Difference between revisions of "Persian curve"
From Encyclopedia of Mathematics
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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 | 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 | ||
| − | + | $$(x^2+y^2+p^2+d^2-r^2)^2=4d^2(x^2+p^2),$$ | |
| − | where | + | 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]]. |
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072400a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072400a.gif" /> | ||
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Figure: p072400a | Figure: p072400a | ||
| − | + | $d>r$. | |
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072400b.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072400b.gif" /> | ||
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Figure: p072400b | Figure: p072400b | ||
| − | + | $d=r$. | |
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072400c.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072400c.gif" /> | ||
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Figure: p072400c | 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. | ||
Revision as of 19:00, 27 April 2014
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
$$(x^2+y^2+p^2+d^2-r^2)^2=4d^2(x^2+p^2),$$
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.
Figure: p072400a
$d>r$.
Figure: p072400b
$d=r$.
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.
References
| [1] | A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian) |
Comments
References
| [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