Difference between revisions of "Dandelin spheres"
From Encyclopedia of Mathematics
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| − | 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 | + | {{TEX|done}} |
| + | 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 $c$ and $c'$ (see Fig.) and let $\pi$ be a certain plane passing through two points $F$ and $F'$. | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d030100a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d030100a.gif" /> | ||
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Figure: d030100a | Figure: d030100a | ||
| − | If an arbitrary point | + | If an arbitrary point $M$ is taken on the intersection line of the cone with $\pi$, and a generatrix $SM$ is drawn intersecting $c$ and $c'$, then if $M$ varies, the points $T$ and $T'$ move around the circles $c$ and $c'$ while preserving the distance $TT'$, i.e. the intersection line will be an ellipse ($MF'+MF=TT'$, $MF'=MT'$ and $MF=MT$). In the case of a hyperbola, Dandelin spheres are located in different sheets. |
Suggested by G. Dandelin in 1822. | Suggested by G. Dandelin in 1822. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Modenov, "Analytic geometry" , Moscow (1969) (In Russian)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> P.S. Modenov, "Analytic geometry" , Moscow (1969) (In Russian)</TD></TR> | ||
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Berger, "Geometry" , '''II''' , Springer (1987) pp. 227</TD></TR> | ||
| + | </table> | ||
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Latest revision as of 11:21, 26 March 2023
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 $c$ and $c'$ (see Fig.) and let $\pi$ be a certain plane passing through two points $F$ and $F'$.
Figure: d030100a
If an arbitrary point $M$ is taken on the intersection line of the cone with $\pi$, and a generatrix $SM$ is drawn intersecting $c$ and $c'$, then if $M$ varies, the points $T$ and $T'$ move around the circles $c$ and $c'$ while preserving the distance $TT'$, i.e. the intersection line will be an ellipse ($MF'+MF=TT'$, $MF'=MT'$ and $MF=MT$). In the case of a hyperbola, Dandelin spheres are located in different sheets.
Suggested by G. Dandelin in 1822.
References
| [1] | P.S. Modenov, "Analytic geometry" , Moscow (1969) (In Russian) |
| [a1] | M. Berger, "Geometry" , II , Springer (1987) pp. 227 |
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How to Cite This Entry:
Dandelin spheres. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dandelin_spheres&oldid=17843
Dandelin spheres. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dandelin_spheres&oldid=17843
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article