Editing Spirograph (section)

==Mathematical basis==
[[File:resonance Cascade.svg|thumb|300px|right|Geometric construction for mathematical explanation of spirograph.]]
Consider a fixed outer circle <math>C_o</math> of radius <math>R</math> centered at the origin. A smaller inner circle <math>C_i</math> of radius <math>r < R</math> is rolling inside <math>C_o</math> and is continuously tangent to it. <math>C_i</math> will be assumed never to slip on <math>C_o</math> (in a real Spirograph, teeth on both circles prevent such slippage). Now assume that a point <math>A</math> lying somewhere inside <math>C_i</math> is located a distance <math>\rho<r</math> from <math>C_i</math>'s center. This point <math>A</math> 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 <math>A</math> was on the <math>X</math> axis. In order to find the trajectory created by a Spirograph, follow point <math>A</math> as the inner circle is set in motion.

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

Now define the new (relative) system of coordinates <math>(X', Y')</math> with its origin at the center of <math>C_i</math> and its axes parallel to <math>X</math> and <math>Y</math>. Let the parameter <math>t</math> be the angle by which the tangent point <math>T</math> rotates on <math>C_o</math>, and <math>t'</math> be the angle by which <math>C_i</math> rotates (i.e. by which <math>B</math> travels) in the relative system of coordinates. Because there is no slipping, the distances traveled by <math>B</math> and <math>T</math> along their respective circles must be the same, therefore

: <math>tR = (t - t')r,</math>

or equivalently,

: <math>t' = -\frac{R - r}{r} t.</math>

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 (<math>t' < 0</math>) accommodates this convention.

Let <math>(x_c, y_c)</math> be the coordinates of the center of <math>C_i</math> in the absolute system of coordinates. Then <math>R - r</math> represents the radius of the trajectory of the center of <math>C_i</math>, which (again in the absolute system) undergoes circular motion thus:

: <math>\begin{align}
 x_c &= (R - r)\cos t,\\
 y_c &= (R - r)\sin t.
\end{align}</math>

As defined above, <math>t'</math> is the angle of rotation in the new relative system. Because point <math>A</math> obeys the usual law of circular motion, its coordinates in the new relative coordinate system <math>(x', y')</math> are

: <math>\begin{align}
 x' &= \rho\cos t',\\
 y' &= \rho\sin t'.
\end{align}</math>

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

: <math>\begin{align}
 x &= x_c + x' = (R - r)\cos t + \rho\cos t',\\
 y &= y_c + y' = (R - r)\sin t + \rho\sin t',\\
\end{align}</math>

where <math>\rho</math> is defined above.

Now, use the relation between <math>t</math> and <math>t'</math> as derived above to obtain equations describing the trajectory of point <math>A</math> in terms of a single parameter <math>t</math>:

: <math>\begin{align}
 x &= x_c + x' = (R - r)\cos t + \rho\cos \frac{R - r}{r}t,\\
 y &= y_c + y' = (R - r)\sin t - \rho\sin \frac{R - r}{r}t\\
\end{align}</math>

(using the fact that function <math>\sin</math> is [[Even and odd functions|odd]]).

It is convenient to represent the equation above in terms of the radius <math>R</math> of <math>C_o</math> and dimensionless
parameters describing the structure of the Spirograph. Namely, let

: <math>l = \frac{\rho}{r}</math>

and

: <math>k = \frac{r}{R}.</math>

The parameter <math>0 \le l \le 1</math> represents how far the point <math>A</math> is located from the center of <math>C_i</math>. At the same time, <math>0 \le k \le 1</math> represents how big the inner circle <math>C_i</math> is with respect to the outer one <math>C_o</math>.

It is now observed that

: <math>\frac{\rho}{R} = lk,</math>

and therefore the trajectory equations take the form

: <math>\begin{align}
 x(t) &= R\left[(1 - k)\cos t + lk\cos \frac{1 - k}{k}t\right],\\
 y(t) &= R\left[(1 - k)\sin t - lk\sin \frac{1 - k}{k}t\right].\\
\end{align}</math>

Parameter <math>R</math> is a scaling parameter and does not affect the structure of the Spirograph. Different values of <math>R</math> would yield [[similarity (geometry)|similar]] Spirograph drawings.

The two extreme cases <math>k = 0</math> and <math>k = 1</math> result in degenerate trajectories of the Spirograph. In the first extreme case, when <math>k = 0</math>, we have a simple circle of radius <math>R</math>, corresponding to the case where <math>C_i</math> has been shrunk into a point. (Division by <math>k = 0</math> in the formula is not a problem, since both <math>\sin</math> and <math>\cos</math> are bounded functions.)

The other extreme case <math>k = 1</math> corresponds to the inner circle <math>C_i</math>'s radius <math>r</math> matching the radius <math>R</math> of the outer circle <math>C_o</math>, i.e. <math>r = R</math>. In this case the trajectory is a single point. Intuitively, <math>C_i</math> is too large to roll inside the same-sized <math>C_o</math> without slipping.

If <math>l = 1</math>, then the point <math>A</math> is on the circumference of <math>C_i</math>. In this case the trajectories are called [[hypocycloid]]s and the equations above reduce to those for a hypocycloid.