Astroid

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

$$x = 3b\cos\phi+b\cos(3\phi)= 4b\cos^3\phi=a\cos^3\phi$$ (1)

$$y = 3b\sin\phi-b\sin(3\phi)= 4b\sin^3\phi=a\sin^3\phi.$$ (2)

In Cartesian Coordinates,

$$x^{2/3}+y^{2/3}=a^{2/3}.$$ (3)

In Pedal Coordinates with the Pedal Point at the center, the equation is

$$r^2+3p^2=a^2.$$ (4)

The Arc Length, Curvature, and Tangential Angle are

$$s(t) = {\textstyle{3\over 2}}\int_0^t \vert\sin(2t')\vert\,dt'={\textstyle{3\over 2}}\sin^2 t$$ (5)

$$\kappa(t) = -{\textstyle{2\over 3}}\csc(2t)$$ (6)

$$\phi(t) = -t.$$ (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

$$n=\left\lfloor{2t\over\pi}\right\rfloor +1,$$ (8)

where $\left\lfloor{x}\right\rfloor$ is the Floor Function, giving the formula

$$s(t)=(-1)^{1+[n{\rm\ (mod\ 2)}]} {\textstyle{3\over 2}}\sin^2 t+3\left\lfloor{{\textstyle{1\over 2}}n}\right\rfloor .$$ (9)

The overall Arc Length of the astroid can be computed from the general Hypocycloid formula

$$s_n={8a(n-1)\over n}$$ (10)

with $n=4$,

$$s_4=6a.$$ (11)

The Area is given by

$$A_n= {(n-1)(n-2)\over n^2} \pi a^2$$ (12)

with $n=4$,

$$A_4 = {\textstyle{3\over 8}} \pi a^2.$$ (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

$$x\sin p+y\cos p = {\textstyle{1\over 2}}a\sin(2p)$$ (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

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

which can be written

$$U(x,y,x_0)=y+{\sqrt{L^2-{x_0}^2}\over x_0}(x-x_0).$$ (16)

The equation of the Envelope is given by the simultaneous solution of

$$\cases{ U(x,y,x_0)=y+{\sqrt{L^2-{x_0}^2}\over x_0}(x-x_0)=0... ...tial x_0}={{x_0}^3-L^2x\over {x_0}^2\sqrt{L^2-{x_0}^2}}=0,\cr}$$ (17)

which is

$$x = {{x_0}^3\over L^2}$$ (18)

$$y = {(L^2-{x_0}^2)^{3/2}\over L^2}.$$ (19)

Noting that

$$x^{2/3} = {{x_0}^2\over L^{4/3}}$$ (20)

$$y^{2/3} = {L^2-{x_0}^2\over L^{4/3}}$$ (21)

allows this to be written implicitly as

$$x^{2/3}+y^{2/3}=L^{2/3},$$ (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

$$x = -\Delta L\cos\theta$$ (23)

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

Using

$$x_0=L\cos\theta$$ (25)

then gives

$$x = -{\Delta L\over L}x_0$$ (26)

$$y = \sqrt{L^2-{x_0}^2}\left({1+{\Delta L\over L}}\right).$$ (27)

Solving (26) for $x_0$, plugging into (27) and squaring then gives

$$y^2=L^2-{L^2x^2\over(\Delta L)^2}\left({1+{\Delta L\over L}}\right)^2.$$ (28)

Rearranging produces the equation

$${x^2\over(\Delta L)^2}+{y^2\over (L+\Delta L)^2}=1,$$ (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

$${x^2\over c^2}+{y^2\over (1-c)^2}-1=0,$$ (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.