Polar Equations

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|ImageName=A polar rose (Rhodonea Curve)
|ImageName=A polar rose (Rhodonea Curve)
|Image=Rose2.gif
|Image=Rose2.gif
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|other=calculus, trigonometry
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|AuthorName=chanj
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|AuthorDesc=The images on this page were created using C++ with OpenGL.
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|SiteName=chanj
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|SiteURL=http://mathforum.org/mathimages/index.php/User:Chanj
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|Field=Algebra
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|Field2=Calculus
|ImageIntro=This polar rose is created with the polar equation: <math> r = cos(\pi\theta) </math>.
|ImageIntro=This polar rose is created with the polar equation: <math> r = cos(\pi\theta) </math>.
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|ImageDescElem=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. See the [[Polar Coordinates]] page for some background information.
|ImageDescElem=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. See the [[Polar Coordinates]] page for some background information.
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|ImageDesc=== Rose ==
|ImageDesc=== Rose ==
The general polar equations form to create a rose is <math>r = a \sin(n \theta)</math> or <math>r = a \cos(n \theta)</math>. Note that the difference between sine and cosine is <math>\sin(\theta) = \cos(\theta-\frac{\pi}{2})</math>, so choosing between sine and cosine affects where the curve starts and ends. <math>a</math> represents the maximum value <math>r</math> can be, i.e. the maximum radius of the rose. <math>n</math> affects the number of petals on the graph:
The general polar equations form to create a rose is <math>r = a \sin(n \theta)</math> or <math>r = a \cos(n \theta)</math>. Note that the difference between sine and cosine is <math>\sin(\theta) = \cos(\theta-\frac{\pi}{2})</math>, so choosing between sine and cosine affects where the curve starts and ends. <math>a</math> represents the maximum value <math>r</math> can be, i.e. the maximum radius of the rose. <math>n</math> affects the number of petals on the graph:
<p>
<p>
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* If <math>n</math> is an odd integer, then there would be <math>n</math> petals, and the curve repeats itself every <math>\pi</math>.<br>'''Examples: '''{{Hide|1=
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* If <math>n</math> is an odd integer, then there would be <math>n</math> petals, and the curve repeats itself every <math>\pi</math>.<br>'''Examples: '''{{hide|1=
:: [[Image:Odd2.jpeg|600px]]}}
:: [[Image:Odd2.jpeg|600px]]}}
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* If <math>n</math> is an even integer, then there would be <math>2n</math> petals, and the curve repeats itself every <math>2 \pi</math>.<br>'''Examples: '''{{Hide|1=
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* If <math>n</math> is an even integer, then there would be <math>2n</math> petals, and the curve repeats itself every <math>2 \pi</math>.<br>'''Examples: '''{{hide|1=
:: [[Image:Even.jpeg|600px]]}}
:: [[Image:Even.jpeg|600px]]}}
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* If <math>n</math> is a rational fraction (<math>p/q</math> where <math>p</math> and <math>q</math> are integers), then the curve repeats at the <math>\theta = \pi q k </math>, where <math>k = 1</math> if <math>pq</math> is odd, and <math>k = 2</math> if <math>pq</math> is even.<br>'''Examples: '''{{Hide|1=
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* If <math>n</math> is a rational fraction (<math>p/q</math> where <math>p</math> and <math>q</math> are integers), then the curve repeats at the <math>\theta = \pi q k </math>, where <math>k = 1</math> if <math>pq</math> is odd, and <math>k = 2</math> if <math>pq</math> is even.<br>'''Examples: '''{{hide|1=
:{{{!}}border="1" cellspacing="2" cellpadding="20"
:{{{!}}border="1" cellspacing="2" cellpadding="20"
{{!}}[[Image:Half.png|400px]]<br><math> r = \cos(\frac{1}{2}\theta) </math> <br> The angle coefficient is <math>\frac{1}{2} = 0.5</math>. <br><math>1 \times 2 = 2</math>, which is '''even'''. Therefore, the curve<br> repeats itself every <math>\pi \times 2 \times 2 \approx 12.566.</math> {{!}}{{!}}[[Image:Third.png|400px]]<br><math> r = \cos(\frac{1}{3}\theta) </math> <br> The angle coefficient is <math>\frac{1}{3} \approx 0.33333</math>. <br><math>1 \times 3 = 3</math>, which is '''odd'''. Therefore, the curve <br>repeats itself every <math>\pi \times 3 \times 1 \approx 9.425.</math>
{{!}}[[Image:Half.png|400px]]<br><math> r = \cos(\frac{1}{2}\theta) </math> <br> The angle coefficient is <math>\frac{1}{2} = 0.5</math>. <br><math>1 \times 2 = 2</math>, which is '''even'''. Therefore, the curve<br> repeats itself every <math>\pi \times 2 \times 2 \approx 12.566.</math> {{!}}{{!}}[[Image:Third.png|400px]]<br><math> r = \cos(\frac{1}{3}\theta) </math> <br> The angle coefficient is <math>\frac{1}{3} \approx 0.33333</math>. <br><math>1 \times 3 = 3</math>, which is '''odd'''. Therefore, the curve <br>repeats itself every <math>\pi \times 3 \times 1 \approx 9.425.</math>
{{!}}}
{{!}}}
}}
}}
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* If <math>n</math> is irrational, then there are an infinite number of petals.<br>'''Examples: '''{{Hide|1=
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* If <math>n</math> is irrational, then there are an infinite number of petals.<br>'''Examples: '''{{hide|1=
:<math> r = \cos(e \theta)</math><br>
:<math> r = \cos(e \theta)</math><br>
:<math> \theta \text{ from } 0 \text{ to...} </math>
:<math> \theta \text{ from } 0 \text{ to...} </math>
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}}
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<br>'''Below is an applet to graph polar roses, which is used to graph the examples above:'''<br>{{Hide|1=
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<br>'''Below is an applet to graph polar roses, which is used to graph the examples above:'''<br>{{hide|1=
<java_applet code="Graph.class" height="471" width="450" archive="Roses.jar" /><br>Source code: [http://code.google.com/p/math-images/source/browse/#svn%2Ftrunk%2Fpage_code%2Frose_graph_applet Rose graphing applet]<br>}}
<java_applet code="Graph.class" height="471" width="450" archive="Roses.jar" /><br>Source code: [http://code.google.com/p/math-images/source/browse/#svn%2Ftrunk%2Fpage_code%2Frose_graph_applet Rose graphing applet]<br>}}
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'''Archimedean Spirals'''<br>
'''Archimedean Spirals'''<br>
{{{!}}border="1" cellspacing="5" cellpadding="20"
{{{!}}border="1" cellspacing="5" cellpadding="20"
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{{!}}Archimedes' Spiral<br> <math> r = a\theta </math> <br> [[Image:Archimedes' spiral.png|350px]]<br>The spiral can be used to square a circle, which is constructing<br> a square with the same area as a given circle, and trisect an<br> angle, which is constructing an angle that is one-third of a given <br>angle (more on these topics can be found under [http://mathforum.org/mathimages/index.php/Polar_Equations#Related_Links related links] ).{{!}}{{!}}Fermat's Spiral<br> <math> r = \pm a\sqrt\theta </math> <br> [[Image:fermat's_spiral.jpg|350px]]<br> This spiral's pattern can be seen in disc phyllotaxis,<br> which is the circular head in the middle of flowers, like <br>[http://mathforum.org/mathimages/index.php/Polar_Equations#Why_It.27s_Interesting sunflowers].
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{{!}}Archimedes' Spiral<br> <math> r = a\theta </math> <br> [[Image:Archimedes' spiral.png|350px]]<br>The spiral can be used to square a circle, which is constructing<br> a square with the same area as a given circle, and trisect an<br> angle, which is constructing an angle that is one-third of a given <br>angle (more on these topics can be found under [http://mathforum.org/mathimages/index.php/Polar_Equations#Related_Links related links] ).{{!}}{{!}}Fermat's Spiral<br> <math> r = \pm a\sqrt\theta </math> <br> [[Image:fermat's_spiral.jpg|350px]]<br> This spiral's pattern can be seen in disc phyllotaxis,<br> which is the circular head in the middle of flowers <br>(e.g. [http://mathforum.org/mathimages/index.php/Polar_Equations#Why_It.27s_Interesting sunflowers]).
{{!}}-
{{!}}-
{{!}}Hyperbolic spiral<br><math> r = \frac{a}{\theta}</math><br> [[Image:Hyperbolic_spiral.png|350px]] <br>It begins at an infinite distance from the pole, and <br>winds faster as it approaches closer to the pole.{{!}}{{!}}Lituus<br> <math> r^2 \theta = a^2 </math><br>[[Image:Lituus.png|400px]]<br>It is asymptotic at the <math>x</math> axis as the distance increases <br>from the pole.
{{!}}Hyperbolic spiral<br><math> r = \frac{a}{\theta}</math><br> [[Image:Hyperbolic_spiral.png|350px]] <br>It begins at an infinite distance from the pole, and <br>winds faster as it approaches closer to the pole.{{!}}{{!}}Lituus<br> <math> r^2 \theta = a^2 </math><br>[[Image:Lituus.png|400px]]<br>It is asymptotic at the <math>x</math> axis as the distance increases <br>from the pole.
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Using the method of finding the derivative of parametric equations and the product rule, we would get:<br>
Using the method of finding the derivative of parametric equations and the product rule, we would get:<br>
<math>\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{\frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)}</math><br><br>Note: It is not necessary to turn the polar equation to parametric equations to find derivatives. You can simply use the formula above.
<math>\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{\frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)}</math><br><br>Note: It is not necessary to turn the polar equation to parametric equations to find derivatives. You can simply use the formula above.
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<br>'''Examples: '''{{Hide|1=
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<br>'''Examples: '''{{hide|1=
Find the derivative of <math>r = 1 + \sin(\theta)</math> at <math>\theta = \frac{\pi}{3}</math>.<br>
Find the derivative of <math>r = 1 + \sin(\theta)</math> at <math>\theta = \frac{\pi}{3}</math>.<br>
<math>\frac{dr}{d\theta} = \cos(\theta)</math><br>
<math>\frac{dr}{d\theta} = \cos(\theta)</math><br>
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:: <math> L = \int\limits_a^b\! \sqrt{r^2 + {\bigg(\frac{dr}{d\theta}\bigg)} ^2} </math> <math>d\theta</math><br>
:: <math> L = \int\limits_a^b\! \sqrt{r^2 + {\bigg(\frac{dr}{d\theta}\bigg)} ^2} </math> <math>d\theta</math><br>
<div>
<div>
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|other=calculus, trigonometry
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|AuthorName=chanj
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|AuthorDesc=The images on this page were created using C++ with OpenGL.
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|SiteName=chanj
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|SiteURL=http://mathforum.org/mathimages/index.php/User:Chanj
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|Field=Algebra
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|Field2=Calculus
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|WhyInteresting=[[Image:PolarStar.png|<math> r = \sin^2(1.2\theta) + \cos^3(6\theta)</math>|thumb|210px|right]][[Image:SunflorwerPolar.png|The disc phyllotaxis of a sunflower.<ref name=sunflower>Lauer, Christoph. http://www.christoph-lauer.de/Homepage/Blog/Eintrage/2009/10/6_Fermats_Spiral_Suflower_Generator.html. Christoph Lauer's Blog.</ref>|thumb|200px|left]]
|WhyInteresting=[[Image:PolarStar.png|<math> r = \sin^2(1.2\theta) + \cos^3(6\theta)</math>|thumb|210px|right]][[Image:SunflorwerPolar.png|The disc phyllotaxis of a sunflower.<ref name=sunflower>Lauer, Christoph. http://www.christoph-lauer.de/Homepage/Blog/Eintrage/2009/10/6_Fermats_Spiral_Suflower_Generator.html. Christoph Lauer's Blog.</ref>|thumb|200px|left]]
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* More details can be written about the different curves, maybe they can get their own pages.
* 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.
* Applets can be made to draw these different curves, like the one on the page for roses.
 +
* A page should be made to expand on squaring a circle and another on trisecting an angle, since they are both ancient math problems.
|FieldLinks=[[Polar Coordinates]]<br>[[Cardioid]]<br>
|FieldLinks=[[Polar Coordinates]]<br>[[Cardioid]]<br>
Source code: [http://code.google.com/p/math-images/source/browse/#svn%2Ftrunk%2Fpage_code%2Frose_graph_applet Rose graphing applet]<br>[http://www-history.mcs.st-and.ac.uk/PrintHT/Squaring_the_circle.html Squaring the Circle]<br>
Source code: [http://code.google.com/p/math-images/source/browse/#svn%2Ftrunk%2Fpage_code%2Frose_graph_applet Rose graphing applet]<br>[http://www-history.mcs.st-and.ac.uk/PrintHT/Squaring_the_circle.html Squaring the Circle]<br>

Current revision


A polar rose (Rhodonea Curve)
Fields: Algebra and Calculus
Image Created By: chanj
Website: chanj

A polar rose (Rhodonea Curve)

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


Contents

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. See the Polar Coordinates page for some background information.

A More Mathematical Explanation

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

Rose

The general polar equations form to create a rose is UNIQ74d0ea2e2e014e66-math-00000001-Q [...]

Rose

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 maximum 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.
    Examples:
  • If n is an even integer, then there would be 2n petals, and the curve repeats itself every 2 \pi.
    Examples:
  • 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.
    Examples:

r = \cos(\frac{1}{2}\theta)
The angle coefficient is \frac{1}{2} = 0.5.
1 \times 2 = 2, which is even. Therefore, the curve
repeats itself every \pi \times 2 \times 2 \approx 12.566.
r = \cos(\frac{1}{3}\theta)
The angle coefficient is \frac{1}{3} \approx 0.33333.
1 \times 3 = 3, which is odd. Therefore, the curve
repeats itself every \pi \times 3 \times 1 \approx 9.425.
  • If n is irrational, then there are an infinite number of petals.
    Examples:
r = \cos(e \theta)
\theta \text{ from } 0 \text{ to...}
...10
...50
...100

\text{Note: }e \approx 2.71828


Below is an applet to graph polar roses, which is used to graph the examples above:

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


Source code: Rose graphing applet

Other Polar Curves

Archimedean Spirals

Archimedes' Spiral
r = a\theta

The spiral can be used to square a circle, which is constructing
a square with the same area as a given circle, and trisect an
angle, which is constructing an angle that is one-third of a given
angle (more on these topics can be found under related links ).		Fermat's Spiral
r = \pm a\sqrt\theta

This spiral's pattern can be seen in disc phyllotaxis,
which is the circular head in the middle of flowers
(e.g. sunflowers).
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.		Lituus
r^2 \theta = a^2

It is asymptotic at the x axis as the distance increases
from the pole.


Limaçon[1]
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 = a/2, then it is a trisectrix (see figure 2).
  • If b = a, then it becomes a cardioid (see figure 3).
  • If 2a > b > a , then it is dimpled (see figure 4).
  • If b \geq 2a, then the curve is convex (see figure 5).
r = \cos(\theta)

1		 r = 0.5 + \cos(\theta)

2 		Cardioid
r = 1 + \cos(\theta)

3		 r = 1.5 + \cos(\theta)

4		 r = 2 + \cos(\theta)

5


Finding Derivatives[2]

A derivative gives the slope of any point in a function.
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{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{\frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)}

Note: It is not necessary to turn the polar equation to parametric equations to find derivatives. You can simply use the formula above.
Examples:

Find the derivative of r = 1 + \sin(\theta) at \theta = \frac{\pi}{3}.
\frac{dr}{d\theta} = \cos(\theta)

\frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)} = \frac{\cos(\theta) \sin(\theta) + (1 + \sin(\theta) ) \cos(\theta)}{\cos(\theta)\cos(\theta) - (1 + \sin(\theta) ) \sin(\theta)}

= \frac{\cos(\theta)\sin(\theta) + \cos(\theta) + \cos(\theta)\sin(\theta)}{\cos^2(\theta) - \sin(\theta) - \sin^2(\theta)}

Note: Using the double-angle formula, we get \cos^2(\theta) - \sin^2(\theta) = 1 - 2\sin^2(\theta)

= \frac{\cos(1+2\sin(\theta))}{1-2\sin^2(\theta)-\sin(\theta)} = \frac{\cos(1+2\sin(\theta))}{(1+\sin(\theta))(1-2\sin(\theta))}

\frac{dy}{dx} \Big |_{\theta=\pi/3} = \frac{\cos(\pi/3)(1+2\sin(\pi/3))}{(1+\sin(\pi/3))(1-2\sin(\pi/3))}

= \frac{\frac{1}{2}(1+\sqrt{3})}{(1+\sqrt{3}/2)(1-\sqrt{3})} = \frac{1+\sqrt{3}}{(2+\sqrt{3})(1-\sqrt{3})} = \frac{1+\sqrt{3}}{-1-\sqrt{3}} = -1

Finding Areas and Arc Lengths[2]

Area of a sector of a circle.
Area of a sector of a circle.
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 .
A = \int_{-\frac{\pi}{4}}^\frac{\pi}{4}\! \frac{1}{2} \cos^2(2\theta) d\theta

Therefore, for r = f(\theta), the formula for the area of a polar region is:

A = \int\limits_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\limits_a^b\! \sqrt{r^2 + {\bigg(\frac{dr}{d\theta}\bigg)} ^2} d\theta


Why It's Interesting

r = \sin^2(1.2\theta) + \cos^3(6\theta)
The disc phyllotaxis of a sunflower.
The disc phyllotaxis of a sunflower.[3]

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 the Cardioid.

Archimedes' spiral can be used for compass and straightedge division of an angle into n parts and circle squaring. [4] Fermat's spiral is a Archimedean spiral that is observed in nature. The pattern happens to appear in the mesh of mature disc phyllotaxis. The florets in sunflowers are arranged in a form of that spiral. Archimedean spirals can also be seen in patterns of solar wind and Catherine's wheel.

As you can see, these equations can create interesting curves and patterns. More complicated patterns can be created with more complicated equations, like the image on the right. Since intriguing patterns can be expressed mathematically, like these curves, they are often used for art and design.

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.
  • A page should be made to expand on squaring a circle and another on trisecting an angle, since they are both ancient math problems.


Teaching Materials

There are currently no teaching materials for this page. Add teaching materials.

About the Creator of this Image

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


Related Links

Additional Resources

Polar Coordinates
Cardioid
Source code: Rose graphing applet
Squaring the Circle
Trisect an Angle Using the Spiral of Archimedes

References

Wolfram MathWorld: Rose, Limacon, Archimedean Spiral
Wikipedia: Polar Coordinate System, Archimedean Spiral, Fermat's Spiral
  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.
  3. ↑ Lauer, Christoph. http://www.christoph-lauer.de/Homepage/Blog/Eintrage/2009/10/6_Fermats_Spiral_Suflower_Generator.html. Christoph Lauer's Blog.
  4. ↑ Weisstein, Eric W. (2011). http://mathworld.wolfram.com/ArchimedesSpiral.html. Wolfram:MathWorld.




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