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''spiric curve''
 
''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 Fig. a, Fig. b, Fig. c). The equation in rectangular coordinates is
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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 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 [[Booth lemniscate|Booth lemniscate]], the [[Cassini oval|Cassini oval]] and the [[Bernoulli lemniscate|Bernoulli lemniscate]].
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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]].
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072400a.gif" />
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<gallery>
 
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File:Courbe persane 042.svg|$d>r$
Figure: p072400a
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File:Courbe persane 022.svg|$d=r$
 
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File:Courbe persane 024.svg|$d<r$
$d>r$.
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</gallery>
 
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072400b.gif" />
 
 
 
Figure: p072400b
 
 
 
$d=r$.
 
 
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072400c.gif" />
 
 
 
Figure: p072400c
 
 
 
$d<r$.
 
  
 
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"> A.A. Savelov,   "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Savelov, "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR>
 
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> F. Gomez Teixeira, "Traité des courbes" , '''1–3''' , Chelsea, reprint (1971)</TD></TR>
 
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<TR><TD valign="top">[a2]</TD> <TD valign="top"> K. Fladt, "Analytische Geometrie spezieller ebener Kurven" , Akad. Verlagsgesell. (1962)</TD></TR>
====Comments====
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</table>
 
 
 
 
====References====
 
<table><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>
 

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.

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
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article