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) |
Trochoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Trochoid&oldid=55774