t R = ( t – t ) r

Consider a fixed outer circle {\displaystyle C_{o}} C_{o} of radius {\displaystyle R} R centered at the origin. A smaller inner circle {\displaystyle C_{i}} C_{i} of radius {\displaystyle r<R} r

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

Now define the new (relative) system of coordinates {\displaystyle ({\hat {X}},{\hat {Y}})} (\hat{X},\hat{Y}) with its origin at the center of {\displaystyle C_{i}} C_{i} and its axes parallel to {\displaystyle X} X and {\displaystyle Y} Y. Let the parameter {\displaystyle t} t be the angle by which the tangent point {\displaystyle T} T rotates on {\displaystyle C_{o}} C_{o} and {\displaystyle {\hat {t}}} {\hat {t}} be the angle by which {\displaystyle C_{i}} C_{i} rotates (i.e. by which {\displaystyle B} B travels) in the relative system of coordinates. Because there is no slipping, the distances traveled by {\displaystyle B} B and {\displaystyle T} T along their respective circles must be the same, therefore

} tR=(t-\hat{t})r

or equivalently

{\displaystyle {\hat {t}}=-{\frac {R-r}{r}}t.} \hat{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 ( {\displaystyle {\hat {t}}<0} \hat{t}<0) accommodates this convention.

Let {\displaystyle (x_{c},y_{c})} (x_c,y_c) be the coordinates of the center of {\displaystyle C_{i}} C_{i} in the absolute system of coordinates. Then {\displaystyle R-r} R-r represents the radius of the trajectory of the center of {\displaystyle C_{i}} C_{i}, which (again in the absolute system) undergoes circular motion thus:

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

As defined above, {\displaystyle {\hat {t}}} {\hat {t}} is the angle of rotation in the new relative system. Because point {\displaystyle A} A obeys the usual law of circular motion, its coordinates in the new relative coordinate system {\displaystyle ({\hat {x}},{\hat {y}})} (\hat{x},\hat{y}) obey:

{\displaystyle {\begin{array}{rcl}{\hat {x}}&=&\rho \cos {\hat {t}},\\{\hat {y}}&=&\rho \sin {\hat {t}}.\end{array}}} \begin{array}{rcl}
\hat{x}&=&\rho\cos \hat{t},\\
\hat{y}&=&\rho\sin \hat{t}.

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

{\displaystyle {\begin{array}{rcrcl}x&=&x_{c}+{\hat {x}}&=&(R-r)\cos t+\rho \cos {\hat {t}},\\y&=&y_{c}+{\hat {y}}&=&(R-r)\sin t+\rho \sin {\hat {t}},\\\end{array}}} \begin{array}{rcrcl}
x&=&x_c+\hat{x}&=&(R-r)\cos t+\rho\cos \hat{t},\\
y&=&y_c+\hat{y}&=&(R-r)\sin t+\rho\sin \hat{t},\\

where {\displaystyle \rho } \rho  is defined above.

Now, use the relation between {\displaystyle t} t and {\displaystyle {\hat {t}}} {\hat {t}} as derived above to obtain equations describing the trajectory of point {\displaystyle A} A in terms of a single parameter {\displaystyle t} t:

{\displaystyle {\begin{array}{rcrcl}x&=&x_{c}+{\hat {x}}&=&(R-r)\cos t+\rho \cos {\frac {R-r}{r}}t,\\[4pt]y&=&y_{c}+{\hat {y}}&=&(R-r)\sin t-\rho \sin {\frac {R-r}{r}}t.\\\end{array}}} \begin{array}{rcrcl}
x&=&x_c+\hat{x}&=&(R-r)\cos t+\rho\cos \frac{R-r}{r}t,\\[4pt]
y&=&y_c+\hat{y}&=&(R-r)\sin t-\rho\sin \frac{R-r}{r}t.\\

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

It is convenient to represent the equation above in terms of the radius {\displaystyle R} R of {\displaystyle C_{o}} C_{o} and dimensionless parameters describing the structure of the Spirograph. Namely, let

{\displaystyle l={\frac {\rho }{r}}} l=\frac{\rho}{r}


{\displaystyle k={\frac {r}{R}}.} k=\frac{r}{R}.

The parameter {\displaystyle 0\leq l\leq 1} 0\le l \le 1 represents how far the point {\displaystyle A} A is located from the center of {\displaystyle C_{i}} C_{i}. At the same time, {\displaystyle 0\leq k\leq 1} 0\le k \le 1 represents how big the inner circle {\displaystyle C_{i}} C_{i} is with respect to the outer one {\displaystyle C_{o}} C_{o}.

It is now observed that

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

and therefore the trajectory equations take the form

{\displaystyle {\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}}} \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].\\

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

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

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

If {\displaystyle l=1} l=1 then the point {\displaystyle A} A is on the circumference of {\displaystyle C_{i}} C_{i}. In this case the trajectories are called hypocycloids and the equations above reduce to those for a hypocycloid.