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=a/b=4 or 4/3 into the equations for a general hypocycloid, giving parametric equations
for 0<=phi<=2pi.
The polar equation can be obtained by computing
and plugging in to r=sqrt(x^2+y^2) to obtain
for 0<=theta<=2pi.
| x^(2/3)+y^(2/3)=a^(2/3). |
(9)
|
A generalization of the curve to
| [画像: (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
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
| [画像: s_n=(8a(n-1))/n ] |
(17)
|
with n=4,
| s=6a. |
(18)
|
For a squashed astroid, the perimeter has length
| [画像: s=(4(a^2+ab+b^2))/(a+b). ] |
(19)
|
The area is given by
| [画像: A_n=((n-1)(n-2))/(n^2)pia^2 ] |
(20)
|
with n=4,
(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.
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
which can be written
The equation of the envelope is given by the simultaneous solution of
which is
Noting that
allows this to be written implicitly as
| x^(2/3)+y^(2/3)=L^(2/3), |
(31)
|
the equation of the astroid, as promised.
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
Using
| x_0=Lcostheta |
(34)
|
then gives
Solving (◇) for x_0, plugging into (◇) and squaring then gives
Rearranging produces the equation
the equation of a (quadrant of an) ellipse with semimajor and semiminor axes of lengths deltal and l+deltal.
the astroid is also the envelope of the family of ellipses
illustrated above (Wells 1991).
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, SuperellipseExplore with Wolfram|Alpha
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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." https://mathshistory.st-andrews.ac.uk/Curves/Astroid/.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. https://www.mathematicaguidebooks.org/.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London, England: 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.Referenced on Wolfram|Alpha
AstroidCite this as:
Weisstein, Eric W. "Astroid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Astroid.html