Difference between revisions of "Hypocycloid"
From Encyclopedia of Mathematics
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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 | ||
| − | + | $$x(\theta)=(R-r)\cos\theta+r\cos\left[(R-r)\frac\theta r\right],$$ | |
| − | + | $$y(\theta)=(R-r)\sin\theta-r\sin\left[(R-r)\frac\theta r\right],$$ | |
| − | where | + | 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 | The radius of the curvature is | ||
| − | + | $$r_k=\frac{4R(m-1)}{m^2(m-2)}\sin\frac\theta2.$$ | |
| − | + | $m=3$. | |
<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 | + | 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 | + | 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
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