Spirograph
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. History. In 1827, Greek-born English architect and engineer Peter Hubert Desvignes developed and advertised a "Speiragraph", a device to create elaborate  spiral drawings. A man named J. Jopling soon claimed to have previously invented similar methods. When working in Vienna between 1845 and 1848, Desvignes constructed a version of the machine that would help prevent banknote forgeries, as any of the nearly endless variations of roulette patterns that it could produce were extremely difficult to reverse engineer. The mathematician Bruno Abakanowicz invented a new Spirograph device 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 definitive Spirograph toy was developed by the British engineer Denys Fisher between 1962 and 1964 by creating drawing machines with Meccano pieces. Fisher exhibited his spirograph 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. Kenner later introduced Spirotot, Magnetic Spirograph, Spiroman, and various refill sets. In 2013  the Spirograph brand was re-launched worldwide, with the original gears and wheels, by Kahootz Toys. The modern products use removable putty in place of pins to hold the stationary pieces in place. The Spirograph was Toy of the Year in 1967, and Toy of the Year finalist, in two categories, in 2014. Operation. 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. Consider a fixed outer circle formula_1 of radius formula_2 centered at the origin. A smaller inner circle formula_3 of radius formula_4 is rolling inside formula_1 and is continuously tangent to it. formula_3 will be assumed never to slip on formula_1 (in a real Spirograph, teeth on both circles  prevent such slippage). Now assume that a point formula_8 lying somewhere inside formula_3 is located a distance formula_10 from formula_3's center. This point formula_8 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 formula_8 was on the formula_14 axis. In order to find the trajectory created by a Spirograph, follow point formula_8 as the inner circle is set in motion. Now mark two points formula_16 on formula_1 and formula_18 on formula_3. The point formula_16 always indicates the location where the  two circles are tangent. Point formula_18, however, will travel on formula_3, and its initial location coincides with formula_16. After setting formula_3 in motion counterclockwise around formula_1, formula_3 has a clockwise rotation with respect to its center. The distance that point formula_18 traverses on formula_3 is the same as that traversed by the tangent point formula_16 on formula_1, due to the absence of slipping. Now define the new (relative) system of coordinates formula_31 with its origin at the center of formula_3 and its axes parallel to formula_14 and formula_34. Let the parameter formula_35 be the angle by which the tangent point  formula_16 rotates on formula_1, and formula_38 be the angle by which formula_3 rotates (i.e. by which formula_18 travels) in the relative system of coordinates. Because there is no slipping, the distances traveled by formula_18 and formula_16 along their respective circles must be the same, therefore or equivalently, 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 (formula_45) accommodates this convention. Let formula_46 be the coordinates of the center of formula_3 in the absolute system of coordinates . Then formula_48 represents the radius of the trajectory of the center of formula_3, which (again in the absolute system) undergoes circular motion thus: As defined above, formula_38 is the angle of rotation in the new relative system. Because point formula_8 obeys the usual law of circular motion, its coordinates in the new relative coordinate system formula_53 are In order to obtain the trajectory of formula_8 in the absolute (old) system of coordinates, add these two motions: where formula_57 is defined above. Now, use the relation between formula_35 and formula_38 as derived above to obtain equations describing the trajectory of point  formula_8 in terms of a single parameter formula_35: (using the fact that function formula_63 is odd). It is convenient to represent the equation above in terms of the radius formula_2 of formula_1 and dimensionless parameters describing the structure of the Spirograph. Namely, let and The parameter formula_68 represents how far the point formula_8 is located from the center of formula_3. At the same time, formula_71 represents how big the inner circle formula_3 is with respect to the outer one formula_1. It is now observed that and therefore the trajectory equations take the form Parameter formula_2 is a scaling parameter and  does not affect the structure of the Spirograph. Different values of formula_2 would yield similar Spirograph drawings. The two extreme cases formula_78 and formula_79 result in degenerate trajectories of the Spirograph. In the first extreme case, when formula_78, we have a simple circle of radius formula_2, corresponding to the case where formula_3 has been shrunk into a point. (Division by formula_78 in the formula is not a problem, since both formula_63 and formula_85 are bounded functions.) The other extreme case formula_79 corresponds to the inner circle formula_3's radius formula_88 matching the radius formula_2 of the outer circle formula_1, i.e.  formula_91. In this case the trajectory is a single point. Intuitively, formula_3 is too large to roll inside the same-sized formula_1 without slipping. If formula_94, then the point formula_8 is on the circumference of formula_3. In this case the trajectories are called hypocycloids and the equations above reduce to those for a hypocycloid.