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 space curves are only distinguished by the way in which they bend and twist. Quantitatively, this is measured by the differential-geometric invariants called the "curvature" and the "torsion" of a curve. The fundamental theorem of curves asserts that the knowledge of these invariants completely determines the curve. Let (i) formula_1, (ii) formula_2, and (iii) formula_3 be a non-empty interval of real numbers. Then a vector-valued function of class formula_5 (i.e., the component-functions of formula_6 are formula_7-times continuously differentiable) is called a parametric formula_5-curve or a formula_5-parametrization. Note that formula_10 is called the image of the parametric curve. It is important