WOLFRAM|DEMONSTRATIONS PROJECT

Epicycloids from an Envelope of Lines

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lines
32
cusps
1
2
3
4
5
6
draw epicycloid
background
white
black
This Demonstration lets you see the construction of an epicycloid with straight lines. Start with the diameter of a unit circle and offset one end by a series of steps of equal arc length along the circumference. At the other end, offset the points by steps of equal arc lengths
n+1
times as large and connect these pairs of points with straight line segments. With enough lines drawn, you can see an epicycloid with
n
cusps ("pinches", at which the curve cannot be differentiated).​