astroid

The astroid is the hypocycloid for which the rolled circle is four times as large as the rolling circle. The curve can be written in a Whewell equation as s = cos 2φ²⁾.

The curve can also be constructed as the envelope of the lines through the two points (cos t, 0) and (0, sin t), being a line piece between the axes of equal length. A mechanical device composed from a fixed bar with endings sliding on two perpendicular tracks is called a trammel of Archimedes.
This is equivalent with a falling ladder, the astroid can also be seen as a glissette.

The astroid is also the envelope of co-axial ellipses whoes sum of major and minor axes is constant.

The length of the this unit astroid curve is 6, and its area is 3π/8.

This sextic curve ³⁾ is also called the regular star curve ⁴⁾. 'Astroid' is an old word for 'asteroid', a celestial object in an orbit around the sun, intermediate in size between a meteoroid and a planet.
The curve acquired its astroid name from a book from Littrow, published in 1836 in Vienna, replacing existing names as cubocycloid, paracycle and four-cusp-curve.
Because of its four cusps it is also called the tetracuspid, and the hypocycloid of four cusps
Abbreviation for a hypocycloid with four cusps (a=1/4) led to the name of H4.

Some relations with other curves:

• its radial, pedal and orthoptic are all the same curve: the quadrifolium
• the curve is the catacaustic of the quadrifolium
• it is the deltoid's catacaustic
• it is the evolute of the ellipse

The first to investigate the curve was Roemer (1674), during his search for gear teeth.
Those who also worked on the curve where:

• Johann Bernoulli (1691)
• Leibniz (who corresponded in 1715)
• Johann Bernoulli (1725)
• d'Alembert (1748)

The curve can be generalized into the super ellipse or Lamé curve.
Some authors call this generalization the astroid.

notes

1) In Italian: astroide.

2) φ: inclination angle of the tangent; s: arc length.

3) In Cartesian coordinates: (x² + y² - 1) + 27 x² y² = 0

4) Astrum (Lat.) = star