus Early Transcendentals. Ohio:Cengage Learning.

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