View source for Dandelin spheres

Spheres forming part of the geometrical constructions relating the planimetric definition of the ellipse, hyperbola or parabola with their stereometric definitions. For instance, let two spheres (also known as Dandelin spheres), inscribed in a circular cone, make contact with the surface of the cone along circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d0301001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d0301002.png" /> (see Fig.) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d0301003.png" /> be a certain plane passing through two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d0301004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d0301005.png" />.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d030100a.gif" />

Figure: d030100a

If an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d0301006.png" /> is taken on the intersection line of the cone with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d0301007.png" />, and a generatrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d0301008.png" /> is drawn intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d0301009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d03010010.png" />, then if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d03010011.png" /> varies, the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d03010012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d03010013.png" /> move around the circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d03010014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d03010015.png" /> while preserving the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d03010016.png" />, i.e. the intersection line will be an ellipse (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d03010017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d03010018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030100/d03010019.png" />). In the case of a hyperbola, Dandelin spheres are located in different sheets.

Suggested by G. Dandelin in 1822.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Modenov,   "Analytic geometry" , Moscow  (1969)  (In Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   "Geometry" , '''II''' , Springer  (1987)  pp. 227</TD></TR></table>