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ASTROID
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 : H_{4} (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 twopanel 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.
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© Robert FERRÉOL 2017