Difference between revisions of "Persian curve"
From Encyclopedia of Mathematics
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| − | A plane algebraic curve of order four that is the line of intersection between the surface of a torus and a plane parallel to its axis (see | + | A plane algebraic curve of order four that is the line of intersection between the surface of a torus and a plane parallel to its axis (see Figures). The equation in rectangular coordinates is |
$$(x^2+y^2+p^2+d^2-r^2)^2=4d^2(x^2+p^2),$$ | $$(x^2+y^2+p^2+d^2-r^2)^2=4d^2(x^2+p^2),$$ | ||
| − | where $r$ is the radius of the circle describing the torus, $d$ is the distance from the origin to its centre and $p$ is the distance from the axis of the torus to the plane. The following are Persian curves: the [[ | + | where $r$ is the radius of the circle describing the torus, $d$ is the distance from the origin to its centre and $p$ is the distance from the axis of the torus to the plane. The following are Persian curves: the [[Booth lemniscate]], the [[Cassini oval]] and the [[Bernoulli lemniscate]]. |
| − | < | + | <gallery> |
| − | + | File:Courbe persane 042.svg|$d>r$ | |
| − | + | File:Courbe persane 022.svg|$d=r$ | |
| − | + | File:Courbe persane 024.svg|$d<r$ | |
| − | $d>r$ | + | </gallery> |
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| − | $d=r$ | ||
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| − | $d<r$ | ||
The name is after the Ancient Greek geometer Persei (2nd century B.C.), who examined it in relation to research on various ways of specifying curves. | The name is after the Ancient Greek geometer Persei (2nd century B.C.), who examined it in relation to research on various ways of specifying curves. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table> |
| − | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)</TD></TR> | |
| − | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Gomez Teixeira, "Traité des courbes" , '''1–3''' , Chelsea, reprint (1971)</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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Latest revision as of 08:45, 12 November 2023
spiric curve
A plane algebraic curve of order four that is the line of intersection between the surface of a torus and a plane parallel to its axis (see Figures). The equation in rectangular coordinates is
$$(x^2+y^2+p^2+d^2-r^2)^2=4d^2(x^2+p^2),$$
where $r$ is the radius of the circle describing the torus, $d$ is the distance from the origin to its centre and $p$ is the distance from the axis of the torus to the plane. The following are Persian curves: the Booth lemniscate, the Cassini oval and the Bernoulli lemniscate.
$d>r$
$d=r$
$d<r$
The name is after the Ancient Greek geometer Persei (2nd century B.C.), who examined it in relation to research on various ways of specifying curves.
References
| [1] | A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian) |
| [a1] | F. Gomez Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971) |
| [a2] | K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962) |
How to Cite This Entry:
Persian curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Persian_curve&oldid=31952
Persian curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Persian_curve&oldid=31952
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article