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[[Image:EpitrochoidOn3-generation.gif|thumb|500px|The red curve is an epicycloid traced as the small circle (radius {{nowrap|''r'' {{=}} 1)}} rolls around the outside of the large circle (radius {{nowrap|''R'' {{=}} 3)}}.]]
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In [[geometry]], an '''epicycloid''' is a plane [[curve]] produced by tracing the path of a chosen point of a [[circle]] — called an ''[[Deferent and epicycle|epicycle]]'' — which rolls without slipping around a fixed circle. It is a particular kind of [[roulette (curve)|roulette]].
 
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:
:<math>x (\theta) = (R + r) \cos \theta - r \cos \left( \frac{R + r}{r} \theta \right)</math>
:<math>y (\theta) = (R + r) \sin \theta - r \sin \left( \frac{R + r}{r} \theta \right),</math>
or:
:<math>x (\theta) = r (k + 1) \cos \theta - r \cos \left( (k + 1) \theta \right) \,</math>
:<math>y (\theta) = r (k + 1) \sin \theta - r \sin \left( (k + 1) \theta \right). \,</math>
 
If ''k'' is an integer, then the curve is closed, and has ''k'' [[cusp (singularity)|cusp]]s (i.e., sharp corners, where the curve is not
[[differentiable]]).
 
If ''k'' is a [[rational number]], say ''k=p/q'' expressed in simplest terms, then the curve has ''p'' cusps.
 
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''.  
 
<gallery caption="Epicycloid examples">
Image:Epicycloid-1.svg| ''k'' = 1
Image:Epicycloid-2.svg| ''k'' = 2
Image:Epicycloid-3.svg| ''k'' = 3
Image:Epicycloid-4.svg| ''k'' = 4
Image:Epicycloid-2-1.svg| ''k'' = 2.1 = 21/10
Image:Epicycloid-3-8.svg| ''k'' = 3.8 = 19/5
Image:Epicycloid-5-5.svg| ''k'' = 5.5 = 11/2
Image:Epicycloid-7-2.svg| ''k'' = 7.2 = 36/5
</gallery>
 
The epicycloid is a special kind of [[epitrochoid]].
 
An epicycle with one cusp is a [[cardioid]].
 
An epicycloid and its [[evolute]] are [[Similarity (geometry)|similar]].[http://mathworld.wolfram.com/EpicycloidEvolute.html]
 
==Proof==
[[Image:epicycloid geometry.svg|right|400px]]
We assume that the position of <math>p</math> is what we want to solve, <math>\alpha</math> is the radian from the tangential point to the moving point <math>p</math>, and <math>\theta</math> is the radian from the starting point to the tangential point.
 
Since there is no sliding between the two cycles<!-- what does "sliding" mean mathematically? -->, then we have that
:<math>\ell_R=\ell_r</math>
By the definition of radian (which is the rate arc over radius), then we have that
:<math>\ell_R= \theta R, \ell_r=\alpha r</math>
From these two conditions, we get the identity
:<math>\theta R=\alpha r</math>
By calculating<!-- what? -->, we get the relation between <math>\alpha</math> and <math>\theta</math>, which is
:<math>\alpha =\frac{R}{r} \theta</math>
 
From the figure, we see the position of the point <math>p</math> clearly.
:<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>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>
 
==See also==
* Special cases: [[Cardioid]], [[Nephroid]]
* [[List of periodic functions]]
* [[Cycloid]]
* [[Hypocycloid]]
* [[Epitrochoid]]
* [[Hypotrochoid]]
* [[Spirograph]]
* [[Deferent and epicycle]]
* [[Epicyclic gearing]]
 
==References==
* {{cite book| author=J. Dennis Lawrence| title=A catalog of special plane curves| publisher=Dover Publications| year=1972| isbn=0-486-60288-5| pages=161,168–170,175}}
 
==External links==
*[http://demonstrations.wolfram.com/Epicycloid/ Epicycloid], [[MathWorld]]
*"[http://demonstrations.wolfram.com/Epicycloid/ Epicycloid]" by Michael Ford, [[The Wolfram Demonstrations Project]], 2007
*{{MacTutor|class=Curves|id=Epicycloid|title=Epicycloid}}
*[http://sourceforge.net/p/geofun/wiki/Home/ Sipirograph -- GeoFun]
*[http://link.springer.com/article/10.1007%2Fs12045-013-0106-3/ Historical note on the application of the epicycloid to the form of Gear Teeth]
[[Category:Curves]]
[[Category:Algebraic curves]]
 
[[nl:Cycloïde#Epicycloïde]]

Latest revision as of 18:04, 1 October 2014

I don't normally discuss myself. I love sharing remarkable written content with other folks. I also want to explore the online world due to the fact I feel there's a great deal to discover on the market. When I am on the internet.
|
Name: Erna Dumolo
Age: 35
Country: Austria
Home town: Kitzbuhel
ZIP: 6370
Street: Laxenburger Strasse 31
|
My name is Erna Dumolo. I life in Kitzbuhel (Austria).
|
I am Erna from Kitzbuhel doing my final year engineering in Dance. I did my schooling, secured 71% and hope to find someone with same interests in Fishing.
|
I like Fishing.
I also to learn Norwegian in my spare time.
|
My name is Erna (35 years old) and my hobbies are Drawing and Martial arts.
|
Hello! I am Erna. I am satisfied that I can unite to the whole world. I live in Austria, in the south region. I dream to go to the different nations, to look for familiarized with interesting people.
|
I'm Erna (27) from Kitzbuhel, Austria.
I'm learning Norwegian literature at a local high school and I'm just about to graduate.
I have a part ipad 2 repair elk grove village time job in a backery.
|
Hello!
My name is Erna and I'm a 29 years old girl from Kitzbuhel.
|
Hi, everybody! My name is Erna.
It is a little about myself: I live in Austria, my city of Kitzbuhel.
It's called often Eastern or cultural capital of . I've married 3 years ago.
I have two children - a son (Lamar) and the daughter (Noble). We all like Fishing.
|
Hi, everybody!
I'm Norwegian female :).
I really like 2 Broke Girls!
|
Hello from Austria. I'm glad to came here. My first name is Erna.
I live in a town called Kitzbuhel in nothern Austria.
I was also born in Kitzbuhel 22 years ago. Married in July year 2000. I'm working at the college.
|
My name's Erna Dumolo but everybody calls me Erna. I'm from Austria. I'm studying at the college (1st year) and I play the Euphonium for 3 years. Usually I choose music from the famous films :D.
I have two sister. I like Conlanging, watching movies and Kayaking.
|
I'm a 43 years old and study at the college (Dance).
In my free time I'm trying to teach myself Norwegian. I have been there and look forward to returning sometime in the future. I love to read, preferably on my kindle. I really love to watch The Big Bang Theory and American Dad as well as documentaries about nature. I love Fishing.
|
I'm Erna and I live in Kitzbuhel.
I'm interested in Dance, Fishing and Norwegian art. I like travelling and reading fantasy.
|
I'm Erna and I live with my husband and our three children in Kitzbuhel, in the south part. My hobbies are Nordic skating, Golfing and Sculling or Rowing.
|
Hello, I'm Erna, a 19 year old from Kitzbuhel, Austria.
My hobbies include (but are not limited to) Juggling, Meteorology and watching 2 Broke Girls.
|
Hi there! :) My name is Erna, I'm a student studying Dance from Kitzbuhel, Austria.
|
I'm Erna and I live in a seaside city in northern Austria, Kitzbuhel. I'm 22 and I'm will soon finish my study at Dance.
|
I am Erna from Kitzbuhel. I am learning to play the Euphonium. Other hobbies are Fishing.
|
Im Erna and was born on 4 July 1974. My hobbies are Metal detecting and Air sports.
|
My name is Erna and I am studying Asian Studies and Asian Studies at Kitzbuhel / Austria.
}

Also visit my web blog - computer repair in hoffman estates

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