Astroid created with Elipses with a plus b const

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an astroid is a particular type of

roulette curve In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes. Definition Informal definition Roughly speaking, a roulette is th ...

: a hypocycloid with four cusps. Specifically, it is the locus of a point on a circle as it

rolls Roll or Rolls may refer to: Movement about the longitudinal axis * Roll angle (or roll rotation), one of the 3 angular degrees of freedom of any stiff body (for example a vehicle), describing motion about the longitudinal axis ** Roll (aviation), ...

inside a fixed circle with four times the

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

. By double generation, it is also the locus of a point on a circle as it rolls inside a fixed circle with 4/3 times the radius. It can also be defined as the

envelope An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card. Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a shor ...

of a line segment of fixed length that moves while keeping an end point on each of the axes. It is therefore the

of the moving bar in the

Trammel of Archimedes A trammel of Archimedes is a mechanism that generates the shape of an ellipse. () It consists of two shuttles which are confined ("trammeled") to perpendicular channels or rails and a rod which is attached to the shuttles by pivots at fixed posi ...

. Its modern name comes from the Greek word for "

star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

". It was proposed, originally in the form of "Astrois", by

Joseph Johann von Littrow Joseph Johann von Littrow (13 March 1781, Horšovský Týn (german: Bischofteinitz) – 30 November 1840, Vienna) was an Austrian astronomer. In 1837, he was ennobled with the title Joseph Johann Edler von Littrow. He was the father of Karl Ludwi ...

in 1838. The curve had a variety of names, including tetracuspid (still used), cubocycloid, and paracycle. It is nearly identical in form to the evolute of an ellipse.

Equations

If the radius of the fixed circle is ''a'' then the equation is given by :

x^ + y^ = a^. \,

This implies that an astroid is also a

superellipse A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape. In the ...

Parametric equation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...

s are :

& y=a\sin^3 t = ( 3\sin t - \sin 3t). \end

The

pedal equation For a plane curve ''C'' and a given fixed point ''O'', the pedal equation of the curve is a relation between ''r'' and ''p'' where ''r'' is the distance from ''O'' to a point on ''C'' and ''p'' is the perpendicular distance from ''O'' to the tangent ...

with respect to the origin is :

r^2 = a^2 - 3p^2,

the

Whewell equation The Whewell equation of a plane curve is an equation that relates the tangential angle () with arclength (), where the tangential angle is the angle between the tangent to the curve and the -axis, and the arc length is the distance along the curv ...

is :

s =  \cos 2\varphi,

and the Cesàro equation is :

R^2 + 4s^2 = \frac.

The

polar equation In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...

is :

r=\frac.

The astroid is a real locus of a plane algebraic curve of

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

zero. It has the equation :

(x^2+y^2-a^2)^3+27a^2x^2y^2=0. \,

The astroid is, therefore, a real algebraic curve of degree six.

Derivation of the polynomial equation

The polynomial equation may be derived from Leibniz's equation by elementary algebra: :

x^ + y^ = a^. \,

Cube both sides: :

x^ + 3x^y^ + 3x^y^ + y^ = a^ \,

x^2 + 3x^y^(x^ + y^) + y^2 = a^2 \,

x^2 + y^2 - a^2 = -3x^y^(x^ + y^) \,

Cube both sides again: :

(x^2 + y^2 - a^2)^3 = -27x^2y^2(x^ + y^)^3 \,

But since: :

x^ + y^ = a^ \,

It follows that :

(x^ + y^)^3 = a^2. \,

Therefore: :

(x^2 + y^2 - a^2)^3 = -27x^2y^2a^2 \,

or :

(x^2 + y^2 - a^2)^3 + 27x^2y^2a^2 = 0. \,

Metric properties

;Area enclosed :

\frac \pi a^2

;Length of curve :

6a

;Volume of the surface of revolution of the enclose area about the ''x''-axis. :

\frac\pi a^3

;Area of surface of revolution about the ''x''-axis :

\frac\pi a^2

Properties

The astroid has four cusp singularities in the real plane, the points on the star. It has two more complex cusp singularities at infinity, and four complex double points, for a total of ten singularities. The dual curve to the astroid is the

cruciform curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree of a polynomial, degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0 ...

with equation

\textstyle x^2 y^2 = x^2 + y^2.

The evolute of an astroid is an astroid twice as large. The astroid has only one tangent line in each oriented direction, making it an example of a hedgehog.

References

* * *

External links