Trochoid

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.

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

[1]	A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[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 (1972) ISBN 0-486-60288-5 Zbl 0257.50002
[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)

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