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Astroid

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Astroid
AstroidFrames

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=a/b=4 or 4/3 into the equations for a general hypocycloid, giving parametric equations

x=3bcost+bcos(3t)
(1)
=4bcos^3t
(2)
=acos^3t
(3)
y=3bsint-bsin(3t)
(4)
=4bsin^3t
(5)
=asin^3t
(6)

for 0<=phi<=2pi.

The polar equation can be obtained by computing

theta=tan^(-1)(y/x)=tan^(-1)(tan^3t),
(7)

and plugging in to r=sqrt(x^2+y^2) to obtain

r=(|sectheta|)/((1+tan^(2/3)theta)^(3/2))
(8)

for 0<=theta<=2pi.

AstroidSquashed

In Cartesian coordinates,

x^(2/3)+y^(2/3)=a^(2/3).
(9)

A generalization of the curve to

(x/a)^(2/3)+(y/b)^(2/3)=1
(10)

gives "squashed" astroids, which are a special case of the superellipse corresponding to parameter r=2/3.

In pedal coordinates with the pedal point at the center, the equation is

r^2+3p^2=a^2,
(11)

and the Cesàro equation is

rho^2+4s^2=6as.
(12)

A further generalization to an equation of the form

|x/a|^r+|y/b|^r=1,
(13)

is known as a superellipse.

The arc length, curvature, and tangential angle are

s(t)=3/2sin^2t
(14)
kappa(t)=-2/3|csc(2t)|
(15)
phi(t)=-t,
(16)

where the formula for s(t) holds for 0<t<pi/2.

The perimeter of the entire astroid can be computed from the general hypocycloid formula

s_n=(8a(n-1))/n
(17)

with n=4,

s=6a.
(18)

For a squashed astroid, the perimeter has length

s=(4(a^2+ab+b^2))/(a+b).
(19)
AstroidArea

The area is given by

A_n=((n-1)(n-2))/(n^2)pia^2
(20)

with n=4,

A=3/8pia^2
(21)
approx 1.178097a^2
(22)

(OEIS A093828).

The evolute of an ellipse is a stretched hypocycloid. The gradient of the tangent T from the point with parameter p is -tanp. The equation of this tangent T is

xsinp+ycosp=1/2asin(2p)
(23)

(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.

AstroidLadders
AstroidLines

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))/(-x_0)(x-x_0),
(24)

which can be written

U(x,y,x_0)=y+(sqrt(L^2-x_0^2))/(x_0)(x-x_0).
(25)

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

{U(x,y,x_0)=y+(sqrt(L^2-x_0^2))/(x_0)(x-x_0)=0; (partialU)/(partialx_0)=(x_0^3-L^2x)/(x_0^2sqrt(L^2-x_0^2))=0,
(26)

which is

x=(x_0^3)/(L^2)
(27)
y=((L^2-x_0^2)^(3/2))/(L^2).
(28)

Noting that

x^(2/3)=(x_0^2)/(L^(4/3))
(29)
y^(2/3)=(L^2-x_0^2)/(L^(4/3))
(30)

allows this to be written implicitly as

x^(2/3)+y^(2/3)=L^(2/3),
(31)

the equation of the astroid, as promised.

AstroidLaddersExtended

The related problem obtained by having the "garage door" of length L with an "extension" of length DeltaL 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=-DeltaLcostheta
(32)
y=sqrt(L^2-x_0^2)+DeltaLsintheta.
(33)

Using

x_0=Lcostheta
(34)

then gives

x=-(DeltaL)/Lx_0
(35)
y=sqrt(L^2-x_0^2)(1+(DeltaL)/L).
(36)

Solving (◇) for x_0, plugging into (◇) and squaring then gives

y^2=L^2-(L^2x^2)/((DeltaL)^2)(1+(DeltaL)/L)^2.
(37)

Rearranging produces the equation

(x^2)/((DeltaL)^2)+(y^2)/((L+DeltaL)^2)=1,
(38)

the equation of a (quadrant of an) ellipse with semimajor and semiminor axes of lengths deltal and l+deltal.

AstroidEllipses

the astroid is also the envelope of the family of ellipses

(x^2)/(c^2)+(y^2)/((1-c)^2)-1=0,
(39)

illustrated above (Wells 1991).

AstroidByTangents

An attractive arrangement of astroids can be constructed as a set of tangents to circular arcs (Trott 2004, pp. 18-19).

See also

Astroidal Ellipsoid, Deltoid, Ellipse Envelope, Hyperbolic Octahedron, Lamé Curve, Nephroid, Ranunculoid, Superellipse

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 219, 1987.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 172-175, 1972.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.Sloane, N. J. A. Sequence A093828 in "The On-Line Encyclopedia of Integer Sequences."Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 146-147, 1999.Trott, M. Graphica 1: The World of Mathematica Graphics. The Imaginary Made Real: The Images of Michael Trott. Champaign, IL: Wolfram Media, pp. 11 and 83, 1999.Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, p. 19, 2004. http://www.mathematicaguidebooks.org/.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 10-11, 1991.Yates, R. C. "Astroid." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 1-3, 1952.

Cite this as:

Weisstein, Eric W. "Astroid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Astroid.html

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