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A plane curve that is the trajectory of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094330/t0943301.png" /> inside or outside a circle that rolls upon another circle. A trochoid is called an epitrochoid (Fig.1a, Fig.1b) or a hypotrochoid (Fig.2a, Fig.2b), depending on whether the rolling circle has external or internal contact with the fixed circle.
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A plane curve that is the trajectory of a point $M$ inside or outside a circle that rolls upon another circle. A trochoid is called an epitrochoid (Fig.1a, Fig.1b) or a hypotrochoid (Fig.2a, Fig.2b), depending on whether the rolling circle has external or internal contact with the fixed circle.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t094330a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/t094330a.gif" />
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The parametric equations of the epitrochoid are:
 
The parametric equations of the epitrochoid 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/t/t094/t094330/t0943302.png" /></td> </tr></table>
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$$x=(R+mR)\cos mt-h\cos(t+mt),$$
  
<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/t/t094/t094330/t0943303.png" /></td> </tr></table>
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$$y=(R+mR)\sin mt-h\sin(t+mt);$$
  
 
and of the hypotrochoid:
 
and of the hypotrochoid:
  
<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/t/t094/t094330/t0943304.png" /></td> </tr></table>
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$$x=(R-mR)\cos mt+h\cos(t-mt),$$
  
<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/t/t094/t094330/t0943305.png" /></td> </tr></table>
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$$y=(R-mR)\sin mt-h\sin(t-mt),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094330/t0943306.png" /> is the radius of the rolling circle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094330/t0943307.png" /> is the radius of the fixed circle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094330/t0943308.png" /> is the modulus of the trochoid, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094330/t0943309.png" /> is the distance from the tracing point to the centre of the rolling circle. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094330/t09433010.png" />, then the trochoid is called elongated (Fig.1a, Fig.2a), when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094330/t09433011.png" /> shortened (Fig.1b, Fig.2b) and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094330/t09433012.png" />, an [[Epicycloid|epicycloid]] or [[Hypocycloid|hypocycloid]].
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where $r$ is the radius of the rolling circle, $R$ is the radius of the fixed circle, $m=R/r$ is the modulus of the trochoid, and $h$ is the distance from the tracing point to the centre of the rolling circle. If $h>r$, then the trochoid is called elongated (Fig.1a, Fig.2a), when $h>r$ shortened (Fig.1b, Fig.2b) and when $h=r$, an [[Epicycloid|epicycloid]] or [[Hypocycloid|hypocycloid]].
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094330/t09433013.png" />, then the trochoid is called a trochoidal rosette; its equation in polar coordinates is
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If $h=R=r$, then the trochoid is called a trochoidal rosette; its equation in polar coordinates 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/t/t094/t094330/t09433014.png" /></td> </tr></table>
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$$\rho=a\sin\mu\phi.$$
  
For rational values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094330/t09433015.png" /> the trochoidal rosette is an [[Algebraic curve|algebraic curve]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094330/t09433016.png" />, then the trochoid is called the [[Pascal limaçon|Pascal limaçon]]; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094330/t09433017.png" />, an [[Ellipse|ellipse]].
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For rational values of $\mu$ the trochoidal rosette is an [[Algebraic curve|algebraic curve]]. If $R=r$, then the trochoid is called the [[Pascal limaçon|Pascal limaçon]]; if $R=2r$, an [[Ellipse|ellipse]].
  
 
Trochoids are related to the so-called cycloidal curves (cf. [[Cycloidal curve|Cycloidal curve]]). Sometimes the trochoid is called a shortened or elongated cycloid.
 
Trochoids are related to the so-called cycloidal curves (cf. [[Cycloidal curve|Cycloidal curve]]). Sometimes the trochoid is called a shortened or elongated cycloid.

Revision as of 10:45, 7 August 2014

A plane curve that is the trajectory of a point $M$ inside or outside a circle that rolls upon another circle. A trochoid is called an epitrochoid (Fig.1a, Fig.1b) or a hypotrochoid (Fig.2a, Fig.2b), depending on whether the rolling circle has external or internal contact with the fixed circle.

Figure: t094330a

Figure: t094330b

Figure: t094330c

Figure: t094330d

The parametric equations of the epitrochoid are:

$$x=(R+mR)\cos mt-h\cos(t+mt),$$

$$y=(R+mR)\sin mt-h\sin(t+mt);$$

and of the hypotrochoid:

$$x=(R-mR)\cos mt+h\cos(t-mt),$$

$$y=(R-mR)\sin mt-h\sin(t-mt),$$

where $r$ is the radius of the rolling circle, $R$ is the radius of the fixed circle, $m=R/r$ is the modulus of the trochoid, and $h$ is the distance from the tracing point to the centre of the rolling circle. If $h>r$, then the trochoid is called elongated (Fig.1a, Fig.2a), when $h>r$ shortened (Fig.1b, Fig.2b) and when $h=r$, an epicycloid or hypocycloid.

If $h=R=r$, then the trochoid is called a trochoidal rosette; its equation in polar coordinates is

$$\rho=a\sin\mu\phi.$$

For rational values of $\mu$ the trochoidal rosette is an algebraic curve. If $R=r$, then the trochoid is called the Pascal limaçon; if $R=2r$, an ellipse.

Trochoids are related to the so-called cycloidal curves (cf. Cycloidal curve). Sometimes the trochoid is called a shortened or elongated cycloid.

References

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


Comments

Trochoids play an important role in kinematics. They are used for the construction of gears and engines (see [a2]). Historically, they were a tool for the description of the movement of the planets before N. Copernicus and J. Kepler succeeded to establish the actual view of the dynamics of the solar system.

References

[a1] K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962)
[a2] H.-R. Müller, "Kinematik" , de Gruyter (1963)
[a3] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
[a4] M. Berger, "Geometry" , 1–2 , Springer (1987) pp. §9.14.34 (Translated from French)
[a5] F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)
How to Cite This Entry:
Trochoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trochoid&oldid=18583
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article