Cassini oval
Encyclopedia
A Cassini oval is a plane curve
defined as the set (or locus
) of points in the plane
such that the product of the distances to two fixed points is constant. This is related to an ellipse
, for which the
sum of the distances is constant, rather than the product. They are the special case of polynomial lemniscate
s when the polynomial used has degree 2.
Cassini ovals are named after the astronomer Giovanni Domenico Cassini
who studied them in 1680. Other names include Cassinian ovals, Cassinian curves and ovals of Cassini.
. Then a Cassini oval with foci
q1 and q2 is defined to be the locus of points p so that the product of the distance from p to q1 and the distance from p to q2 is b2. That is, if we define the function dist(x,y) to be the distance from a point x to a point y, then all points p on a Cassini oval satisfy the equation
When expanded this becomes
The equivalent polar equation is
having the shape of a sideways figure eight with a double point (specifically, a crunode
) at the origin. The limiting case of a → 0 (hence e →
), in which case the foci coincide with each other, is a circle.
The curve always has x-intercepts at ±c where c2=a2+b2. When e<1 there are two additional real x-intercepts and when e>1 there are two real y-intercepts, all other x and y-intercepts being imaginary.
The curve has double points at the circular points at infinity
, in other words the curve is bicircular
. 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 and Plücker's formulas
it is possible to deduce the Plücker numbers for the case e≠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.
The tangents at the circular points are given by x±iy=±a which have real points of intersection at (±a, 0). So the foci are, in fact, foci in the sense defined by Plücker. The circular points are points of inflection so these are triple foci. When e≠1 the curve has class eight, which implies that there should be at 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 the x-axis when the curve has two loops and on the y-axis when the curve has a single loop.
Curves orthogonal to the Cassini ovals: Formed when the foci of the Cassini ovals are the points (a,0) and (-a,0), equilateral hyperbolas centered at (0,0) after a rotation around (0,0) are made to pass through the foci.
Plane curve
In mathematics, a plane curve is a curve in a Euclidean plane . The most frequently studied cases are smooth plane curves , and algebraic plane curves....
defined as the set (or locus
Locus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....
) of points in the plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
such that the product of the distances to two fixed points is constant. This is related to an ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
, for which the
sum of the distances is constant, rather than the product. They are the special case of polynomial lemniscate
Polynomial lemniscate
In mathematics, a polynomial lemniscate or polynomial level curve is a plane algebraic curve of degree 2n, constructed from a polynomial p with complex coefficients of degree n....
s when the polynomial used has degree 2.
Cassini ovals are named after the astronomer Giovanni Domenico Cassini
Giovanni Domenico Cassini
This article is about the Italian-born astronomer. For his French-born great-grandson, see Jean-Dominique Cassini.Giovanni Domenico Cassini was an Italian/French mathematician, astronomer, engineer, and astrologer...
who studied them in 1680. Other names include Cassinian ovals, Cassinian curves and ovals of Cassini.
Formal definition
Let q1 and q2 be two fixed points in the plane and let b be a constantConstant (mathematics)
In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition...
. Then a Cassini oval with foci
Focus (geometry)
In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...
q1 and q2 is defined to be the locus of points p so that the product of the distance from p to q1 and the distance from p to q2 is b2. That is, if we define the function dist(x,y) to be the distance from a point x to a point y, then all points p on a Cassini oval satisfy the equation
Equations
If the foci are (a, 0) and (−a, 0), then the equation of the curve isWhen expanded this becomes
The equivalent polar equation is
Form of the curve
The shape of the curve depends, up to similarity, on e=b/a. When e>1, the curve is a single, connected loop enclosing both foci. When e<1, the curve consists of two disconnected loops, each of which contains a focus. When e=1, the curve is the lemniscate of BernoulliLemniscate of Bernoulli
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F1 and F2, known as foci, at distance 2a from each other as the locus of points P so that PF1·PF2 = a2. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscus, which is...
having the shape of a sideways figure eight with a double point (specifically, a crunode
Crunode
A crunode is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines.For a plane curve, defined as the locus of points f = 0, where f is a smooth function of variables x and y ranging over the real numbers, a crunode of the curve is a singularity of...
) at the origin. The limiting case of a → 0 (hence e →
), in which case the foci coincide with each other, is a circle.The curve always has x-intercepts at ±c where c2=a2+b2. When e<1 there are two additional real x-intercepts and when e>1 there are two real y-intercepts, all other x and y-intercepts being imaginary.
The curve has double points at the circular points at infinity
Circular points at infinity
In projective geometry, the circular points at infinity in the complex projective plane are and ....
, in other words the curve is bicircular
Circular algebraic curve
In geometry, a circular algebraic curve is a type of plane algebraic curve determined by an equation F = 0, where F is a polynomial with real coefficients and the highest-order terms of F form a polynomial divisible by x2 + y2...
. 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 and Plücker's formulas
Plücker formula
In mathematics, a Plücker formula, named after Julius Plücker, is one of a family of formulae, of a type first developed by Plücker in the 1830s, that relate certain numeric invariants of algebraic curves to corresponding invariants of their dual curves. The invariant called the genus, common to...
it is possible to deduce the Plücker numbers for the case e≠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.
The tangents at the circular points are given by x±iy=±a which have real points of intersection at (±a, 0). So the foci are, in fact, foci in the sense defined by Plücker. The circular points are points of inflection so these are triple foci. When e≠1 the curve has class eight, which implies that there should be at 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 the x-axis when the curve has two loops and on the y-axis when the curve has a single loop.
Curves orthogonal to the Cassini ovals: Formed when the foci of the Cassini ovals are the points (a,0) and (-a,0), equilateral hyperbolas centered at (0,0) after a rotation around (0,0) are made to pass through the foci.
Examples
Second lemniscate of Mandelbrot set is Cassini oval with equation