Astroid
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A 4-cusped hypocycloid which is sometimes also called a tetracuspid, cubocycloid, or paracycle. The parametric
equations of the astroid can be obtained by plugging in 
or 
into the equations for a general hypocycloid,
giving parametric equations
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(1)
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(2)
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(3)
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(4)
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(5)
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(6)
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for 
.
The polar equation can be obtained by computing
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(7)
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and plugging in to 
to obtain
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(8)
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for 
.
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(9)
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A generalization of the curve to
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(10)
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gives "squashed" astroids, which are a special case of the superellipse corresponding to parameter 
.
In pedal coordinates with the pedal point at the center, the equation is
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(11)
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and the Cesàro equation is
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(12)
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A further generalization to an equation of the form
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(13)
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is known as a superellipse.
The arc length, curvature, and tangential angle are
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(14)
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(15)
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(16)
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where the formula for 
holds for 
.
The perimeter of the entire astroid can be computed from the general hypocycloid formula
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(17)
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with 
,
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(18)
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For a squashed astroid, the perimeter has length
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(19)
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The area is given by
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(20)
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with 
,
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(21)
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(22)
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(OEIS A093828).
The evolute of an ellipse is a stretched hypocycloid. The gradient of the tangent 
from the point
with parameter 
is 
. The equation
of this tangent 
is
 |
(23)
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(MacTutor Archive). Let 
cut the x-axis
and the y-axis at 
and 
, respectively.
Then the length 
is a constant and is equal to 
.
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The astroid can also be formed as the envelope produced when a line segment is moved with each end on one
of a pair of perpendicular axes (e.g., it is the
curve enveloped by a ladder sliding against a wall or a garage door with the top
corner moving along a vertical track; left figure above). The astroid is therefore
a glissette. To see this, note that for a ladder of
length 
, the points of contact with the wall and floor
are 
and 
,
respectively. The equation of the line made by the ladder
with its foot at 
is therefore
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(24)
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which can be written
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(25)
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The equation of the envelope is given by the simultaneous solution of
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(26)
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which is
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(27)
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(28)
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Noting that
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(29)
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(30)
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allows this to be written implicitly as
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(31)
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the equation of the astroid, as promised.
The related problem obtained by having the "garage door" of length 
with an "extension" of length 
move up and
down a slotted track also gives a surprising answer. In this case, the position of
the "extended" end for the foot of the door at horizontal position 
and angle 
is given by
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(32)
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(33)
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Using
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(34)
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then gives
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(35)
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(36)
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Solving (◇) for 
, plugging into (◇) and squaring
then gives
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(37)
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Rearranging produces the equation
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(38)
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the equation of a (quadrant of an) ellipse with semimajor and semiminor
axes of lengths 
and 
.
the astroid is also the envelope of the family of ellipses
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(39)
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illustrated above (Wells 1991).
An attractive arrangement of astroids can be constructed as a set of tangents to circular arcs (Trott 2004, pp. 18-19).
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astroid
