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.
The parametric equations of the epitrochoid are:
and of the hypotrochoid:
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
Trochoids are related to the so-called cycloidal curves (cf. Cycloidal curve). Sometimes the trochoid is called a shortened or elongated cycloid.
|||A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)|
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.
|[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)|
Hypotrochoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hypotrochoid&oldid=42489