Spirograph

A Hypotrochoid generated by a fixed point on a Circle rolling inside a fixed Circle. It has parametric equations,

$$x = (R+r)\cos\theta-(r+\rho)\cos\left({{R+r\over r}\theta}\right)$$ (1)

$$y = (R+r)\sin\theta-(r+\rho)\sin\left({{R+r\over r}\theta}\right),$$ (2)

where $R$ is the radius of the fixed circle, $r$ is the radius of the rotating circle, and $\rho$ is the offset of the edge of the rotating circle. The figure closes only if $R$, $r$, and $\rho$ are Rational. The equations can also be written

$$x = x_0[m\cos t+a\cos(n t)]-y_0[m\sin t-a\sin(nt)]$$ (3)

$$y = y_0[m\cos t+a\cos(n t)]+x_0[m\sin t-a\sin(nt)],$$ (4)

where the outer wheel has radius 1, the inner wheel a radius $p/q$, the pen is placed $a$ units from the center, the beginning is at $\theta$ radians above the $x$-axis, and

$$m \equiv {q-p\over q}$$ (5)

$$n \equiv {q-p\over p}$$ (6)

$$x_0 \equiv \cos\theta$$ (7)

$$y_0 \equiv \sin\theta.$$ (8)

The following curves are for $a=i/10$, with $i=1$, 2, ..., 10, and $\theta=0$.

Additional attractive designs such as the following can also be made by superposing individual spirographs.

See also Epitrochoid, Hypotrochoid, Maurer Rose, Spirolateral