by and 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 () accommodates this convention. Let be the coordinates of the center of in the absolute system of coordinates. Then represents the radius of the trajectory of the center of , which (again in the absolute system) undergoes circular motion thus: As defined above, is the angle of rotation in the new relative system. Because point obeys the usual law of circular motion, its coordinates in the new relative coordinate system are In order to obtain the trajectory of in the absolute (old) system of coordinates, add these two motions: where is defined above. Now, use the relation between and as derived above to obtain equations describing the trajectory of point in terms of a single parameter : (using the fact that function is odd). It is convenient to represent the equation above in terms of the radius of and dimensionless parameters describing the structure of the Spirograph. Namely, let and The parameter represents how far the point is located from the center of . At the same time, represents how big the inner circle is with respect to the outer one . It is now observed that and therefore the trajectory equations take the form Parameter is a scaling parameter and does not affect the structure of the Spirograph. Different values of would yield similar Spirograph drawings. The two extreme cases and result in degenerate trajectories of the Spirograph. In the first extreme case, when , we have a simple circle of radius , corresponding to the case where has been shrunk into a point. (Division by in the formula is not a problem, since both and are bounded functions.) The other extreme case corresponds to the inner circle 's radius matching the radius of the outer circle , i.e. . In this case the trajectory is a single point. Intuitively, is too large to roll inside the same-sized without slipping. If , then the point is on the circumference of . In this case the trajectories are called hypocycloids and the equations above reduce to those for a hypocycloid. See also Cardioid Apsidal precession Cyclograph Geometric lathe Guilloché Harmonograph Hypotrochoid Lissajous curve List