# spirograph

the Spirograph is a geometric drawing toy that produces a variety of mathematical roulette curves. it was invented and developed by the English engineer Denys Fisher. in 1960, Fisher set up his own company, Denys Fisher Engineering, in Leeds. the company won a contract with NATO to supply springs and precision component for its 20 mm cannon. between 1962 and 1964 he developed various drawing machines, eventually producing a prototype Spirograph.

Spirograph was patented in 16 countries and was first sold in the UK in 1965. a year later, Fisher licensed Spirograph to Kenner Products in the United States. in 1967 Spirograph was chosen as the UK Toy of the Year. it consisted of different-sized plastic rings, with gear teeth on both the inside and outside of their circumferences. they were pinned to a cardboard backing with pins and any of several provided gearwheels, which had holes provided for a ballpoint pen, could be spun around to make geometric shapes on paper. later, the Super-Spirograph consisted of a set of plastic gears and other interlocking shape-segments such as rings, triangles, or straight bars.  the curves produced by the Spirograph toy are technically known as hypotrochoids and epitrochoids.

a hypotrochoid curve is traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the centre of the interior circle.

The parametric equations for a hypotrochoid are: $x (\theta) = (R - r)\cos\theta + d\cos\left({R - r \over r}\theta\right)$ $y (\theta) = (R - r)\sin\theta - d\sin\left({R - r \over r}\theta\right).$

where θ is the angle formed by the horizontal and the centre of the rotating circle.  a epitrochoid curve is traced by a point attached to a circle of radius r rolling around the outside of a fixed circle of radius R, where the point is a distance d from the center of the exterior circle.

The parametric equations for a epitrochoid are: $x (\theta) = (R + r)\cos\theta - d\cos\left({R + r \over r}\theta\right),\,$ $y (\theta) = (R + r)\sin\theta - d\sin\left({R + r \over r}\theta\right).\,$

where θ is the angle formed by the horizontal and the centre of the rotating circle.  ## 2 thoughts on “spirograph”

1. akismetuser151508432 says:

Wow great post Maria. Great to see the different spirograph shapes that can be achieved. The triangular one (top left) is surprising! Looking forward to seeing how the physics simulation and time based parametric model will allow you to go one step further than what is possible with a 2d spirograph!

1. maria.vergop says:

thank you arthur! really glad you like it, can’t wait to explore this in more dimensions!