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\begin{figure}\BoxedEPSF{asteroid.epsf scaled 500}\end{figure}

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 $n\equiv a/b=4$ or $4/3$ into the equations for a general Hypocycloid, giving

$\displaystyle x$ $\textstyle =$ $\displaystyle 3b\cos\phi+b\cos(3\phi)= 4b\cos^3\phi=a\cos^3\phi$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle 3b\sin\phi-b\sin(3\phi)= 4b\sin^3\phi=a\sin^3\phi.$ (2)

In Cartesian Coordinates,
\end{displaymath} (3)

In Pedal Coordinates with the Pedal Point at the center, the equation is
\end{displaymath} (4)

\begin{figure}\begin{center}\BoxedEPSF{AstroidInfo.epsf scaled 750}\end{center}\end{figure}

The Arc Length, Curvature, and Tangential Angle are

$\displaystyle s(t)$ $\textstyle =$ $\displaystyle {\textstyle{3\over 2}}\int_0^t \vert\sin(2t')\vert\,dt'={\textstyle{3\over 2}}\sin^2 t$ (5)
$\displaystyle \kappa(t)$ $\textstyle =$ $\displaystyle -{\textstyle{2\over 3}}\csc(2t)$ (6)
$\displaystyle \phi(t)$ $\textstyle =$ $\displaystyle -t.$ (7)

As usual, care must be taken in the evaluation of $s(t)$ for $t>\pi/2$. Since (5) comes from an integral involving the Absolute Value of a function, it must be monotonic increasing. Each Quadrant can be treated correctly by defining
n=\left\lfloor{2t\over\pi}\right\rfloor +1,
\end{displaymath} (8)

where $\left\lfloor{x}\right\rfloor $ is the Floor Function, giving the formula
s(t)=(-1)^{1+[n{\rm\ (mod\ 2)}]} {\textstyle{3\over 2}}\sin^2 t+3\left\lfloor{{\textstyle{1\over 2}}n}\right\rfloor .
\end{displaymath} (9)

The overall Arc Length of the astroid can be computed from the general Hypocycloid formula
s_n={8a(n-1)\over n}
\end{displaymath} (10)

with $n=4$,
\end{displaymath} (11)

The Area is given by
A_n= {(n-1)(n-2)\over n^2} \pi a^2
\end{displaymath} (12)

with $n=4$,
A_4 = {\textstyle{3\over 8}} \pi a^2.
\end{displaymath} (13)

The Evolute of an Ellipse is a stretched Hypocycloid. The gradient of the Tangent $T$ from the point with parameter $p$ is $-\tan p$. The equation of this Tangent $T$ is
x\sin p+y\cos p = {\textstyle{1\over 2}}a\sin(2p)
\end{displaymath} (14)

(MacTutor Archive). Let $T$ cut the x-Axis and the y-Axis at $X$ and $Y$, respectively. Then the length $XY$ is a constant and is equal to $a$.

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 $L$, the points of contact with the wall and floor are $(x_0, 0)$ and $(0, \sqrt{L^2-{x_0}^2}\,)$, respectively. The equation of the Line made by the ladder with its foot at $(x_0, 0)$ is therefore

y-0={\sqrt{L^2-{x_0}^2}\over -x_0}(x-x_0)
\end{displaymath} (15)

which can be written
U(x,y,x_0)=y+{\sqrt{L^2-{x_0}^2}\over x_0}(x-x_0).
\end{displaymath} (16)

The equation of the Envelope is given by the simultaneous solution of
U(x,y,x_0)=y+{\sqrt{L^2-{x_0}^2}\over x_0}(x-x_0)=0...
...tial x_0}={{x_0}^3-L^2x\over {x_0}^2\sqrt{L^2-{x_0}^2}}=0,\cr}
\end{displaymath} (17)

which is
$\displaystyle x$ $\textstyle =$ $\displaystyle {{x_0}^3\over L^2}$ (18)
$\displaystyle y$ $\textstyle =$ $\displaystyle {(L^2-{x_0}^2)^{3/2}\over L^2}.$ (19)

Noting that
$\displaystyle x^{2/3}$ $\textstyle =$ $\displaystyle {{x_0}^2\over L^{4/3}}$ (20)
$\displaystyle y^{2/3}$ $\textstyle =$ $\displaystyle {L^2-{x_0}^2\over L^{4/3}}$ (21)

allows this to be written implicitly as
\end{displaymath} (22)

the equation of the astroid, as promised.


The related problem obtained by having the ``garage door'' of length $L$ with an ``extension'' of length $\Delta L$ 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 $x_0$ and Angle $\theta$ is given by

$\displaystyle x$ $\textstyle =$ $\displaystyle -\Delta L\cos\theta$ (23)
$\displaystyle y$ $\textstyle =$ $\displaystyle \sqrt{L^2-{x_0}^2}+\Delta L\sin\theta.$ (24)

\end{displaymath} (25)

then gives
$\displaystyle x$ $\textstyle =$ $\displaystyle -{\Delta L\over L}x_0$ (26)
$\displaystyle y$ $\textstyle =$ $\displaystyle \sqrt{L^2-{x_0}^2}\left({1+{\Delta L\over L}}\right).$ (27)

Solving (26) for $x_0$, plugging into (27) and squaring then gives
y^2=L^2-{L^2x^2\over(\Delta L)^2}\left({1+{\Delta L\over L}}\right)^2.
\end{displaymath} (28)

Rearranging produces the equation
{x^2\over(\Delta L)^2}+{y^2\over (L+\Delta L)^2}=1,
\end{displaymath} (29)

the equation of a (Quadrant of an) Ellipse with Semimajor and Semiminor Axes of lengths $\Delta L$ and $L+\Delta L$.


The astroid is also the Envelope of the family of Ellipses

{x^2\over c^2}+{y^2\over (1-c)^2}-1=0,
\end{displaymath} (30)

illustrated above.

See also Deltoid, Ellipse Envelope, Lamé Curve, Nephroid, Ranunculoid


Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 172-175, 1972.

Lee, X. ``Astroid.''

Lockwood, E. H. ``The Astroid.'' Ch. 6 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 52-61, 1967.

MacTutor History of Mathematics Archive. ``Astroid.''

Yates, R. C. ``Astroid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 1-3, 1952.

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© 1996-9 Eric W. Weisstein