
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 Cartesian
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 o ...
is a
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic ...
consisting of points that have the same
linear combination of distances from 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 ...
). These curves are named after French mathematician
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathe ...
, who used them in
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultra ...
.
Definition
Let and be fixed points in the plane, and let and denote the
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore o ...
s from these points to a third variable point . Let and be arbitrary
real number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
s. Then the Cartesian oval is 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 ''S'' satisfying . The two ovals formed by the four equations and are closely related; together they form 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 o ...
called the ovals of Descartes.
Special cases
In the equation , when and the resulting shape is an
ellipse. In the
limiting case in which ''P'' and ''Q'' coincide, the ellipse becomes 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 ...
. When
it is a
limaçon
In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. I ...
of Pascal. If
and
the equation gives a branch of a
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, c ...
and thus is not a closed oval.
Polynomial equation
The
set of points satisfying the
quartic polynomial equation
:
where is the distance
between the two fixed
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 ...
and , forms two ovals, the sets of points satisfying the two of the following four equations
:
:
that have real solutions. The two ovals are generally
disjoint, except in the case that or belongs to them. At least one of the two perpendiculars to through points and cuts this quartic curve in four real points; it follows from this that they are necessarily nested, with at least one of the two points and contained in the interiors of both of them.
[.] For a different parametrization and resulting quartic, see Lawrence.
Applications in optics
As Descartes discovered, Cartesian ovals may be used in
lens
A lens is a transmissive optical device which focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses (''elements'' ...
design. By choosing the ratio of distances from and to match the ratio of
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
s in
Snell's law
Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through ...
, and using the
surface of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.
Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending o ...
of one of these ovals, it is possible to design a so-called
aplanatic lens An aplanatic lens is a lens that is free of both spherical and coma aberrations. Aplanatic lenses can be made by combining two or three lens elements. A single-element aplanatic lens is an aspheric lens whose surfaces are surfaces of revolution of a ...
, that has no
spherical aberration
In optics, spherical aberration (SA) is a type of aberration found in optical systems that have elements with spherical surfaces. Lenses and curved mirrors are prime examples, because this shape is easier to manufacture. Light rays that strike a ...
.
Additionally, if a spherical wavefront is refracted through a spherical lens, or reflected from a concave spherical surface, the refracted or reflected wavefront takes on the shape of a Cartesian oval. The
caustic
Caustic most commonly refers to:
* Causticity, a property of various corrosive substances
** Sodium hydroxide, sometimes called ''caustic soda''
** Potassium hydroxide, sometimes called ''caustic potash''
** Calcium oxide, sometimes called ''caust ...
formed by spherical aberration in this case may therefore be described as the
evolute
In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that c ...
of a Cartesian oval.
History
The ovals of Descartes were first studied by René Descartes in 1637, in connection with their applications in optics.
These curves were also studied by
Newton beginning in 1664. One method of drawing certain specific Cartesian ovals, already used by Descartes, is analogous to a standard construction of an
ellipse by a pinned thread. If one stretches a thread from a pin at one
focus
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 ...
to wrap around a pin at a second focus, and ties the free end of the thread to a pen, the path taken by the pen, when the thread is stretched tight, forms a Cartesian oval with a 2:1 ratio between the distances from the two foci.
However, Newton rejected such constructions as insufficiently
rigorous
Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as m ...
. He defined the oval as the solution to a
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
, constructed its
subnormals, and again investigated its optical properties.
The French mathematician
Michel Chasles
Michel Floréal Chasles (; 15 November 1793 – 18 December 1880) was a French mathematician.
Biography
He was born at Épernon in France and studied at the École Polytechnique in Paris under Siméon Denis Poisson. In the War of the Sixth Coa ...
discovered in the 19th century that, if a Cartesian oval is defined by two points and , then there is in general a third point on the same line such that the same oval is also defined by any pair of these three points.
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
rediscovered these curves, generalized them to curves defined by keeping constant the weighted sum of distances from three or more foci, and wrote a paper titled ''Observations on Circumscribed Figures Having a Plurality of Foci, and Radii of Various Proportions''. An account of his results, titled ''On the description of oval curves, and those having a plurality of foci'', was written by
J.D. Forbes and presented to the
Royal Society of Edinburgh
The Royal Society of Edinburgh is Scotland's national academy of science and letters. It is a registered charity that operates on a wholly independent and non-partisan basis and provides public benefit throughout Scotland. It was established i ...
in 1846, when Maxwell was at the young age of 14 (almost 15).
[.]MacTutor History of Mathematics archive
/ref>
See also
*Cassini oval
In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points (foci) is constant. This may be contrasted with an ellipse, for which the ''sum'' of the d ...
*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
External links
*{{mathworld, title=Cartesian Ovals, urlname=CartesianOvals
Benjamin Williamson, An Elementary Treatise on the Differential Calculus, Containing the Theory of Plane Curves (1884)
Algebraic curves