Ingeometry, aCassinioval is aquartic plane curve defined as thelocus of points in theplane such that theproduct of the distances to two fixed points (foci) is constant. This may be contrasted with anellipse, for which thesum of the distances is constant, rather than the product. Cassini ovals are the special case ofpolynomial lemniscates when thepolynomial used hasdegree 2.

Cassini ovals are named after the astronomerGiovanni Domenico Cassini who studied them in the late 17th century.^[1] Cassini believed that a planetorbiting around another body traveled on one of these ovals, with the body it orbited around at one focus of the oval.^[2]Other names includeCassinian ovals,Cassinian curves andovals of Cassini.

ACassini oval is a set of points, such that for any point${\displaystyle P}$  of the set, theproduct of the distances${\displaystyle |P{P}_1|,|P{P}_2|}$  to two fixed points${\displaystyle P_1,P_2}$  is a constant, usually written as${\displaystyle b^2}$  where${\displaystyle b>0}$ :

${\displaystyle \{P:|P{P}_1|\times |P{P}_2|=b^2\}.}$ 

As with an ellipse, the fixed points${\displaystyle P_1,P_2}$  are called thefoci of the Cassini oval.

If the foci are (a, 0) and (−a, 0), then the equation of the curve is

${\displaystyle ((x-a{)}^2+y^2)((x+a{)}^2+y^2)=b^4.}$ 

When expanded this becomes

${\displaystyle (x^2+y^2{)}^2-2a^2(x^2-y^2)+a^4=b^4.}$ 

The equivalentpolar equation is

${\displaystyle r^4-2a^2{r}^2\cos 2\theta =b^4-a^4.}$ 

The curve depends, up to similarity, one =b/a. Whene < 1, the curve consists of two disconnected loops, each of which contains a focus. Whene = 1, the curve is thelemniscate of Bernoulli having the shape of a sideways figure eight with adouble point (specifically, acrunode) at the origin.^[3][4] Whene > 1, the curve is a single, connected loop enclosing both foci. It is peanut-shaped for  and convex for${\displaystyle e\ge \sqrt{2}.}$ ^[5] The limiting case ofa → 0 (hencee → ∞), in which case the foci coincide with each other, is acircle.

The curve always hasx-intercepts at± c wherec² =a² +b². Whene < 1 there are two additionalrealx-intercepts and whene > 1 there are two realy-intercepts, all otherx- andy-intercepts being imaginary.^[6]

The curve has double points at thecircular points at infinity, in other words the curve isbicircular. These points are biflecnodes, meaning that the curve has two distinct tangents at these points and each branch of the curve has a point of inflection there. From this information andPlücker's formulas it is possible to deduce the Plücker numbers for the casee ≠ 1: degree = 4, class = 8, number of nodes = 2, number of cusps = 0, number of double tangents = 8, number of points of inflection = 12, genus = 1.^[7]

The tangents at the circular points are given byx ±iy = ± a which have real points of intersection at(± a, 0). So the foci are, in fact, foci in the sense defined by Plücker.^[8] The circular points are points of inflection so these are triple foci. Whene ≠ 1 the curve has class eight, which implies that there should be a total of eight real foci. Six of these have been accounted for in the two triple foci and the remaining two are at So the additional foci are on thex-axis when the curve has two loops and on they-axis when the curve has a single loop.^[9]

Orthogonal trajectories of a givenpencil of curves are curves which intersect all given curves orthogonally. For example the orthogonal trajectories of a pencil ofconfocal ellipses are the confocalhyperbolas with the same foci. For Cassini ovals one has:

• The orthogonal trajectories of the Cassini curves with foci${\displaystyle P_1,P_2}$  are theequilateral hyperbolas containing${\displaystyle P_1,P_2}$  with the same center as the Cassini ovals (see picture).

Proof:
For simplicity one chooses${\displaystyle P_1=(1,0),P_2=(-1,0)}$ .

The Cassini ovals have the equation$${\displaystyle f(x,y)=(x^2+y^2{)}^2-2(x^2-y^2)+1-b^4=0.}$$ 

Theequilateral hyperbolas (theirasymptotes are rectangular) containing${\displaystyle (1,0),(-1,0)}$  with center${\displaystyle (0,0)}$  can be described by the equation$${\displaystyle x^2-y^2-\lambda xy-1=0,\lambda \in \mathbb{R}.}$$ 

These conic sections have no points with they-axis in common and intersect thex-axis at${\displaystyle (\pm 1,0)}$ . Theirdiscriminants show that these curves are hyperbolas. A more detailed investigation reveals that the hyperbolas are rectangular. In order to get normals, which are independent from parameter${\displaystyle \lambda }$  the following implicit representation is more convenient$${\displaystyle g(x,y)=\frac{x^2-y^2-1}{xy}-\lambda =\frac{x}{y}-\frac{y}{x}-\frac{1}{xy}-\lambda =0.}$$ A simple calculation shows that${\displaystyle \mathrm{grad}f(x,y)\cdot \mathrm{grad}g(x,y)=0}$  for all${\displaystyle (x,y),x\ne 0\ne y}$ . Hence the Cassini ovals and the hyperbolas intersect orthogonally.

Remark:
The image depicting the Cassini ovals and the hyperbolas looks like theequipotential curves of two equalpoint charges together with the lines of the generatedelectrical field. But for the potential of two equal point charges one has${\displaystyle 1/|P{P}_1|+1/|P{P}_2|=\text{constant}}$ . (SeeImplicit curve.) Instead these curves actually correspond to the (plane sections of) equipotential sets of two infinite wires with equal constant line charge density, or alternatively, to the level sets of the sums of theGreen’s functions for the Laplacian in two dimensions centered at the foci.

The single-loop and double loop Cassini curves can be represented as the orthogonal trajectories of each other when each family is coaxal but not confocal. If the single-loops are described by${\displaystyle (x^2+y^2)-1=axy}$  then the foci are variable on the axis${\displaystyle y=x}$  if${\displaystyle a>0}$ ,${\displaystyle y=-x}$  if ; if the double-loops are described by${\displaystyle (x^2+y^2)+1=b(x^2-y^2)}$  then the axes are, respectively,${\displaystyle y=0}$  and${\displaystyle x=0}$ . Each curve, up to similarity, appears twice in the image, which now resembles the field lines and potential curves for four equal point charges, located at${\displaystyle (\pm 1,0)}$  and${\displaystyle (0,\pm 1)}$ . Further, the portion of this image in the upper half-plane depicts the following situation: The double-loops are a reduced set of congruence classes for the central Steiner conics in the hyperbolic plane produced by direct collineations;^[10] and each single-loop is the locus of points${\displaystyle P}$  such that the angle${\displaystyle OPQ}$  is constant, where${\displaystyle O=(0,1)}$  and${\displaystyle Q}$  is the foot of the perpendicular through${\displaystyle P}$  on the line described by${\displaystyle x^2+y^2=1}$ .

The secondlemniscate of the Mandelbrot set is a Cassini oval defined by the equation${\displaystyle L_2=\{c:\mathrm{abs}(c^2+c)=ER\}.}$  Its foci are at the pointsc on thecomplex plane that have orbits where every second value ofz is equal to zero, which are the values 0 and −1.

Cassini ovals appear as planar sections oftori, but only when the cutting plane is parallel to the axis of the torus and its distance to the axis equals the radius of the generating circle (see picture).

The intersection of the torus with equation

${\displaystyle {\left(x^2+y^2+z^2+R^2-r^2\right)}^2=4R^2\left(x^2+y^2\right)}$ 

and the plane${\displaystyle y=r}$  yields

${\displaystyle {\left(x^2+z^2+R^2\right)}^2=4R^2\left(x^2+r^2\right).}$ 

After partially resolving the first bracket one gets the equation

${\displaystyle {\left(x^2+z^2\right)}^2-2R^2(x^2-z^2)=4R^2{r}^2-R^4,}$ 

which is the equation of a Cassini oval with parameters${\displaystyle b^2=2Rr}$  and${\displaystyle a=R}$ .

Cassini's method is easy to generalize to curves and surfaces with an arbitrarily many defining points:

• ${\displaystyle |P{P}_1|\times |P{P}_2|\times \cdots \times |P{P}_n|=b^n}$ 

describes in the planar case animplicit curve and in 3-space animplicit surface.

•  
  curve with 3 defining points
•  
  surface with 6 defining points

• Two-center bipolar coordinates

• J.-D. Cassini (1693).De l'Origine et du progrès de l'astronomie et de son usage dans la géographie et dans la navigation. L’Imprimerie Royale. pp. 36.
• Cohen, I. Bernard (1962). "Leibniz on elliptical orbits: as seen in his correspondence with the Académie Royale des Sciences in 1700".Journal of the History of Medicine and Allied Sciences.17 (1):72–82.doi:10.1093/jhmas/xvii.1.72.JSTOR 24620858.
• J. Dennis Lawrence (1972).A catalog of special plane curves.Dover Publications. pp. 5, 153–155.ISBN 0-486-60288-5.
• A. B. Basset (1901).An Elementary Treatise on Cubic and Quartic Curves. London: Deighton Bell and Co. pp. 162 ff.
• Lawden, D. F., "Families of ovals and their orthogonal trajectories",Mathematical Gazette 83, November 1999, 410–420.

• "Cassini oval".Encyclopedia of Mathematics.EMS Press. 2001 [1994].
• MacTutor description
• Weisstein, Eric W."Cassini Ovals".MathWorld.
• 2Dcurves.com description
• "MacTutor History of Mathematics" Famous Curves