# Spirograph (Y-coordinate of the traiectory of the pen-hole in the inner disk of a Spirograph)

## Description

Spirograph is a geometric drawing toy that produces mathematical roulette curves as hypotrochoids and epitrochoids. A fixed outer circle of radius R is centered at the origin and a smaller inner circle of radius r<R is rolling inside and is continuously tangent to it. A point A lying somewhere inside the rolling circle is located at a distance <r from small circle’s center. This point A corresponds to the pen-hole in the inner disk of a real Spirograph. In order to find the trajectory created by a Spirograph, we follow point A as the inner circle is set in motion. ( We assume that a counterclockwise motion corresponds to a positive change of angle and a clockwise one to a negative change of angle).

Related formulas## Variables

y | Y-coordinate of the pen-hole point (m) |

R | The radius of the fixed outer circle (m) |

r | Radius of the circle rolling around the inside of the fixed circle (m) |

θ_{0} | The angle by which the tangent point rotates on the fixed circle (m) |

ρ | The distance of the pen-hole point from small circle's center (m) |