Epicicloidali

Epicicloidali refers to a family of plane curves generated by the motion of a circle rolling around the outside of another circle. The curves produced in this way are epicycloids, and the adjective epicicloidal describes properties or shapes related to these curves. They have been studied in classical geometry and appear in various applications in design and mechanics.

Mathematically, consider a circle of radius r rolling externally around a fixed circle of radius R. A point on the circumference of the rolling circle traces the epicycloid with parametric equations:

x(t) = (R + r) cos t − r cos(((R + r) / r) t)

y(t) = (R + r) sin t − r sin(((R + r) / r) t)

The shape of the curve depends on the ratio R/r. If the ratio is rational, the tracing is periodic and the curve closes after a finite number of revolutions; if the ratio is irrational, the path does not close and can densely fill a region.

Notable special cases include the cardioid, which occurs when R = r, a heart-shaped curve with a single cusp. Another example is the nephroid, obtained when R = 2r, which has two cusps. The family thus spans a range from simple to highly intricate outlines as the radii vary.

In addition to pure mathematics, epicicloidal forms appear in gear design, decorative patterns, and computer graphics. The term derives from Greek roots meaning “upon” and “circle,” reflecting the construction process of rolling one circle around another.