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Spirograph
Spirograph set (UK Palitoy early 1980s) (perspective fixed).jpg
Spirograph set (early 1980s UK version)
Inventor(s) Denys Fisher
Company Hasbro
Country United Kingdom
Availability 1965–present
Materials Plastic

Spirograph is a geometric drawing device that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. The well known toy version was developed by British engineer Denys Fisher and first sold in 1965.

The name has been a registered trademark of Hasbro Inc. since 1998 following purchase of the company that had acquired the Denys Fisher company. The Spirograph brand was relaunched worldwide in 2013, with its original product configurations, by Kahootz Toys.

Operation

Spiograph Animation
Animation of a Spirograph
Various Spirograph Designs
Several Spirograph designs drawn with a Spirograph set using multiple different colored pens

The original US-released Spirograph consisted of two differently sized plastic rings (or stators), with gear teeth on both the inside and outside of their circumferences. Once either of these rings were held in place (either by pins, with an adhesive, or by hand) any of several provided gearwheels (or rotors)—each having holes for a ballpoint pen—could be spun around the ring to draw geometric shapes. Later, the Super-Spirograph introduced additional shapes such as rings, triangles, and straight bars. All edges of each piece have teeth to engage any other piece; smaller gears fit inside the larger rings, but they also can rotate along the rings' outside edge or even around each other. Gears can be combined in many different arrangements. Sets often included variously colored pens, which could enhance a design by switching colors, as seen in the examples shown here.

Beginners often slip the gears, especially when using the holes near the edge of the larger wheels, resulting in broken or irregular lines. Experienced users may learn to move several pieces in relation to each other (say, the triangle around the ring, with a circle "climbing" from the ring onto the triangle).

Mathematical basis

Resonance Cascade

Consider a fixed outer circle C_o of radius R centered at the origin. A smaller inner circle C_i of radius r < R is rolling inside C_o and is continuously tangent to it. C_i will be assumed never to slip on C_o (in a real Spirograph, teeth on both circles prevent such slippage). Now assume that a point A lying somewhere inside C_i is located a distance \rho<r from C_i's center. This point A corresponds to the pen-hole in the inner disk of a real Spirograph. Without loss of generality it can be assumed that at the initial moment the point A was on the X axis. In order to find the trajectory created by a Spirograph, follow point A as the inner circle is set in motion.

Now mark two points T on C_o and B on C_i. The point T always indicates the location where the two circles are tangent. Point B, however, will travel on C_i, and its initial location coincides with T. After setting C_i in motion counterclockwise around C_o, C_i has a clockwise rotation with respect to its center. The distance that point B traverses on C_i is the same as that traversed by the tangent point T on C_o, due to the absence of slipping.

Now define the new (relative) system of coordinates (X', Y') with its origin at the center of C_i and its axes parallel to X and Y. Let the parameter t be the angle by which the tangent point T rotates on C_o, and t' be the angle by which C_i rotates (i.e. by which B travels) in the relative system of coordinates. Because there is no slipping, the distances traveled by B and T along their respective circles must be the same, therefore

tR = (t - t')r,

or equivalently,

t' = -\frac{R - r}{r} t.

It is common to assume that a counterclockwise motion corresponds to a positive change of angle and a clockwise one to a negative change of angle. A minus sign in the above formula (t' < 0) accommodates this convention.

Let (x_c, y_c) be the coordinates of the center of C_i in the absolute system of coordinates. Then R - r represents the radius of the trajectory of the center of C_i, which (again in the absolute system) undergoes circular motion thus:

\begin{align}
 x_c &= (R - r)\cos t,\\
 y_c &= (R - r)\sin t.
\end{align}

As defined above, t' is the angle of rotation in the new relative system. Because point A obeys the usual law of circular motion, its coordinates in the new relative coordinate system (x', y') are

\begin{align}
 x' &= \rho\cos t',\\
 y' &= \rho\sin t'.
\end{align}

In order to obtain the trajectory of A in the absolute (old) system of coordinates, add these two motions:

\begin{align}
 x &= x_c + x' = (R - r)\cos t + \rho\cos t',\\
 y &= y_c + y' = (R - r)\sin t + \rho\sin t',\\
\end{align}

where \rho is defined above.

Now, use the relation between t and t' as derived above to obtain equations describing the trajectory of point A in terms of a single parameter t:

\begin{align}
 x &= x_c + x' = (R - r)\cos t + \rho\cos \frac{R - r}{r}t,\\
 y &= y_c + y' = (R - r)\sin t - \rho\sin \frac{R - r}{r}t\\
\end{align}

(using the fact that function \sin is odd).

It is convenient to represent the equation above in terms of the radius R of C_o and dimensionless parameters describing the structure of the Spirograph. Namely, let

l = \frac{\rho}{r}

and

k = \frac{r}{R}.

The parameter 0 \le l \le 1 represents how far the point A is located from the center of C_i. At the same time, 0 \le k \le 1 represents how big the inner circle C_i is with respect to the outer one C_o.

It is now observed that

\frac{\rho}{R} = lk,

and therefore the trajectory equations take the form

\begin{align}
 x(t) &= R\left[(1 - k)\cos t + lk\cos \frac{1 - k}{k}t\right],\\
 y(t) &= R\left[(1 - k)\sin t - lk\sin \frac{1 - k}{k}t\right].\\
\end{align}

Parameter R is a scaling parameter and does not affect the structure of the Spirograph. Different values of R would yield similar Spirograph drawings.

The two extreme cases k = 0 and k = 1 result in degenerate trajectories of the Spirograph. In the first extreme case, when k = 0, we have a simple circle of radius R, corresponding to the case where C_i has been shrunk into a point. (Division by k = 0 in the formula is not a problem, since both \sin and \cos are bounded functions).

The other extreme case k = 1 corresponds to the inner circle C_i's radius r matching the radius R of the outer circle C_o, i.e. r = R. In this case the trajectory is a single point. Intuitively, C_i is too large to roll inside the same-sized C_o without slipping.

If l = 1, then the point A is on the circumference of C_i. In this case the trajectories are called hypocycloids and the equations above reduce to those for a hypocycloid.

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