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).
Geometric calculus the correspondences of the differential form its derivative and its Hodge dual embed the theory of differential forms within geometric calculus. Following is a diagram summarizing the history of geometric calculus. Geometric calculus In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including differential geometry and differential forms. With a geometric algebra given, let "a" and "b" be vectors and let "F(a)" be a multivector-valued function. The directional derivative of "F(a)" along "b" is defined as provided that the limit exists, where the