Spirograph (X-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

xX-coordinate of the pen-hole point (m)
RThe radius of the fixed outer circle (m)
rRadius of the circle rolling around the inside of the fixed circle (m)
θ0The angle by which the tangent point rotates on the fixed circle (radians)
ρThe distance of the pen-hole point from small circle's center (m)