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A plane curve for which the distances <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313402.png" /> between any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313403.png" /> of the curve and two fixed points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313405.png" /> (the foci) are related by the non-homogeneous linear equation
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A plane curve for which the distances $r_1$ and $r_2$ between any point $P$ of the curve and two fixed points $F_1$ and $F_2$ (the foci) are related by the non-homogeneous linear equation
  
<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/d/d031/d031340/d0313406.png" /></td> </tr></table>
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$$r_1+mr_2=a.$$
  
 
A Descartes oval may be defined by means of the homogeneous linear equation
 
A Descartes oval may be defined by means of the homogeneous linear equation
  
<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/d/d031/d031340/d0313407.png" /></td> </tr></table>
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$$r_1+mr_2+nr_3=0,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313408.png" /> is the distance to the third focus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d0313409.png" /> located on the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134010.png" />. In the general case, a Descartes oval consists of two closed curves, one enclosing the other (see Fig.). The equation of a Descartes oval in Cartesian coordinates is
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where $r_3$ is the distance to the third focus $F_3$ located on the straight line $F_1F_2$. In the general case, a Descartes oval consists of two closed curves, one enclosing the other (see Fig.). The equation of a Descartes oval in 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/d/d031/d031340/d03134011.png" /></td> </tr></table>
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$$\sqrt{x^2+y^2}+m\sqrt{(x-d)^2+y^2}=a,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134012.png" /> is the length of the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134015.png" />, the Descartes oval is an ellipse; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134017.png" />, it is a hyperbola; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134018.png" />, it is a [[Pascal limaçon|Pascal limaçon]]. First studied by R. Descartes in the context of problems of optics [[#References|[1]]].
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where $d$ is the length of the segment $F_1F_2$. If $m=1$ and $a>d$, the Descartes oval is an ellipse; if $m=-1$ and $a<d$, it is a hyperbola; if $m=a/d$, it is a [[Pascal limaçon|Pascal limaçon]]. First studied by R. Descartes in the context of problems of optics [[#References|[1]]].
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031340a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d031340a.gif" />
  
 
Figure: d031340a
 
Figure: d031340a
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Descartes,  "Géométrie" , Leyden  (1637)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>
 
 
 
  
 
====Comments====
 
====Comments====
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The [[Caustic|caustic]] of a circle with respect to a point light source in the plane of the circle is the [[Evolute|evolute]] of a Descartes oval.
 
The [[Caustic|caustic]] of a circle with respect to a point light source in the plane of the circle is the [[Evolute|evolute]] of a Descartes oval.
  
There are several ways in which a Cartesian oval can arise. One is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134020.png" /> be two cones with axes parallel to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134021.png" />-axis and with circular intersections with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134022.png" />-plane. Then the vertical projection onto the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134023.png" />-plane of their intersection is a Cartesian oval. This is a result of A. Quetelet. Suppose the apexes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134025.png" /> are at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134027.png" />, respectively, and the intersections with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134028.png" />-plane are circles of radius 1 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134029.png" /> (see below). Then in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134031.png" /> the projection of the intersection is a [[Cardioid|cardioid]].
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There are several ways in which a Cartesian oval can arise. One is as follows. Let $C$ and $C_1$ be two cones with axes parallel to the $z$-axis and with circular intersections with the $xy$-plane. Then the vertical projection onto the $xy$-plane of their intersection is a Cartesian oval. This is a result of A. Quetelet. Suppose the apexes of $C$ and $C_1$ are at $(0,0,1)$ and $(c,0,a)$, respectively, and the intersections with the $xy$-plane are circles of radius 1 and $b$ (see below). Then in the case $b=(1-a)^{-1}a(a+1)$, $c=-(1-a)^{-1}(a+1)^2$ the projection of the intersection is a [[Cardioid|cardioid]].
  
In [[Polar coordinates|polar coordinates]] the equation for a Cartesian oval is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134032.png" />. Hence when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134033.png" /> the equation becomes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134034.png" /> (plus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134035.png" />) and one obtains the [[Pascal limaçon|Pascal limaçon]]; when in addition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134036.png" />, the [[Cardioid|cardioid]] results, whose equation can also be written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031340/d03134037.png" />.
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In [[Polar coordinates|polar coordinates]] the equation for a Cartesian oval is $r^2-2(a+b\cos\phi)r+c^2=0$. Hence when $c=0$ the equation becomes $r=a+b\cos\phi$ (plus $r=0$) and one obtains the [[Pascal limaçon|Pascal limaçon]]; when in addition $a=b$, the [[Cardioid|cardioid]] results, whose equation can also be written as $\sqrt r=\sqrt m\cos(\phi/2)$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Fladt,  "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell.  (1962)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Descartes,  "Géométrie" , Leyden  (1637)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Fladt,  "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell.  (1962)</TD></TR></table>
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{{OldImage}}

Latest revision as of 11:37, 26 March 2023

A plane curve for which the distances $r_1$ and $r_2$ between any point $P$ of the curve and two fixed points $F_1$ and $F_2$ (the foci) are related by the non-homogeneous linear equation

$$r_1+mr_2=a.$$

A Descartes oval may be defined by means of the homogeneous linear equation

$$r_1+mr_2+nr_3=0,$$

where $r_3$ is the distance to the third focus $F_3$ located on the straight line $F_1F_2$. In the general case, a Descartes oval consists of two closed curves, one enclosing the other (see Fig.). The equation of a Descartes oval in Cartesian coordinates is

$$\sqrt{x^2+y^2}+m\sqrt{(x-d)^2+y^2}=a,$$

where $d$ is the length of the segment $F_1F_2$. If $m=1$ and $a>d$, the Descartes oval is an ellipse; if $m=-1$ and $a<d$, it is a hyperbola; if $m=a/d$, it is a Pascal limaçon. First studied by R. Descartes in the context of problems of optics [1].

Figure: d031340a

Comments

A Descartes oval is also called a Cartesian oval, or simply a Cartesian.

The caustic of a circle with respect to a point light source in the plane of the circle is the evolute of a Descartes oval.

There are several ways in which a Cartesian oval can arise. One is as follows. Let $C$ and $C_1$ be two cones with axes parallel to the $z$-axis and with circular intersections with the $xy$-plane. Then the vertical projection onto the $xy$-plane of their intersection is a Cartesian oval. This is a result of A. Quetelet. Suppose the apexes of $C$ and $C_1$ are at $(0,0,1)$ and $(c,0,a)$, respectively, and the intersections with the $xy$-plane are circles of radius 1 and $b$ (see below). Then in the case $b=(1-a)^{-1}a(a+1)$, $c=-(1-a)^{-1}(a+1)^2$ the projection of the intersection is a cardioid.

In polar coordinates the equation for a Cartesian oval is $r^2-2(a+b\cos\phi)r+c^2=0$. Hence when $c=0$ the equation becomes $r=a+b\cos\phi$ (plus $r=0$) and one obtains the Pascal limaçon; when in addition $a=b$, the cardioid results, whose equation can also be written as $\sqrt r=\sqrt m\cos(\phi/2)$.

References

[1] R. Descartes, "Géométrie" , Leyden (1637)
[2] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[a1] J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972)
[a2] K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962)


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How to Cite This Entry:
Descartes oval. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Descartes_oval&oldid=16712
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article