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A planar curve given by the trajectory of a point on a circle rolling on the exterior side of another circle. The parametric equations are:
 
A planar curve given by the trajectory of a point on a circle rolling on the exterior side of another circle. The parametric equations are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e0358601.png" /></td> </tr></table>
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$$x=(r+R)\cos\theta-r\cos\left[(r+R)\frac\theta r\right],$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e0358602.png" /></td> </tr></table>
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$$y=(r+R)\sin\theta-r\sin\left[(r+R)\frac\theta r\right],$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e0358603.png" /> is the radius of the rolling and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e0358604.png" /> that of the fixed circle, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e0358605.png" /> is the angle between the radius vector of the point of contact of the circles (see Fig. a, Fig. b) and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e0358606.png" />-axis.
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where $r$ is the radius of the rolling and $R$ that of the fixed circle, and $\theta$ is the angle between the radius vector of the point of contact of the circles (see Fig. a, Fig. b) and the $x$-axis.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e035860a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e035860a.gif" />
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Figure: e035860a
 
Figure: e035860a
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e0358607.png" />
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$m=3$
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e035860b.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/e035860b.gif" />
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Figure: e035860b
 
Figure: e035860b
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e0358608.png" />
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$m=3/2$
  
Depending on the value of the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e0358609.png" />, the resulting epicycloid has different forms. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586010.png" /> it is a cardioid, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586011.png" /> is an integer, the curve consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586012.png" /> distinct branches. The cusps <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586013.png" /> have the polar coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586016.png" />. The vertices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586017.png" /> of the curve have the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586019.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586020.png" /> is a rational fraction, the branches intersect each other in the interior; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586021.png" /> is irrational there are infinitely many branches and the curve does not return to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586022.png" /> describing a position obtained previously; for rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586023.png" /> the epicycloid is a closed algebraic curve. The arc length from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586024.png" /> is:
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Depending on the value of the modulus $m=R/r$, the resulting epicycloid has different forms. For $m=1$ it is a cardioid, and if $m$ is an integer, the curve consists of $m$ distinct branches. The cusps $A_1,\ldots,A_m$ have the polar coordinates $\rho=R$, $\phi=\pi2k/m$, $k=0,\ldots,m-1$. The vertices $B_1,\ldots,B_m$ of the curve have the coordinates $\rho=R+2r$, $\phi=\pi(2k+1)/m$. When $m$ is a rational fraction, the branches intersect each other in the interior; when $m$ is irrational there are infinitely many branches and the curve does not return to a point $M$ describing a position obtained previously; for rational $m$ the epicycloid is a closed algebraic curve. The arc length from the point $A_1$ is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586025.png" /></td> </tr></table>
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$$s=8Rm(1+m)\sin^2\frac\theta4,$$
  
and from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586026.png" /> it is
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and from $B_1$ it is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586027.png" /></td> </tr></table>
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$$s=4Rm(1+m)\cos\frac\theta2.$$
  
 
The area of a sector bounded by two radius vectors of the curve and its arc is
 
The area of a sector bounded by two radius vectors of the curve and its arc is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586028.png" /></td> </tr></table>
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$$S=m\pi(R+mR)(R+2mR).$$
  
 
The radius of curvature is
 
The radius of curvature is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035860/e03586029.png" /></td> </tr></table>
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$$r_k=\frac{4Rm(1+m)}{2m+1}\sin\frac\theta2.$$
  
 
When the point is not situated on the rolling circle, but lies in its exterior (or interior) region, then the curve is called an elongated (respectively, shortened) epicycloid or epitrochoid (see [[Trochoid|Trochoid]]). Epicycloids belong to the so-called cycloidal curves (cf. [[Cycloidal curve|Cycloidal curve]]).
 
When the point is not situated on the rolling circle, but lies in its exterior (or interior) region, then the curve is called an elongated (respectively, shortened) epicycloid or epitrochoid (see [[Trochoid|Trochoid]]). Epicycloids belong to the so-called cycloidal curves (cf. [[Cycloidal curve|Cycloidal curve]]).

Revision as of 10:40, 7 August 2014

A planar curve given by the trajectory of a point on a circle rolling on the exterior side of another circle. The parametric equations are:

$$x=(r+R)\cos\theta-r\cos\left[(r+R)\frac\theta r\right],$$

$$y=(r+R)\sin\theta-r\sin\left[(r+R)\frac\theta r\right],$$

where $r$ is the radius of the rolling and $R$ that of the fixed circle, and $\theta$ is the angle between the radius vector of the point of contact of the circles (see Fig. a, Fig. b) and the $x$-axis.

Figure: e035860a

$m=3$

Figure: e035860b

$m=3/2$

Depending on the value of the modulus $m=R/r$, the resulting epicycloid has different forms. For $m=1$ it is a cardioid, and if $m$ is an integer, the curve consists of $m$ distinct branches. The cusps $A_1,\ldots,A_m$ have the polar coordinates $\rho=R$, $\phi=\pi2k/m$, $k=0,\ldots,m-1$. The vertices $B_1,\ldots,B_m$ of the curve have the coordinates $\rho=R+2r$, $\phi=\pi(2k+1)/m$. When $m$ is a rational fraction, the branches intersect each other in the interior; when $m$ is irrational there are infinitely many branches and the curve does not return to a point $M$ describing a position obtained previously; for rational $m$ the epicycloid is a closed algebraic curve. The arc length from the point $A_1$ is:

$$s=8Rm(1+m)\sin^2\frac\theta4,$$

and from $B_1$ it is

$$s=4Rm(1+m)\cos\frac\theta2.$$

The area of a sector bounded by two radius vectors of the curve and its arc is

$$S=m\pi(R+mR)(R+2mR).$$

The radius of curvature is

$$r_k=\frac{4Rm(1+m)}{2m+1}\sin\frac\theta2.$$

When the point is not situated on the rolling circle, but lies in its exterior (or interior) region, then the curve is called an elongated (respectively, shortened) epicycloid or epitrochoid (see Trochoid). Epicycloids belong to the so-called cycloidal curves (cf. Cycloidal curve).

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)


Comments

Epicycloids (and hypocycloids, cf. Hypocycloid) have many equivalent definitions. See, e.g., [a3], pp. 273-277. Epicycloids and, more generally, trochoids are important for kinematical constructions, cf. [a1].

References

[a1] H.-R. Müller, "Kinematik" , de Gruyter (1963)
[a2] K. Strubecker, "Differential geometry" , I , de Gruyter (1964)
[a3] M. Berger, "Geometry" , I , Springer (1977)
[a4] M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)
How to Cite This Entry:
Epicycloid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Epicycloid&oldid=13845
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article