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Difference between revisions of "Confocal conics"

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''co-focal curves''
 
''co-focal curves''
  
Curves of the second order with common foci. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c0247101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c0247102.png" /> are two given points in the plane, then through every point of the plane there are one ellipse and one hyperbola that have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c0247103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c0247104.png" /> as their foci (Fig. a).
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Curves of the second order with common foci. If $F$ and $F_1$ are two given points in the plane, then through every point of the plane there are one ellipse and one hyperbola that have $F$ and $F_1$ as their foci (Fig. a).
  
 
Every ellipse is orthogonal to every hyperbola confocal with it, that is, they intersect (in four points) at right angles. In a suitable coordinate system, all the confocal ellipses and hyperbolas can be given by the equation
 
Every ellipse is orthogonal to every hyperbola confocal with it, that is, they intersect (in four points) at right angles. In a suitable coordinate system, all the confocal ellipses and hyperbolas can be given by the 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/c/c024/c024710/c0247105.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\frac{x^2}{\lambda}+\frac{y^2}{\lambda-c^2}=1,\tag{*}$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c0247106.png" /> is the distance of the foci from the coordinate origin and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c0247107.png" /> is a variable parameter. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c0247108.png" />, this equation defines an ellipse, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c0247109.png" />, it defines a hyperbola (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c02471010.png" />, it is an imaginary curve of the second order). If one of the foci tends to infinity, then in the limit one obtains two families of confocal parabolas (Fig. b).
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where $c$ is the distance of the foci from the coordinate origin and $\lambda$ is a variable parameter. If $\lambda>c^2$, this equation defines an ellipse, and if $0<\lambda<c^2$, it defines a hyperbola (if $\lambda<0$, it is an imaginary curve of the second order). If one of the foci tends to infinity, then in the limit one obtains two families of confocal parabolas (Fig. b).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024710a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024710a.gif" />
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Figure: c024710b
 
Figure: c024710b
  
Any two parabolas from different families are also orthogonal to one another. Confocal ellipses and hyperbolas can be used to introduce a so-called elliptic coordinate system in the plane as follows. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c02471011.png" /> is any point of the plane, then by substituting its coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c02471012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c02471013.png" /> into (*), one obtains a quadratic equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c02471014.png" />; its roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c02471015.png" /> are the elliptic coordinates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c02471016.png" />. The confocal ellipses and hyperbolas themselves form the coordinate net of this coordinate system, that is, they are defined by the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c02471017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024710/c02471018.png" />.
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Any two parabolas from different families are also orthogonal to one another. Confocal ellipses and hyperbolas can be used to introduce a so-called elliptic coordinate system in the plane as follows. If $M(x,y)$ is any point of the plane, then by substituting its coordinates $x$ and $y$ into \ref{*}, one obtains a quadratic equation for $\lambda$; its roots $\lambda_1,\lambda_2$ are the elliptic coordinates of $M$. The confocal ellipses and hyperbolas themselves form the coordinate net of this coordinate system, that is, they are defined by the equations $\lambda_1=\text{const}$ and $\lambda_2=\text{const}$.
  
  

Revision as of 09:01, 29 August 2014

co-focal curves

Curves of the second order with common foci. If $F$ and $F_1$ are two given points in the plane, then through every point of the plane there are one ellipse and one hyperbola that have $F$ and $F_1$ as their foci (Fig. a).

Every ellipse is orthogonal to every hyperbola confocal with it, that is, they intersect (in four points) at right angles. In a suitable coordinate system, all the confocal ellipses and hyperbolas can be given by the equation

$$\frac{x^2}{\lambda}+\frac{y^2}{\lambda-c^2}=1,\tag{*}$$

where $c$ is the distance of the foci from the coordinate origin and $\lambda$ is a variable parameter. If $\lambda>c^2$, this equation defines an ellipse, and if $0<\lambda<c^2$, it defines a hyperbola (if $\lambda<0$, it is an imaginary curve of the second order). If one of the foci tends to infinity, then in the limit one obtains two families of confocal parabolas (Fig. b).

Figure: c024710a

Figure: c024710b

Any two parabolas from different families are also orthogonal to one another. Confocal ellipses and hyperbolas can be used to introduce a so-called elliptic coordinate system in the plane as follows. If $M(x,y)$ is any point of the plane, then by substituting its coordinates $x$ and $y$ into \ref{*}, one obtains a quadratic equation for $\lambda$; its roots $\lambda_1,\lambda_2$ are the elliptic coordinates of $M$. The confocal ellipses and hyperbolas themselves form the coordinate net of this coordinate system, that is, they are defined by the equations $\lambda_1=\text{const}$ and $\lambda_2=\text{const}$.


Comments

References

[a1] D.J. Struik, "Lectures on analytic and projective geometry" , Addison-Wesley (1953) pp. 157–160
How to Cite This Entry:
Confocal conics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Confocal_conics&oldid=13848
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article