Spirograph Spirograph Spirograph is a geometric drawing toy that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. It was developed by British engineer Denys Fisher and first sold in 1965. The name has been a registered trademark of Hasbro Inc. since 1998 following purchase of the company that had acquired the Denys Fisher company. The Spirograph brand was relaunched worldwide with original product configurations in 2013 by Kahootz Toys. In 1827 English architect and engineer Peter Hubert Desvignes developed his "Speiragraph", a machine to create elaborate spiral drawings, intended to prevent bank note forgeries. The mathematician
Differential geometry of curves for any formula_39. Given the image of a parametric curve, one can define several different parametrizations of the parametric curve. Differential geometry aims to describe the properties of parametric curves that are invariant under certain re-parametrizations. We thus have to define a suitable equivalence relation on the set of all parametric curves. The differential-geometric properties of a parametric curve (e.g., its length, its Frenet frame and its generalized curvature) are invariant under re-parametrization and therefore properties of the equivalence class itself. The equivalence classes are called formula_5-curves and are central objects studied in the differential geometry of curves. Two parametric