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
In Cartesian Coordinates,
![\begin{displaymath}
x^{2/3}+y^{2/3}=a^{2/3}.
\end{displaymath}](a_1809.gif) |
(3) |
In Pedal Coordinates with the Pedal Point at the center, the equation is
![\begin{displaymath}
r^2+3p^2=a^2.
\end{displaymath}](a_1810.gif) |
(4) |
The Arc Length, Curvature, and Tangential Angle are
As usual, care must be taken in the evaluation of
for
. 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
![\begin{displaymath}
n=\left\lfloor{2t\over\pi}\right\rfloor +1,
\end{displaymath}](a_1819.gif) |
(8) |
where
is the Floor Function, giving the formula
![\begin{displaymath}
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}](a_1820.gif) |
(9) |
The overall Arc Length of the astroid can be computed from the general Hypocycloid formula
![\begin{displaymath}
s_n={8a(n-1)\over n}
\end{displaymath}](a_1821.gif) |
(10) |
with
,
![\begin{displaymath}
s_4=6a.
\end{displaymath}](a_1822.gif) |
(11) |
The Area is given by
![\begin{displaymath}
A_n= {(n-1)(n-2)\over n^2} \pi a^2
\end{displaymath}](a_1823.gif) |
(12) |
with
,
![\begin{displaymath}
A_4 = {\textstyle{3\over 8}} \pi a^2.
\end{displaymath}](a_1824.gif) |
(13) |
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
![\begin{displaymath}
x\sin p+y\cos p = {\textstyle{1\over 2}}a\sin(2p)
\end{displaymath}](a_1826.gif) |
(14) |
(MacTutor Archive).
Let
cut the x-Axis and the y-Axis at
and
, respectively. Then the length
is a constant
and is equal to
.
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
![\begin{displaymath}
y-0={\sqrt{L^2-{x_0}^2}\over -x_0}(x-x_0)
\end{displaymath}](a_1830.gif) |
(15) |
which can be written
![\begin{displaymath}
U(x,y,x_0)=y+{\sqrt{L^2-{x_0}^2}\over x_0}(x-x_0).
\end{displaymath}](a_1831.gif) |
(16) |
The equation of the Envelope is given by the simultaneous solution of
![\begin{displaymath}
\cases{
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}](a_1832.gif) |
(17) |
which is
Noting that
allows this to be written implicitly as
![\begin{displaymath}
x^{2/3}+y^{2/3}=L^{2/3},
\end{displaymath}](a_1839.gif) |
(22) |
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
Using
![\begin{displaymath}
x_0=L\cos\theta
\end{displaymath}](a_1844.gif) |
(25) |
then gives
Solving (26) for
, plugging into (27) and squaring then gives
![\begin{displaymath}
y^2=L^2-{L^2x^2\over(\Delta L)^2}\left({1+{\Delta L\over L}}\right)^2.
\end{displaymath}](a_1847.gif) |
(28) |
Rearranging produces the equation
![\begin{displaymath}
{x^2\over(\Delta L)^2}+{y^2\over (L+\Delta L)^2}=1,
\end{displaymath}](a_1848.gif) |
(29) |
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
![\begin{displaymath}
{x^2\over c^2}+{y^2\over (1-c)^2}-1=0,
\end{displaymath}](a_1851.gif) |
(30) |
illustrated above.
See also Deltoid, Ellipse Envelope, Lamé Curve, Nephroid, Ranunculoid
References
Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 172-175, 1972.
Lee, X. ``Astroid.''
http://www.best.com/~xah/SpecialPlaneCurves_dir/Astroid_dir/astroid.html.
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.''
http://www-groups.dcs.st-and.ac.uk/~history/Curves/Astroid.html.
Yates, R. C. ``Astroid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 1-3, 1952.
© 1996-9 Eric W. Weisstein
1999-05-25