In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a Cassini
oval
An oval () is a closed curve in a plane which resembles the outline of an egg. The term is not very specific, but in some areas (projective geometry, technical drawing, etc.) it is given a more precise definition, which may include either one or ...
is a
quartic plane curve
In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation:
:Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0,
with at least one of ...
defined as the
locus
Locus (plural loci) is Latin for "place". It may refer to:
Entertainment
* Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front
* ''Locus'' (magazine), science fiction and fantasy magazine
** ''Locus Award' ...
of points in the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
such that the
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of the distances to two fixed points (
foci
Focus, or its plural form foci may refer to:
Arts
* Focus or Focus Festival, former name of the Adelaide Fringe arts festival in South Australia Film
*''Focus'', a 1962 TV film starring James Whitmore
* ''Focus'' (2001 film), a 2001 film based ...
) is constant. This may be contrasted with an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
, for which the ''sum'' of the distances is constant, rather than the product. Cassini ovals are the special case of
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''.
For any such polynomial ''p'' and positive real number ' ...
s when the
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
used has
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
2.
Cassini ovals are named after the astronomer
Giovanni Domenico Cassini
Giovanni Domenico Cassini, also known as Jean-Dominique Cassini (8 June 1625 – 14 September 1712) was an Italian (naturalised French) mathematician, astronomer and engineer. Cassini was born in Perinaldo, near Imperia, at that time in the C ...
who studied them in the late 17th century.
Cassini believed that the
Sun
The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
traveled around the
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
on one of these ovals, with the Earth at one focus of the oval.
Other names include Cassinian ovals, Cassinian curves and ovals of Cassini.
Formal definition
A Cassini oval is a set of points, such that for any point
of the set, the ''product'' of the distances
to two fixed points
is a constant, usually written as
where
:
:
As with an ellipse, the fixed points
are called the ''foci'' of the Cassini oval.
Equations
If the foci are (''a'', 0) and (−''a'', 0), then the equation of the curve is
:
When expanded this becomes
:
The equivalent
polar
Polar may refer to:
Geography
Polar may refer to:
* Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates
* Polar climate, the c ...
equation is
:
Shape
The curve depends, up to similarity, on ''e'' = ''b''/''a''. 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 Bernoulli
In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. I ...
having the shape of a sideways figure eight with a
double point
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.
Algebraic curves in the plane
Algebraic curv ...
(specifically, a
crunode
In mathematics, a crunode (archaic) or node is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection. A crunode is also known as an ''ordinary double point''.
For a ...
) at the origin. When ''e'' > 1, the curve is a single, connected loop enclosing both foci. It is peanut-shaped for
and convex for
. The limiting case of ''a'' → 0 (hence ''e'' →
), in which case the foci coincide with each other, is a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
.
The curve always has ''x''-intercepts at ± ''c'' where ''c''
2 = ''a''
2 + ''b''
2. When ''e'' < 1 there are two additional
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
''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 projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the complexification of every real circle.
Coordinates ...
, 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 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 the ''x''-axis when the curve has two loops and on the ''y''-axis when the curve has a single loop.
Cassini ovals and orthogonal trajectories
''
Orthogonal trajectories
In mathematics an orthogonal trajectory is a curve, which intersects any curve of a given pencil of (planar) curves ''orthogonally''.
For example, the orthogonal trajectories of a pencil of ''concentric circles'' are the lines through their commo ...
'' of a given
pencil
A pencil () is a writing or drawing implement with a solid pigment core in a protective casing that reduces the risk of core breakage, and keeps it from marking the user's hand.
Pencils create marks by physical abrasion, leaving a trail ...
of curves are curves which intersect all given curves orthogonally. For example the orthogonal trajectories of a pencil of
confocal In geometry, confocal means having the same foci: confocal conic sections.
* For an optical cavity consisting of two mirrors, confocal means that they share their foci. If they are identical mirrors, their radius of curvature, ''R''mirror, equals ' ...
ellipses are the confocal
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
s with the same foci. For Cassini ovals one has:
*The orthogonal trajectories of the Cassini curves with foci
are the
equilateral hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, call ...
s containing
with the same center as the Cassini ovals (see picture).
Proof:
For simplicity one chooses
.
:The Cassini ovals have the equation
::
:The
equilateral hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, call ...
s (their
asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
s are rectangular) containing
with center
can be described by the equation
::
These conic sections have no points with the ''y''-axis in common and intersect the ''x''-axis at
. Their
discriminants 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
the following implicit representation is more convenient
::
A simple calculation shows that
for all
. Hence the Cassini ovals and the hyperbolas intersect orthogonally.
''Remark:''
The image depicting the Cassini ovals and the hyperbolas looks like the
equipotential
In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential), although it can al ...
curves of two equal
point charge
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
s together with the lines of the generated
electrical field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
. But for the potential of two equal point charges one has
. (See
Implicit curve
In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly ''x'' and ''y''. For example, the unit circle is defined by the implicit equation x^2+y^2=1. In general, every implic ...
.)
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
then the foci are variable on the axis
if
,
if
; if the double-loops are described by
then the axes are, respectively,
and
. 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
and
. 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;
and each single-loop is the locus of points
such that the angle
is constant, where
and
is the foot of the perpendicular through
on the line described by
.
Examples
The second
lemniscate of the Mandelbrot set is a Cassini oval defined by the equation
Its foci are at the points ''c'' on the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
that have orbits where every second value of ''z'' is equal to zero, which are the values 0 and −1.
Cassini ovals on tori
Cassini ovals appear as planar sections of
tori, 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
:
and the plane
yields
:
After partially resolving the first bracket one gets the equation
:
which is the equation of a Cassini oval with parameters
and
.
Generalizations
Cassini's method is easy to generalize to curves and surfaces with an arbitrarily many defining points:
*
describes in the planar case an
implicit curve
In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly ''x'' and ''y''. For example, the unit circle is defined by the implicit equation x^2+y^2=1. In general, every implic ...
and in 3-space an
implicit surface
In mathematics, an implicit surface is a surface in Euclidean space defined by an equation
: F(x,y,z)=0.
An ''implicit surface'' is the set of zeros of a function of three variables. ''Implicit'' means that the equation is not solved for o ...
.
Cassini-3p.svg, curve with 3 defining points
Cassinifl-6p-holz.png, surface with 6 defining points
See also
*
Two-center bipolar coordinates
In mathematics, two-center bipolar coordinates is a coordinate system based on two coordinates which give distances from two fixed centers c_1 and c_2. This system is very useful in some scientific applications (e.g. calculating the electric fi ...
References
;Bibliography
*
*
*
*Lawden, D. F., "Families of ovals and their orthogonal trajectories", ''
Mathematical Gazette
''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...
'' 83, November 1999, 410–420.
External links
*
MacTutor description* {{MathWorld , urlname=CassiniOvals , title=Cassini Ovals
Plane curves
Algebraic curves
Spiric sections
Giovanni Domenico Cassini