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A plane curve which is the trajectory of a point on a circle rolling along a second circle while osculating it from inside. The parametric equations are
 
A plane curve which is the trajectory of a point on a circle rolling along a second circle while osculating it from inside. 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/h/h048/h048530/h0485301.png" /></td> </tr></table>
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$$x(\theta)=(R-r)\cos\theta+r\cos\left[(R-r)\frac\theta r\right],$$
  
<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/h/h048/h048530/h0485302.png" /></td> </tr></table>
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$$y(\theta)=(R-r)\sin\theta-r\sin\left[(R-r)\frac\theta r\right],$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h0485303.png" /> is the radius of the moving circle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h0485304.png" /> is the radius of the fixed circle and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h0485305.png" /> is the angle between the radius vector of the centre of the moving circle with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h0485306.png" />-axis (assuming the trajectory passes through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h0485307.png" />). Depending on the size of the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h0485308.png" />, hypocycloids of different forms are obtained. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h0485309.png" /> is an integer, the curve consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853010.png" /> non-intersecting branches (Fig. a). The points of return <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853011.png" /> have polar coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853015.png" /> is irrational, the number of branches is infinite, and the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853016.png" /> does not return to its initial location; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853017.png" /> is rational, the hypocycloid is a closed algebraic curve. The arc length from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853018.png" /> is
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where $r$ is the radius of the moving circle, $R$ is the radius of the fixed circle and $\theta$ is the angle between the radius vector of the centre of the moving circle with the $x$-axis (assuming the trajectory passes through $(0,R)$). Depending on the size of the modulus $m=R/r$, hypocycloids of different forms are obtained. If $m$ is an integer, the curve consists of $m$ non-intersecting branches (Fig. a). The points of return $A_1,\ldots,A_m$ have polar coordinates $\rho=R$, $\phi=2k\pi/m$, $k=0,\ldots,m-1$. If $m$ is irrational, the number of branches is infinite, and the point $M$ does not return to its initial location; if $m$ is rational, the hypocycloid is a closed algebraic curve. The arc length from the point $\theta=0$ 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/h/h048/h048530/h04853019.png" /></td> </tr></table>
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$$l=\frac{8R(m-1)}{m^2}\sin^2\frac\theta4.$$
  
 
The radius of the curvature is
 
The radius of the 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/h/h048/h048530/h04853020.png" /></td> </tr></table>
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$$r_k=\frac{4R(m-1)}{m^2(m-2)}\sin\frac\theta2.$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853021.png" />.
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$m=3$.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h048530a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h048530a.gif" />
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Figure: h048530a
 
Figure: h048530a
  
If the point is not located on the rolling circle, but outside (or inside) it, the curve is said to be a lengthened (shortened) hypocycloid, or hypotrochoid (cf. [[Trochoid|Trochoid]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853022.png" /> the hypocycloid is a segment of a straight line; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853023.png" />, it is a [[Steiner curve|Steiner curve]]; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853024.png" />, it is an [[Astroid|astroid]]. Hypocycloids belong to the so-called cycloidal curves (cf. [[Cycloidal curve|Cycloidal curve]]).
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If the point is not located on the rolling circle, but outside (or inside) it, the curve is said to be a lengthened (shortened) hypocycloid, or hypotrochoid (cf. [[Trochoid|Trochoid]]). If $m=2$ the hypocycloid is a segment of a straight line; if $m=3$, it is a [[Steiner curve|Steiner curve]]; if $m=4$, it is an [[Astroid|astroid]]. Hypocycloids belong to the so-called cycloidal curves (cf. [[Cycloidal curve|Cycloidal curve]]).
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
Every hypocycloid which is generated by circles with radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853026.png" /> can also be generated by circles with radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048530/h04853028.png" /> ([[#References|[a2]]], [[#References|[a3]]]). Hypocycloids, and more generally trochoids, play an important role in plane kinematics.
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Every hypocycloid which is generated by circles with radii $R$ and $r$ can also be generated by circles with radii $R$ and $R-r$ ([[#References|[a2]]], [[#References|[a3]]]). Hypocycloids, and more generally trochoids, play an important role in plane kinematics.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''I''' , Springer  (1987)  pp. 273–276</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Fladt,  "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell.  (1962)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.R. Müller,  "Kinematik" , de Gruyter  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''I''' , Springer  (1987)  pp. 273–276</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Fladt,  "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell.  (1962)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.R. Müller,  "Kinematik" , de Gruyter  (1963)</TD></TR></table>

Revision as of 10:09, 7 August 2014

A plane curve which is the trajectory of a point on a circle rolling along a second circle while osculating it from inside. The parametric equations are

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

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

where $r$ is the radius of the moving circle, $R$ is the radius of the fixed circle and $\theta$ is the angle between the radius vector of the centre of the moving circle with the $x$-axis (assuming the trajectory passes through $(0,R)$). Depending on the size of the modulus $m=R/r$, hypocycloids of different forms are obtained. If $m$ is an integer, the curve consists of $m$ non-intersecting branches (Fig. a). The points of return $A_1,\ldots,A_m$ have polar coordinates $\rho=R$, $\phi=2k\pi/m$, $k=0,\ldots,m-1$. If $m$ is irrational, the number of branches is infinite, and the point $M$ does not return to its initial location; if $m$ is rational, the hypocycloid is a closed algebraic curve. The arc length from the point $\theta=0$ is

$$l=\frac{8R(m-1)}{m^2}\sin^2\frac\theta4.$$

The radius of the curvature is

$$r_k=\frac{4R(m-1)}{m^2(m-2)}\sin\frac\theta2.$$

$m=3$.

Figure: h048530a

If the point is not located on the rolling circle, but outside (or inside) it, the curve is said to be a lengthened (shortened) hypocycloid, or hypotrochoid (cf. Trochoid). If $m=2$ the hypocycloid is a segment of a straight line; if $m=3$, it is a Steiner curve; if $m=4$, it is an astroid. Hypocycloids belong to the so-called cycloidal curves (cf. Cycloidal curve).

References

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


Comments

Every hypocycloid which is generated by circles with radii $R$ and $r$ can also be generated by circles with radii $R$ and $R-r$ ([a2], [a3]). Hypocycloids, and more generally trochoids, play an important role in plane kinematics.

References

[a1] M. Berger, "Geometry" , I , Springer (1987) pp. 273–276
[a2] K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962)
[a3] H.R. Müller, "Kinematik" , de Gruyter (1963)
How to Cite This Entry:
Hypocycloid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hypocycloid&oldid=14707
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article