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
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 (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:
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
In order to obtain the trajectory of A in the absolute (old) system of coordinates, add these two motions:
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:
(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
and
The parameter represents how far the point A is located from the center of C_i. At the same time, represents how big the inner circle C_i is with respect to the outer one C_o.
It is now observed that
and therefore the trajectory equations take the form
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.
