Difference between revisions of "Involute of a Circle"
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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 <math>\pi</math>. This is because the imaginary string would have unwound a quarter of the circle's circumfrence. So <math>\frac{4\pi}{4}</math> is <math>\pi</math> | 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 <math>\pi</math>. This is because the imaginary string would have unwound a quarter of the circle's circumfrence. So <math>\frac{4\pi}{4}</math> is <math>\pi</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> | + | 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> |
Revision as of 11:34, 20 April 2012
Involute of a Circle |
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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 . This is because the imaginary string would have unwound a quarter of the circle's circumfrence. So is
The radius is 2, so using those two measurements we can find using the pythagorean theorem(or the distance from the origin to the point on the involute curve. So would equal. Then, to find one would have to subtract the angle DAC from . In this case it would be
So if you call the radius of the circle (2 in this case) and
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.
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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