Namespaces
Variants
Actions

Difference between revisions of "Astroid"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX; I suppose the X should be an x)
Line 1: Line 1:
A plane algebraic curve of order six, described by a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135401.png" /> on a circle of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135402.png" /> rolling on the inside of a circle of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135403.png" />; a [[Hypocycloid|hypocycloid]] with module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135404.png" />. Its equation in orthogonal Cartesian coordinates is
+
{{TEX|done}}
 +
A plane algebraic curve of order six, described by a point $M$ on a circle of radius $r$ rolling on the inside of a circle of radius $R=4r$; a [[Hypocycloid|hypocycloid]] with module $m=4$. Its equation in orthogonal Cartesian coordinates is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135405.png" /></td> </tr></table>
+
$$x^{2/3}+y^{2/3}=R^{2/3};$$
  
 
and a parametric representation is
 
and a parametric representation is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135406.png" /></td> </tr></table>
+
$$x=R\cos^3\frac t4,\quad y=R\sin^3\frac t4.$$
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/a013540a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/a013540a.gif" />
Line 11: Line 12:
 
Figure: a013540a
 
Figure: a013540a
  
There are four cusps (see Fig.). The length of the arc from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135407.png" /> is
+
There are four cusps (see Fig.). The length of the arc from the point $A$ is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135408.png" /></td> </tr></table>
+
$$l=\frac32R\sin^2\frac t4.$$
  
The length of the entire curve is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a0135409.png" />. The radius of curvature is
+
The length of the entire curve is $6R$. The radius of curvature is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a01354010.png" /></td> </tr></table>
+
$$r_k=\frac32R\sin\frac t2.$$
  
 
The area bounded by the curve is
 
The area bounded by the curve is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013540/a01354011.png" /></td> </tr></table>
+
$$S=\frac38\pi R^2.$$
  
 
The astroid is the envelope of a family of segments of constant length, the ends of which are located on two mutually perpendicular straight lines. This property of the astroid is connected with one of its generalizations — the so-called oblique astroid, which is the envelope of the segments of constant length with their ends located on two straight lines intersecting at an arbitrary angle.
 
The astroid is the envelope of a family of segments of constant length, the ends of which are located on two mutually perpendicular straight lines. This property of the astroid is connected with one of its generalizations — the so-called oblique astroid, which is the envelope of the segments of constant length with their ends located on two straight lines intersecting at an arbitrary angle.

Revision as of 22:32, 11 April 2014

A plane algebraic curve of order six, described by a point $M$ on a circle of radius $r$ rolling on the inside of a circle of radius $R=4r$; a hypocycloid with module $m=4$. Its equation in orthogonal Cartesian coordinates is

$$x^{2/3}+y^{2/3}=R^{2/3};$$

and a parametric representation is

$$x=R\cos^3\frac t4,\quad y=R\sin^3\frac t4.$$

Figure: a013540a

There are four cusps (see Fig.). The length of the arc from the point $A$ is

$$l=\frac32R\sin^2\frac t4.$$

The length of the entire curve is $6R$. The radius of curvature is

$$r_k=\frac32R\sin\frac t2.$$

The area bounded by the curve is

$$S=\frac38\pi R^2.$$

The astroid is the envelope of a family of segments of constant length, the ends of which are located on two mutually perpendicular straight lines. This property of the astroid is connected with one of its generalizations — the so-called oblique astroid, which is the envelope of the segments of constant length with their ends located on two straight lines intersecting at an arbitrary angle.

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)


Comments

References

[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
[a2] E.A. Lockwood, "A book of curves" , Cambridge Univ. Press (1961)
How to Cite This Entry:
Astroid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Astroid&oldid=13095
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article