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).
Twists of curves its quadratic twist, and only the other two are really new. Also in this case, twisted curves are isomorphic over the field extension given by the twist degree. Analogously to the quartic twist case, an elliptic curve over formula_10 with j-invariant equal to zero can be twisted by cubic characters. The curves obtained are isomorphic to the starting curve over the field extension given by the twist degree. Twists of curves In the mathematical field of algebraic geometry, an elliptic curve E over a field K has an associated quadratic twist, that is another elliptic curve which is isomorphic to