Difference between revisions of "Bernoulli lemniscate"
From Encyclopedia of Mathematics
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A plane algebraic curve of order four, the equation of which in orthogonal Cartesian coordinates is: | A plane algebraic curve of order four, the equation of which in orthogonal Cartesian coordinates is: | ||
| − | + | $$(x^2+y^2)^2-2a^2(x^2-y^2)=0;$$ | |
and in polar coordinates | and in polar coordinates | ||
| − | + | $$\rho^2=2a^2\cos2\phi.$$ | |
| − | The Bernoulli lemniscate is symmetric about the coordinate origin (Fig.), which is a node with tangents | + | The Bernoulli lemniscate is symmetric about the coordinate origin (Fig.), which is a node with tangents $y=\pm x$ and the point of inflection. |
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b015620a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b015620a.gif" /> | ||
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Figure: b015620a | Figure: b015620a | ||
| − | The product of the distances of any point | + | The product of the distances of any point $M$ to the two given points $F_1(-a,0)$ and $F_2(a,0)$ is equal to the square of the distance between the points $F_1$ and $F_2$. The Bernoulli lemniscate is a special case of the Cassini ovals, the [[Lemniscates|lemniscates]], and the sinusoidal spirals (cf. [[Cassini oval|Cassini oval]]; [[Sinusoidal spiral|Sinusoidal spiral]]). |
| − | The Bernoulli spiral was named after Jakob Bernoulli, who gave its equation in 1694. | + | The Bernoulli spiral was named after [[Bernoulli, Jakob|Jakob Bernoulli]], who gave its equation in 1694. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)</TD></TR></table> | + | <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"> E. Brieskorn, H. Knörrer, "Plane algebraic curves" , Birkhäuser (1986) (Translated from German)</TD></TR> | ||
| + | </table> | ||
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Latest revision as of 19:07, 17 April 2024
A plane algebraic curve of order four, the equation of which in orthogonal Cartesian coordinates is:
$$(x^2+y^2)^2-2a^2(x^2-y^2)=0;$$
and in polar coordinates
$$\rho^2=2a^2\cos2\phi.$$
The Bernoulli lemniscate is symmetric about the coordinate origin (Fig.), which is a node with tangents $y=\pm x$ and the point of inflection.
Figure: b015620a
The product of the distances of any point $M$ to the two given points $F_1(-a,0)$ and $F_2(a,0)$ is equal to the square of the distance between the points $F_1$ and $F_2$. The Bernoulli lemniscate is a special case of the Cassini ovals, the lemniscates, and the sinusoidal spirals (cf. Cassini oval; Sinusoidal spiral).
The Bernoulli spiral was named after Jakob Bernoulli, who gave its equation in 1694.
References
| [1] | A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian) |
| [a1] | E. Brieskorn, H. Knörrer, "Plane algebraic curves" , Birkhäuser (1986) (Translated from German) |
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How to Cite This Entry:
Bernoulli lemniscate. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernoulli_lemniscate&oldid=12542
Bernoulli lemniscate. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bernoulli_lemniscate&oldid=12542
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article