Polar Equations
From Math Images
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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>== | ||
A derivative gives the slope of any point in a function.<br> | A derivative gives the slope of any point in a function.<br> | ||
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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. | ||
- | + | <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)}</math> | ||
+ | :<math>= \frac{\cos(\theta) \sin(\theta) + (1 + \sin(\theta) ) \cos(\theta)}{\cos(\theta)\cos(\theta) - (1 + \sin(\theta) ) \sin(\theta)}</math> | ||
+ | :<math>= \frac{\cos(\theta)\sin(\theta) + \cos(\theta) + \cos(\theta)\sin(\theta)}{\cos^2(\theta) - \sin(\theta) - \sin^2(\theta)}</math> | ||
+ | }} | ||
==Finding Areas and Arc Lengths<ref name=Textbook />== | ==Finding Areas and Arc Lengths<ref name=Textbook />== | ||
[[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> | [[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> |
Revision as of 17:32, 14 July 2011
- This polar rose is created with the polar equation: .
A polar rose (Rhodonea Curve) |
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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 UNIQ18caec7e2fd21242-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
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 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
- 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
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.
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