Spirograph



Spirograph is a geometric drawing toy that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. It 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 with original product configurations in 2013 by Kahootz Toys.

History
The mathematician Bruno Abakanowicz invented the Spirograph between 1881 and 1900. It was used for calculating an area delimited by curves. Drawing toys based on gears have been around since at least 1908, when The Marvelous Wondergraph was advertised in the Sears catalog. An article describing how to make a Wondergraph drawing machine appeared in the Boys Mechanic publication in 1913. The Spirograph itself was developed by the British engineer Denys Fisher, who exhibited at the 1965 Nuremberg International Toy Fair. It was subsequently produced by his company. US distribution rights were acquired by Kenner, Inc., which introduced it to the United States market in 1966 and promoted it as a creative children's toy.

In 2013 the Spirograph brand was re-launched worldwide by Kahootz Toys with products that returned to the use of the original gears and wheels. The modern products use removable putty in place of pins or are held down by hand to keep the stationary pieces in place on the paper. The Spirograph was a 2014 Toy of the Year finalist in two categories, over 45 years after the toy was named Toy of the Year in 1967.

Operation
The original US-released Spirograph consisted of two different-sized plastic rings, with gear teeth on both the inside and outside of their circumferences. They were pinned to a cardboard backing with pins, and any of several provided gearwheels, which had holes provided for a ballpoint pen to extend through them to an underlying paper writing surface. It could be spun around to make geometric shapes on the underlying paper medium. Later, the Super-Spirograph consisted of a set of plastic gears and other interlocking shape-segments such as rings, triangles, or straight bars. It has several sizes of gears and shapes, and all edges have teeth to engage any other piece. For instance, smaller gears fit inside the larger rings, but also can engage the outside of the rings in such a fashion that they rotate around the inside or along the outside edge of the rings. Kenner also introduced Spirotot, Magnetic Spirograph, Spiroman and various refill sets.

To use it, a sheet of paper is placed on a heavy cardboard backing, and one of the plastic pieces—known as a stator—is secured via pins or reusable adhesive to the paper and cardboard. Another plastic piece—called the rotor—is placed so that its teeth engage with those of the pinned piece. For example, a ring may be pinned to the paper and a small gear placed inside the ring. The number of arrangements possible by combining different gears is very large. The point of a pen is placed in one of the holes of the rotor. As the rotor is moved, the pen traces out a curve. The pen is used both to draw and to provide locomotive force; some practice is required before the Spirograph can be operated without inadvertently disengaging the stator and rotor, particularly when using the holes situated near the edge of the larger rotors. More intricate and unusual-shaped patterns may be made through the use of both hands, one to draw and one to guide the pieces. It is possible 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), but this requires concentration or even assistance from other people. Pens of various colors were frequently included with the sets, or one could obtain them elsewhere, to add texture and color to the design by switching colors in a set pattern or a random fashion, as seen in the image above.

Mathematical basis
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:

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

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 $$(x',y')$$ obey:

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

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

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

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 $$t$$:

$$\begin{array}{rcrcl} x&=&x_c+x'&=&(R-r)\cos t+\rho\cos \frac{R-r}{r}t,\\[4pt] y&=&y_c+y'&=&(R-r)\sin t-\rho\sin \frac{R-r}{r}t.\\ \end{array}$$

(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

$$\begin{array}{rcl} x(t)&=&R\left[(1-k)\cos t+lk\cos \frac{1-k}{k}t\right],\\[4pt] y(t)&=&R\left[(1-k)\sin t-lk\sin \frac{1-k}{k}t\right].\\ \end{array}$$

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

It is interesting to note that 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.