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Spirograph

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

$\displaystyle x$ $\textstyle =$ $\displaystyle (R+r)\cos\theta-(r+\rho)\cos\left({{R+r\over r}\theta}\right)$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle (R+r)\sin\theta-(r+\rho)\sin\left({{R+r\over r}\theta}\right),$ (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
$\displaystyle x$ $\textstyle =$ $\displaystyle x_0[m\cos t+a\cos(n t)]-y_0[m\sin t-a\sin(nt)]$  
      (3)
$\displaystyle y$ $\textstyle =$ $\displaystyle y_0[m\cos t+a\cos(n t)]+x_0[m\sin t-a\sin(nt)],$  
      (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
$\displaystyle m$ $\textstyle \equiv$ $\displaystyle {q-p\over q}$ (5)
$\displaystyle n$ $\textstyle \equiv$ $\displaystyle {q-p\over p}$ (6)
$\displaystyle x_0$ $\textstyle \equiv$ $\displaystyle \cos\theta$ (7)
$\displaystyle y_0$ $\textstyle \equiv$ $\displaystyle \sin\theta.$ (8)

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


\begin{figure}\begin{center}\BoxedEPSF{Spirograph1-3.epsf scaled 801}\end{center}\end{figure}

\begin{figure}\begin{center}$(p,q)=(1,3)$\end{center}\end{figure}


\begin{figure}\begin{center}\BoxedEPSF{Spirograph1-4.epsf scaled 800}\end{center}\end{figure}

\begin{figure}\begin{center}$(p,q)=(1,4)$\end{center}\end{figure}


\begin{figure}\begin{center}\BoxedEPSF{Spirograph1-5.epsf scaled 800}\end{center}\end{figure}

\begin{figure}\begin{center}$(p,q)=(1,5)$\end{center}\end{figure}


\begin{figure}\begin{center}\BoxedEPSF{Spirograph2-5.epsf scaled 800}\end{center}\end{figure}

\begin{figure}\begin{center}$(p,q)=(2,5)$\end{center}\end{figure}


\begin{figure}\begin{center}\BoxedEPSF{Spirograph2-7.epsf scaled 800}\end{center}\end{figure}

\begin{figure}\begin{center}$(p,q)=(2,7)$\end{center}\end{figure}


\begin{figure}\begin{center}\BoxedEPSF{Spirograph3-7.epsf scaled 800}\end{center}\end{figure}

\begin{figure}\begin{center}$(p,q)=(3,7)$\end{center}\end{figure}


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

\begin{figure}\begin{center}\BoxedEPSF{SpirographTriangle.epsf scaled 700}\end{center}\end{figure}

See also Epitrochoid, Hypotrochoid, Maurer Rose, Spirolateral



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© 1996-9 Eric W. Weisstein
1999-05-26