Hypocycloid prove that an epicycloid with "k" cusps moves snugly inside one with "k"+1 cusps. The evolute of a hypocycloid is an enlarged version of the hypocycloid itself, while the involute of a hypocycloid is a reduced copy of itself. The pedal of a hypocycloid with pole at the center of the hypocycloid is a rose curve. The isoptic of a hypocycloid is a hypocycloid. Curves similar to hypocyloids can be drawn with the Spirograph toy. Specifically, the Spirograph can draw hypotrochoids and epitrochoids. The Pittsburgh Steelers' logo, which is based on the Steelmark, includes three astroids (hypocycloids of four cusps).
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