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 formulasVariables
| 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) |
| θ0 | 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) |