PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (194)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: Very high
[parent] Cassini oval (Definition)

Cassini oval is the locus of the point $ P$ in the plane having a constant product of the distances $ PF_1$ and $ PF_2$ measured from two fixed points $ F_1$ and $ F_2$ of the plane.

One obtains the simplest equation for the Cassini oval by choosing $ F_1$ and $ F_2$ on the other coordinate axis and equidistant ($ = c > 0$) from the origin. Let $ F_1 = (-c,\,0)$, $ F_2 = (c,\,0)$ and the locus condition

$\displaystyle PF_1 \cdot PF_2 = a^2 \quad (a > 0).$
This reads in the Cartesian coordinates
$\displaystyle \sqrt{(x+c)^2+y^2}\sqrt{(x-c)^2+y^2} = a^2,$ (1)

which after squaring may be written
$\displaystyle a^4 = (x^2+y^2+c^2+2cx)(x^2+y^2+c^2-2cx) \equiv (x^2+y^2+c^2)^2-(2cx)^2,$
i.e.
$\displaystyle (x^2+y^2+c^2)^2-4c^2x^2 = a^4.$ (2)

One sees that the curve is symmetric both in regard to $ x$-axis and in regard to $ y$-axis, whence it suffices to examine it in the first quadrant ($ x \geqq 0$, $ y \geqq 0$). If (2) is written as
$\displaystyle y^2 = \sqrt{a^4+4c^2x^2}-(x^2+c^2),$ (3)

it appears that $ y$ is real only for $ \sqrt{a^4+4c^2x^2} \geqq x^2+c^2$, which condition can be simplified to
$\displaystyle \vert x^2-c^2\vert \leqq a^2.$ (4)

In order to $ y$ being real, (4) gives the three cases:
$ 1^{\underline{o}}$. $ a < c$. We have $ \sqrt{c^2-a^2} \leqq x \leqq \sqrt{c^2+a^2}$; thus the curve consists of two separate loops.
$ 2^{\underline{o}}$. $ a = c$. Now $ 0 \leqq x \leqq c\sqrt{2}$; the two loops meet in the origin (the lemniscate of Bernoulli).
$ 3^{\underline{o}}$. $ a > c$. Then $ 0 \leqq x \leqq \sqrt{c^2+a^2}$; there is one loop surrounding the origin.

\begin{pspicture}(-3,-2.2)(3,2.5) \psset{unit=2cm} \psaxes[Dx=1,Dy=1]{->}(0,0)(-... ...\rput(3,2.2){.} \rput(0,-1.7){Cassinoids with\, $c = 1$\ (blue)} \end{pspicture}

When $ a$ gets different values (the parametre $ c$ being unchanged), (2) represents a family of curves. For any point $ P$ of the plane (except $ (\pm c,\,0)$), there is one representant of the family passing through $ P$, corresponding the value $ a = \sqrt{PF_1 \cdot PF_2}$.

As a matter of fact, the common name for all members of the family is Cassini curve, and only the special case, where $ a = c\sqrt{2}$, is the Cassini oval proper; it and the other members with $ a \geqq c\sqrt{2}$ have the property of having only one highest point $ (0,\,\sqrt{a^2-c^2})$. All other members (with $ a < c\sqrt{2}$) have two distinct highest points.

The locus of the highest and lowest points of any member with $ a \leqq c\sqrt{2}$ is obtained by solving (2) with respect to $ x^2$,

$\displaystyle x^2 = c^2-y^2\pm\sqrt{a^4-4c^2y^2},$
whence $ y^2 \leqq \frac{a^4}{4c^2}$. When the radicand vanishes, $ \vert y\vert$ gets its maximum value and then we have $ x^2 = c^2-y^2$, which means a circle centered in the origin (green in the picture).

Note. The astronomer Domenico Cassini found in 1680 the curve named after him; he thought that the orbit of Earth relative to the Sun was a cassinoid with the Sun in the other “focus”.

Bibliography

1
F. IVERSEN: Analyyttisen geometrian oppikirja. Second edition. Kustannusosakeyhtiö Otava, Helsinki (1963).



"Cassini oval" is owned by pahio.
(view preamble)

View style:

See Also: Euclidean distance, polar curve

Other names:  cassinoid, oval of Cassini, Cassini curve
Also defines:  lemniscate of Bernoulli

This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: circle, vanishes, radicand, property, passing through, parametre, meet, loops, real, quadrant, symmetric, curve, Cartesian coordinates, origin, axis, coordinate, equation, distances, product, plane, point, locus
There are 2 references to this entry.

This is version 23 of Cassini oval, born on 2008-01-16, modified 2008-08-13.
Object id is 10196, canonical name is CassiniOval.
Accessed 1059 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)