Curve studied by Rømer in 1674, Jean Bernoulli in 1691, Leibniz in 1715 and D'Alembert in 1748. Named by Littrow in 1838. "Astroid" comes from the Greek word "star" : "astron" (like the word "asteroid" designing star like objects). Other names : H4 (hypocycloid with 4 cusps), cubocycloid.

 

Parametric equation: . Cartesian equation: (Lamé's curve) or . Rational algebraic curve of degree six. Complex parametric equation: . Cartesian tangential angle: 1) or 2). Curvilinear abscissa: 1) or 2) . Radius of curvature: 1) or 2) . Cesaro equation: . Whewell equation: . Pedal equation: . Length of curve: 6a. Area enclosed: .

The astroid is a hypocycloid with four cusps (a circle having a/4 (or 3a/4) for radius running in a circle (C) having a for radius).
 

Animation of the double generation

Then, it is the envelope of a chord (PQ) of the circle having the center O and a/2 for radius (inscribed circle of the astroid), P and Q running on this circle with opposite directions, one being three times faster than the other (Cremona generation).

Above, the point n is linked to the point -3n modulo 30.

And it is also the envelope of the diameter of a circle having a/2 for radius running inside (C).

The ends of this diameter trace two perpendicular segments.
 
 

Therefore, the astroid is also the envelope of a segment [AB] with a length of a whose ends move along two perpendicular lines. The contact point M is the projection of the C vertex of the (OACB) rectangle on [AB] (and the points of the line trace an ellipse). The most general case where the lines are not perpendicular gives a tetracuspid. See also the Milk Carton which is the three dimensional generalisation of this envelope.

 

Based on this property, the edge of the mark of a two-panel sliding door of an autobus on the running board is an eighth of an astroid, followed by a circular arc.

Note: the curve determined by a string art built on two perpendicular segments is not an astroid, but an other Lamé curve with 4 parabolic segments (the [AB] segment does not have a fixed length).
 

The astroid is also the envelope of the ellipses described by , with  (two dimensionnal tracks of the ends of the [AB] segment above).

Finally, it is the evolute, by reflection, of a deltoid, with parallel incident rays of any direction(needing to understand how a symmetric curve of degree 3 can generate a symmetric curve of degree 4).
 
 

When incident rays follow every possible direction, the vertices of the astroid's envelope describe an epicycloid with 3 cusps.  There is a similar property with theTschirnhausen's cubic.

 
 

As for any cycloid curve, the evolute of an astroid is a similar astroid (in a ratio of 2):
One of the involutes is then an astroid; there are two others (see also Maltese cross):
The curve whose Cartesian equation is:  is formed by an astroid and two evolutes of rectangular hyperbolas. It can be easily defined as the envelopes of the straight lines crossing Ox and Oy in P and Q while the sum or difference of OP² and OQ² stays constant.

Astroid pedals are beetle curves ; in particular, the pedal with respect to the centre is the quadrifolium, and the polar is the rectangular cross curve.

See also tetracuspid.

© Robert FERRÉOL  2017