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).
Fisher's geometric model to obtain a sharp image has little chance of obtaining an optimally functioning image by randomly turning the knobs. The chances of a clear image are not so bad if the number of knobs is low, but the chances will decrease dramatically if the number of adjustable parameters (knobs) is larger than two or three. Fisher introduced a geometric metaphor, which eventually became known as Fisher's geometric model. In his model, Fisher argues that the functioning of the microscope is analogous to the fitness of an organism. The performance of the microscope depends on the state of various knobs that