View source for Confocal conics

''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).

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>

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).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024710a.gif" />

Figure: c024710a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c024710b.gif" />

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" />.



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J. Struik,   "Lectures on analytic and projective geometry" , Addison-Wesley  (1953)  pp. 157–160</TD></TR></table>