Special Epi- and Hypo- Cycloids

Some Epicycloids and Hypocycloids are so special that we give them special names. Three of these are the Astroid, the Deltoid, and the Nephroid. Each of these can be generated in a different way, rather than as an Epi- or Hypocycloid.

The Astroid

The Astroid is a hypocycloid whose ratio of the radii is four, and as such has four points. It is so named because it appears as a star. It can also be generated in the following manner:

1. Draw a segment AB.
2. Construct its midpoint C
3. Draw the circle centered at C and passing through A and B.
4. Draw a perpendicular to AB at C
5. Draw the intersections of the perpendicular with the circle D and E.
6. Draw point F on segment AB.
7. Mark vector CB.
8. Translate F by CB and draw circle FF'.
9. Construct the intersections G and H of circle FF' with the perpendicular line.
10. Draw segments FG and FH and trace them.
11. Animate point F along segment AB.

The Deltoid

The Deltoid is a hypocycloid with three points. It can be generated by a circle rolling inside a circle of triple its radius, or of one-and-a-half times its radius (the ratio is either 3 or 1.5). It can also be generated as the envelope of the diameter of the latter rolling circle, as well as in this way:

1. Draw a circle AB.
2. Draw a point C on the circle.
3. Mark Angle CAB.
4. Mark Center A.
5. Rotate B by the marked angle.
6. Rotate B' by the marked angle.
7. Draw line CB'' and trace it.
8. Animate point C around circle AB.

The Nephroid

The Nephroid is an epicycloid with two points (i.e. the ratio is two). It can also be generated as the envelope of a circle, as follows:

1. Draw a line AB.
2. Draw the circle AB.
3. Draw a point C on circle AB.
4. Draw the perpendicular to line AB through point C, intersecting AB at point D.
5. Draw the circle with center C passing through D and trace it.
6. Animate point C around circle AB.

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