Epicycloid: Difference between revisions
→Equations: radians mentioned above |
→Proof: rp = origin to point p distance |
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[[Differentiable function|differentiable]]). |
[[Differentiable function|differentiable]]). |
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If ''k'' is a [[rational number]], say ''k=p/q'' expressed as [[irreducible fraction]], then the curve has ''p'' cusps. Radians is 0 to q * 2 PI. p and q are positive integers. |
If ''k'' is a [[rational number]], say ''k=p/q'' expressed as [[irreducible fraction]], then the curve has ''p'' cusps. Radians is (0 to q * 2 PI) to close the curve. p and q are positive integers. |
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If ''k'' is an [[irrational number]], then the curve never closes, and forms a [[dense set|dense subset]] of the space between the larger circle and a circle of radius ''R'' + 2''r''. |
If ''k'' is an [[irrational number]], then the curve never closes, and forms a [[dense set|dense subset]] of the space between the larger circle and a circle of radius ''R'' + 2''r''. |
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The distance rp from (x=0,y=0) origin to (the point <math>p</math> on the small circle) varies up and down as |
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R <= rp <= (R + 2r) |
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<gallery caption="Epicycloid examples"> |
<gallery caption="Epicycloid examples"> |
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:<math>\alpha =\frac{R}{r} \theta</math> |
:<math>\alpha =\frac{R}{r} \theta</math> |
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From the figure, we see the position of the point <math>p</math> clearly. |
From the figure, we see the position of the point <math>p</math> on the small circle clearly. |
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:<math> x=\left( R+r \right)\cos \theta -r\cos\left( \theta+\alpha \right) =\left( R+r \right)\cos \theta -r\cos\left( \frac{R+r}{r}\theta \right)</math> |
:<math> x=\left( R+r \right)\cos \theta -r\cos\left( \theta+\alpha \right) =\left( R+r \right)\cos \theta -r\cos\left( \frac{R+r}{r}\theta \right)</math> |
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:<math>y=\left( R+r \right)\sin \theta -r\sin\left( \theta+\alpha \right) =\left( R+r \right)\sin \theta -r\sin\left( \frac{R+r}{r}\theta \right)</math> |
:<math>y=\left( R+r \right)\sin \theta -r\sin\left( \theta+\alpha \right) =\left( R+r \right)\sin \theta -r\sin\left( \frac{R+r}{r}\theta \right)</math> |
Revision as of 11:30, 1 December 2019
![](http://upload.wikimedia.org/wikipedia/commons/a/ae/EpitrochoidOn3-generation.gif)
In geometry, an epicycloid or hypercycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an epicycle—which rolls without slipping around a fixed circle. It is a particular kind of roulette.
Equations
If the smaller circle has radius r, and the larger circle has radius R = kr, then the parametric equations for the curve can be given by either:
or:
(Assuming the initial point lies on the larger circle.)
If k is a positive integer, then the curve is closed, and has k cusps (i.e., sharp corners, where the curve is not differentiable).
If k is a rational number, say k=p/q expressed as irreducible fraction, then the curve has p cusps. Radians is (0 to q * 2 PI) to close the curve. p and q are positive integers.
If k is an irrational number, then the curve never closes, and forms a dense subset of the space between the larger circle and a circle of radius R + 2r.
The distance rp from (x=0,y=0) origin to (the point on the small circle) varies up and down as
R <= rp <= (R + 2r)
- Epicycloid examples
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k = 1
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k = 2
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k = 3
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k = 4
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k = 2.1 = 21/10
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k = 3.8 = 19/5
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k = 5.5 = 11/2
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k = 7.2 = 36/5
The epicycloid is a special kind of epitrochoid.
An epicycle with one cusp is a cardioid, two cusps is a nephroid.
An epicycloid and its evolute are similar.[1]
Proof
![](http://upload.wikimedia.org/wikipedia/commons/thumb/6/61/Epizykloide_herleitung.svg/440px-Epizykloide_herleitung.svg.png)
We assume that the position of is what we want to solve, is the radian from the tangential point to the moving point , and is the radian from the starting point to the tangential point.
Since there is no sliding between the two cycles, then we have that
By the definition of radian (which is the rate arc over radius), then we have that
From these two conditions, we get the identity
By calculating, we get the relation between and , which is
From the figure, we see the position of the point on the small circle clearly.
See also
- List of periodic functions
- Cycloid
- Cyclogon
- Deferent and epicycle
- Epicyclic gearing
- Epitrochoid
- Hypocycloid
- Hypotrochoid
- Multibrot set
- Roulette (curve)
- Spirograph
References
- J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 161, 168–170, 175. ISBN 978-0-486-60288-2.
External links
- Weisstein, Eric W. "Epicycloid". MathWorld.
- "Epicycloid" by Michael Ford, The Wolfram Demonstrations Project, 2007
- O'Connor, John J.; Robertson, Edmund F., "Epicycloid", MacTutor History of Mathematics Archive, University of St Andrews
- Animation of Epicycloids, Pericycloids and Hypocycloids
- Spirograph -- GeoFun
- Historical note on the application of the epicycloid to the form of Gear Teeth