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).
Cone of curves "cone of curves" of formula_1 to be where the formula_6 are irreducible, reduced, proper curves on formula_1, and formula_18 their classes in formula_13. It is not difficult to see that formula_20 is indeed a convex cone in the sense of convex geometry. One useful application of the notion of the cone of curves is the Kleiman condition, which says that a (Cartier) divisor formula_11 on a complete variety formula_1 is ample if and only if formula_23 for any nonzero element formula_24 in formula_25, the closure of the cone of curves in the usual real topology. (In general, formula_20 need not