Difference between revisions of "Involute of a Circle"

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|other=Alegbra 2, Geometry, Pre-Calculus
|other=Alegbra 2, Geometry, Pre-Calculus
|AuthorName=Wyatt S.C.
|AuthorName=Wyatt S.C.

Revision as of 16:13, 21 April 2012

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

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