Difference between revisions of "Involute of a Circle"

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The radius is 2, so using those two measurements we can find  <math>r</math> using the pythagorean theorem(or the distance from the origin to the point on the involute curve. So  <math>r</math> would equal<math>\sqrt{2^2 + \pi^2}</math>. Then, to find  <math>\theta</math> one would have to subtract the angle DAC from  <math>\frac{\pi}{2}</math>. In this case it would be <math>\tan^{-1}\left(\frac{\pi}{2}\right)</math>
 
The radius is 2, so using those two measurements we can find  <math>r</math> using the pythagorean theorem(or the distance from the origin to the point on the involute curve. So  <math>r</math> would equal<math>\sqrt{2^2 + \pi^2}</math>. Then, to find  <math>\theta</math> one would have to subtract the angle DAC from  <math>\frac{\pi}{2}</math>. In this case it would be <math>\tan^{-1}\left(\frac{\pi}{2}\right)</math>
  
Now we can put these into variables, instead <math>\pi</math> for the length of the string from the point tangent
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Now to form an equation, we just use variables. We name ''a'' as the radius, ''d'' as the length of the tangent generating out point, ''r'' as the distance from the origin to our point (also the hypotenuse of our triangle), <math>\theta</math> is the angle of our point, and <math>\theta '</math> is the angle to the base of the triangle.
 +
 
 +
So from what we said before, using the pythagorean theorem
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<math>d=\sqrt{r^2-a^2}</math>
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So if you call the radius of the circle <math>a</math> (2 in this case) and
 
 
|other=Alegbra 2, Geometry, Pre-Calculus
 
|other=Alegbra 2, Geometry, Pre-Calculus
 
|AuthorName=Wyatt S.C.
 
|AuthorName=Wyatt S.C.

Revision as of 16:02, 21 April 2012

Inprogress.png
Involute of a Circle
Involute of a circle.gif
Field: Geometry
Image Created By: Wyatt S.C.

Involute of a Circle

The involute of a circle is a curve formed by an imaginary string attached at fix point pulled taut either unwinding or winding around a circle.


A More Mathematical Explanation

Note: understanding of this explanation requires: *Alegbra 2, Geometry, Pre-Calculus

When deriving the equation to graph the involute of a circle, it actually has to do with measuring ri [...]

When deriving the equation to graph the involute of a circle, it actually has to do with measuring right triangles.

If you take a point on the involute of a circle with radius 2, where the imaginary string is unwinding and starts at point (2,0), and the string is parallel to the x axis for the first time, that length would be \pi. This is because the imaginary string would have unwound a quarter of the circle's circumfrence. So \frac{4\pi}{4} is \pi

The radius is 2, so using those two measurements we can find r using the pythagorean theorem(or the distance from the origin to the point on the involute curve. So r would equal\sqrt{2^2 + \pi^2}. Then, to find \theta one would have to subtract the angle DAC from \frac{\pi}{2}. In this case it would be \tan^{-1}\left(\frac{\pi}{2}\right)

Now to form an equation, we just use variables. We name a as the radius, d as the length of the tangent generating out point, r as the distance from the origin to our point (also the hypotenuse of our triangle), \theta is the angle of our point, and \theta ' is the angle to the base of the triangle.

So from what we said before, using the pythagorean theorem d=\sqrt{r^2-a^2}


Why It's Interesting

This is very interesting for many reasons. It is amazing that what looks to be a very complex figure's equation can easily be derived using understanding of just geometry and some pre calculus.


The involute of a circle appears commonly in every day life. Other than the simple tetherball which is more of a model for the involute of a circle. The most commonly used gear system utilizes the involute of a circle. The teeth of the gear are involutes.

This allows the contact of the two interlocking teeth to occur at a single point that moves along the tooth. This allows the transfer of energy to one powered gear to a powerless gear smooth and not require as much energy.

Involute wheel.gif


Teaching Materials

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References

http://en.wikipedia.org/wiki/Involute#Involute_of_a_circle http://en.wikipedia.org/wiki/Involute_gear





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