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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