In geometry, a trochoid is a roulette curve formed by a circle rolling along a line. It is the curve traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle) as it rolls along a straight line. If the point is on the circle, the trochoid is called common (also known as a cycloid); if the point is inside the circle, the trochoid is curtate; and if the point is outside the circle, the trochoid is prolate. The word "trochoid" was coined by Gilles de Roberval, referring to the special case of a cycloid.[1]
As a circle of radius rolls without slipping along a line, the center moves parallel to, and every other point in the rotating plane rigidly attached to the circle traces the curve called the trochoid. Let . Parametric equations of the trochoid for which is the -axis are
\begin{align} &x=a\theta-b\sin\theta\\ &y=a-b\cos\theta \end{align}
If lies inside the circle, on its circumference, or outside, the trochoid is described as being curtate ("contracted"), common, or prolate ("extended"), respectively.[2] A curtate trochoid is traced by a pedal (relative to the ground) when a normally geared bicycle is pedaled along a straight line.[3] A prolate trochoid is traced by the tip of a paddle (relative to the water's surface) when a boat is driven with constant velocity by paddle wheels; this curve contains loops. A common trochoid, also called a cycloid, has cusps at the points where touches the line .
A more general approach would define a trochoid as the locus of a point
(x,y)
(x',y')
x=x'+r1\cos(\omega1t+\phi1), y=y'+r1\sin(\omega1t+\phi1), r1>0,
\begin{array}{lcl} x'=x0+v2xt, y'=y0+v2yt\\ \thereforex=x0+r1\cos(\omega1t+\phi1)+v2xt, y=y0+r1\sin(\omega1t+\phi1)+v2yt,\\ \end{array}
(x0,y0)
\begin{array}{lcl} x'=x0+r2\cos(\omega2t+\phi2), y'=y0+r2\sin(\omega2t+\phi2), r2\ge0\\ \thereforex=x0+r1\cos(\omega1t+\phi1)+r2\cos(\omega2t+\phi2), y=y0+r1\sin(\omega1t+\phi1)+r2\sin(\omega2t+\phi2),\\ \end{array}
The ratio of the rates of motion and whether the moving axis translates in a straight or circular path determines the shape of the trochoid. In the case of a straight path, one full rotation coincides with one period of a periodic (repeating) locus. In the case of a circular path for the moving axis, the locus is periodic only if the ratio of these angular motions,
\omega1/\omega2
p/q
p
q
p
q
(x0,y0)
r1
R
\begin{array}{lcl} epicycloid:&\omega1/\omega2&=p/q=r2/r1=R/r1+1, |p-q|cusps\\ hypocycloid:&\omega1/\omega2&=p/q=-r2/r1=-(R/r1-1), |p-q|=|p|+|q|cusps \end{array}
r2