Polar Equations
From Math Images
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- | {{Image Description | + | {{Image Description Ready |
|ImageName=A polar rose (Rhodonea Curve) | |ImageName=A polar rose (Rhodonea Curve) | ||
|Image=Rose2.gif | |Image=Rose2.gif | ||
+ | |other=calculus, trigonometry | ||
+ | |AuthorName=chanj | ||
+ | |AuthorDesc=The images on this page were created using C++ with OpenGL. | ||
+ | |SiteName=chanj | ||
+ | |SiteURL=http://mathforum.org/mathimages/index.php/User:Chanj | ||
+ | |Field=Algebra | ||
+ | |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>. | ||
+ | |||
|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. | ||
+ | |||
|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 | + | 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> | ||
- | * 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: '''{{ | + | * 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]]}} | ||
- | * 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: '''{{ | + | * 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]]}} | ||
- | * 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: '''{{ | + | * 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> | ||
{{!}}} | {{!}}} | ||
}} | }} | ||
- | * If <math>n</math> is irrational, then there are an infinite number of petals.<br>'''Examples: '''{{ | + | * 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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}} | }} | ||
- | <br>'''Below is an applet to graph polar roses, which is used to graph the examples above:'''<br>{{ | + | <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" /> | + | <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>}} |
==Other Polar Curves== | ==Other Polar Curves== | ||
'''Archimedean Spirals'''<br> | '''Archimedean Spirals'''<br> | ||
{{{!}}border="1" cellspacing="5" cellpadding="20" | {{{!}}border="1" cellspacing="5" cellpadding="20" | ||
- | {{!}}Archimedes' Spiral<br> <math> r = a\theta </math> <br> [[Image:Archimedes' spiral.png|350px]]<br>The spiral can be used to square a circle and trisect an angle.{{!}}{{!}}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. | + | {{!}}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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{{!}}} | {{!}}} | ||
<br> | <br> | ||
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==Finding Derivatives<ref name=Textbook> Stewert, James. (2009). Calculus Early Transcendentals. Ohio:Cengage Learning.</ref>== | ==Finding Derivatives<ref name=Textbook> Stewert, James. (2009). Calculus Early Transcendentals. Ohio:Cengage Learning.</ref>== | ||
- | Consider the polar curve <math>r = f(\theta)</math>. | + | A derivative gives the slope of any point in a function.<br> |
- | If we turn it into parametric equations, we would get: | + | Consider the polar curve <math>r = f(\theta)</math>. If we turn it into parametric equations, we would get: |
* <math>x = r \cos(\theta) = f(\theta) \cos(\theta)</math> | * <math>x = r \cos(\theta) = f(\theta) \cos(\theta)</math> | ||
* <math>y = r \sin(\theta) = f(\theta) \sin(\theta)</math> | * <math>y = r \sin(\theta) = f(\theta) \sin(\theta)</math> | ||
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{dr}{d\theta} \sin(\theta) + r \cos(\theta)}{\frac{dr}{d\theta} \cos(\theta) - r \sin(\theta)}</math><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. |
- | + | <br>'''Examples: '''{{hide|1= | |
+ | 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{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)}</math><br><br> | ||
+ | :::<math>= \frac{\cos(\theta)\sin(\theta) + \cos(\theta) + \cos(\theta)\sin(\theta)}{\cos^2(\theta) - \sin(\theta) - \sin^2(\theta)}</math><br><br> | ||
+ | ::::'''Note:''' Using the double-angle formula, we get <math>\cos^2(\theta) - \sin^2(\theta) = 1 - 2\sin^2(\theta)</math><br><br> | ||
+ | :::<math>= \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))}</math><br><br> | ||
+ | <math>\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))}</math><br><br> | ||
+ | :::<math>= \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</math><br> | ||
+ | }} | ||
==Finding Areas and Arc Lengths<ref name=Textbook />== | ==Finding Areas and Arc Lengths<ref name=Textbook />== | ||
- | To find the '''area''' of a sector of a circle, where <math> r </math> is the radius, you would use <math> A = \frac{1}{2} r^2 \theta </math>. <br> | + | [[Image:CircleArea.png|Area of a sector of a circle.|thumb|200px|left]]To find the '''area''' of a sector of a circle, where <math> r </math> is the radius, you would use <math> A = \frac{1}{2} r^2 \theta </math>. [[Image:PolarArea.png|<math>A = \int_{-\frac{\pi}{4}}^\frac{\pi}{4}\! \frac{1}{2} \cos^2(2\theta) d\theta</math>|thumb|200px|right]]<br> |
Therefore, for <math> r = f(\theta)</math>, the formula for the area of a polar region is: <br> | Therefore, for <math> r = f(\theta)</math>, the formula for the area of a polar region is: <br> | ||
- | :: <math> A = \ | + | :: <math> A = \int\limits_a^b\! \frac{1}{2} r^2 d\theta</math><br><br> |
The formula to find the '''arc length''' for <math>r = f(\theta)</math> and assuming <math>r</math> is continuous is: | The formula to find the '''arc length''' for <math>r = f(\theta)</math> and assuming <math>r</math> is continuous is: | ||
- | :: <math> L = \ | + | :: <math> L = \int\limits_a^b\! \sqrt{r^2 + {\bigg(\frac{dr}{d\theta}\bigg)} ^2} </math> <math>d\theta</math><br> |
<div> | <div> | ||
- | + | ||
- | | | + | |
- | | | + | |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]] |
- | | | + | |
- | + | 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]]. | |
- | | | + | </p>Archimedes' spiral can be used for compass and straightedge division of an angle into <math>n</math> parts and circle squaring. <ref name=spiral> Weisstein, Eric W. (2011). http://mathworld.wolfram.com/ArchimedesSpiral.html. Wolfram:MathWorld.</ref> 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. |
- | | | + | <p>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== | ==Possible Future work== | ||
* 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. | ||
- | |FieldLinks=[[Polar Coordinates]]<br>[[Cardioid]] | + | * 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> | ||
+ | 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> | ||
+ | [http://www.youtube.com/watch?v=U-v0CBhgq4k Trisect an Angle Using the Spiral of Archimedes] | ||
+ | |||
|References=Wolfram MathWorld: [http://mathworld.wolfram.com/Rose.html Rose], [http://mathworld.wolfram.com/Limacon.html Limacon], [http://mathworld.wolfram.com/ArchimedeanSpiral.html Archimedean Spiral]<br>Wikipedia: [http://en.wikipedia.org/wiki/Polar_coordinate_system Polar Coordinate System], [http://en.wikipedia.org/wiki/Archimedean_spiral Archimedean Spiral], [http://en.wikipedia.org/wiki/Fermat%27s_spiral Fermat's Spiral]<br><references /> | |References=Wolfram MathWorld: [http://mathworld.wolfram.com/Rose.html Rose], [http://mathworld.wolfram.com/Limacon.html Limacon], [http://mathworld.wolfram.com/ArchimedeanSpiral.html Archimedean Spiral]<br>Wikipedia: [http://en.wikipedia.org/wiki/Polar_coordinate_system Polar Coordinate System], [http://en.wikipedia.org/wiki/Archimedean_spiral Archimedean Spiral], [http://en.wikipedia.org/wiki/Fermat%27s_spiral Fermat's Spiral]<br><references /> | ||
|Pre-K=No | |Pre-K=No | ||
|Elementary=No | |Elementary=No | ||
|MiddleSchool=No | |MiddleSchool=No | ||
- | |InProgress= | + | |InProgress=No |
}} | }} |
Current revision
A polar rose (Rhodonea Curve) |
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A polar rose (Rhodonea Curve)
- This polar rose is created with the polar equation: .
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 UNIQ70f9c463170861fb-math-00000001-Q [...]Rose
The general polar equations form to create a rose is or . Note that the difference between sine and cosine is , so choosing between sine and cosine affects where the curve starts and ends. represents the maximum value can be, i.e. the maximum radius of the rose. affects the number of petals on the graph:
- If is an odd integer, then there would be petals, and the curve repeats itself every .
Examples:
- If is an even integer, then there would be petals, and the curve repeats itself every .
Examples:
- If is a rational fraction ( where and are integers), then the curve repeats at the , where if is odd, and if is even.
Examples:
- If is irrational, then there are an infinite number of petals.
Examples:
Below is an applet to graph polar roses, which is used to graph the examples above:
Other Polar Curves
Archimedean Spirals
Archimedes' Spiral 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 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 It begins at an infinite distance from the pole, and winds faster as it approaches closer to the pole. | Lituus It is asymptotic at the 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 .
- If , then it is a trisectrix (see figure 2).
- If , then it becomes a cardioid (see figure 3).
- If , then it is dimpled (see figure 4).
- If , then the curve is convex (see figure 5).
1 | 2 | Cardioid 3 | 4 | 5 |
Finding Derivatives^{[2]}
A derivative gives the slope of any point in a function.
Consider the polar curve . If we turn it into parametric equations, we would get:
Using the method of finding the derivative of parametric equations and the product rule, we would get:
Note: It is not necessary to turn the polar equation to parametric equations to find derivatives. You can simply use the formula above.
Examples:
Finding Areas and Arc Lengths^{[2]}
To find the area of a sector of a circle, where is the radius, you would use .Therefore, for , the formula for the area of a polar region is:
The formula to find the arc length for and assuming is continuous is:
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 the Cardioid.
Archimedes' spiral can be used for compass and straightedge division of an angle into 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 SpiralWikipedia: Polar Coordinate System, Archimedean Spiral, Fermat's Spiral
- ↑ Weisstein, Eric W. (2011). http://mathworld.wolfram.com/Limacon.html. Wolfram:MathWorld.
- ↑ ^{2.0} ^{2.1} Stewert, James. (2009). Calculus Early Transcendentals. Ohio:Cengage Learning.
- ↑ Lauer, Christoph. http://www.christoph-lauer.de/Homepage/Blog/Eintrage/2009/10/6_Fermats_Spiral_Suflower_Generator.html. Christoph Lauer's Blog.
- ↑ Weisstein, Eric W. (2011). http://mathworld.wolfram.com/ArchimedesSpiral.html. Wolfram:MathWorld.
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