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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, an ellipse is a
plane curve In mathematics, a plane curve is a curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought o ...
surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a
circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

circle
, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off- center, in geometry * Eccentricity (graph theory) of a ...
e, a number ranging from e = 0 (the limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a
parabola In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

parabola
). An ellipse has a simple
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
ic solution for its area, but only approximations for its
perimeter A perimeter is either a path that encompasses/surrounds/outlines a shape (in two dimensions) or its length ( one-dimensional). The perimeter of a circle A circle is a shape consisting of all point (geometry), points in a plane (mathema ...

perimeter
(also known as
circumference In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
), for which integration is required to obtain an exact solution. Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is: : \frac+\frac = 1 . Assuming a \ge b, the foci are (\pm c, 0) for c = \sqrt. The standard parametric equation is: : (x,y) = (a\cos(t),b\sin(t)) \quad \text \quad 0\leq t\leq 2\pi. Ellipses are the closed type of
conic section In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
: a plane curve tracing the intersection of a cone with a
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
(see figure). Ellipses have many similarities with the other two forms of conic sections, parabolas and
hyperbola In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it ...

hyperbola
s, both of which are
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...
and unbounded. An angled
cross section Cross section may refer to: * Cross section (geometry), the intersection of a 3-dimensional body with a plane * Cross section (electronics), a common sample preparation technique in electronics * Cross section (geology), the intersection of a 3-dim ...
of a
cylinder A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditi ...

cylinder
is also an ellipse. An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the
focus FOCUS is a fourth-generation programming language (4GL) computer programming programming language, language and development environment that is used to build database queries. Produced by Information Builders Inc., it was originally developed for d ...
and the distance to the directrix is a constant. This constant ratio is the above-mentioned eccentricity: : e = \frac = \sqrt. Ellipses are common in
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
,
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, phys ...
and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

engineering
. For example, the
orbit In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or po ...

orbit
of each planet in the
Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...

Solar System
is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the
barycenter In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses ...
of the Sunplanet pair). The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are often well described by
ellipsoid An ellipsoid is a surface that may be obtained from a sphere A sphere (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country loca ...

ellipsoid
s. A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under
parallel Parallel may refer to: Computing * Parallel algorithm In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their a ...
or
perspective projection Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, g ...
. The ellipse is also the simplest
Lissajous figure A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations : x=A\sin(at+\delta),\quad y=B\sin(bt), which describe complex harmonic motion. This family of curve In mathematics, a ...
formed when the horizontal and vertical motions are
sinusoid A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in both pure and applied mathemat ...

sinusoid
s with the same frequency: a similar effect leads to
elliptical polarization In electrodynamics, elliptical polarization is the Polarization (waves), polarization of electromagnetic radiation such that the tip of the electric field vector (geometry), vector describes an ellipse in any fixed plane intersecting, and Surface no ...
of light 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 optical instruments, instruments that use or Photodetector, detect it. Optics usually describes t ...

optics
. The name, (, "omission"), was given by
Apollonius of Perga Apollonius of Perga ( grc-gre, Ἀπολλώνιος ὁ Περγαῖος ''Apollonios o Pergeos''; la, Apollonius Pergaeus; ) was an Ancient Greece, Ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the ...
in his ''Conics''.


Definition as locus of points

An ellipse can be defined geometrically as a set or
locus of points In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...
in the Euclidean plane: : Given two fixed points F_1, F_2 called the foci and a distance 2a which is greater than the distance between the foci, the ellipse is the set of points P such that the sum of the distances , PF_1, ,\ , PF_2, is equal to 2a:E = \left\\ . The midpoint C of the line segment joining the foci is called the ''center'' of the ellipse. The line through the foci is called the ''major axis'', and the line perpendicular to it through the center is the ''minor axis''. The major axis intersects the ellipse at two '' vertices'' V_1,V_2, which have distance a to the center. The distance c of the foci to the center is called the ''focal distance'' or linear eccentricity. The quotient e=\tfrac is the ''eccentricity''. The case F_1=F_2 yields a circle and is included as a special type of ellipse. The equation , PF_2, + , PF_1 , = 2a can be viewed in a different way (see figure): : If c_2 is the circle with center F_2 and radius 2a, then the distance of a point P to the circle c_2 equals the distance to the focus F_1: :: , PF_1, =, Pc_2, . c_2 is called the ''circular directrix'' (related to focus F_2) of the ellipse. This property should not be confused with the definition of an ellipse using a directrix line below. Using
Dandelin spheres In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of fi ...

Dandelin spheres
, one can prove that any plane section of a cone with a plane is an ellipse, assuming the plane does not contain the apex and has slope less than that of the lines on the cone.


In Cartesian coordinates


Standard equation

The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, the ''x''-axis is the major axis, and: : the foci are the points F_1 = (c,\, 0),\ F_2=(-c,\, 0), : the vertices are V_1 = (a,\, 0),\ V_2 = (-a,\, 0). For an arbitrary point (x,y) the distance to the focus (c,0) is \sqrt and to the other focus \sqrt. Hence the point (x,\, y) is on the ellipse whenever: :\sqrt + \sqrt = 2a\ . Removing the radicals by suitable squarings and using b^2 = a^2-c^2 (see diagram) produces the standard equation of the ellipse: :\frac + \frac = 1, or, solved for ''y:'' :y = \pm\frac\sqrt = \pm \sqrt. The width and height parameters a,\; b are called the
semi-major and semi-minor axes In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
. The top and bottom points V_3 = (0,\, b),\; V_4 = (0,\, -b) are the ''co-vertices''. The distances from a point (x,\, y) on the ellipse to the left and right foci are a + ex and a - ex. It follows from the equation that the ellipse is ''symmetric'' with respect to the coordinate axes and hence with respect to the origin.


Parameters


Principal axes

Throughout this article, the
semi-major and semi-minor axes In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
are denoted a and b, respectively, i.e. a \ge b > 0 \ . In principle, the canonical ellipse equation \tfrac + \tfrac = 1 may have a < b (and hence the ellipse would be taller than it is wide). This form can be converted to the standard form by transposing the variable names x and y and the parameter names a and b.


Linear eccentricity

This is the distance from the center to a focus: c = \sqrt.


Eccentricity

The eccentricity can be expressed as: : e = \frac = \sqrt, assuming a > b. An ellipse with equal axes (a = b) has zero eccentricity, and is a circle.


Semi-latus rectum

The length of the chord through one focus, perpendicular to the major axis, is called the ''latus rectum''. One half of it is the ''semi-latus rectum'' \ell. A calculation shows: : \ell = \fraca = a \left(1 - e^2\right). The semi-latus rectum \ell is equal to the
radius of curvature In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a co ...

radius of curvature
at the vertices (see section
curvature In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
).


Tangent

An arbitrary line g intersects an ellipse at 0, 1, or 2 points, respectively called an ''exterior line'', ''tangent'' and ''secant''. Through any point of an ellipse there is a unique tangent. The tangent at a point (x_1,\, y_1) of the ellipse \tfrac + \tfrac = 1 has the coordinate equation: :\fracx + \fracy = 1. A vector
parametric equation In mathematics, a parametric equation defines a group of quantities as Function (mathematics), functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that m ...
of the tangent is: : \vec x = \beginx_1 \\ y_1\end + s\begin \;\! -y_1 a^2 \\ \;\ \ \ x_1 b^2 \end\ with \ s \in \mathbb\ . Proof: Let (x_1,\, y_1) be a point on an ellipse and \vec = \beginx_1 \\ y_1\end + s\beginu \\ v\end be the equation of any line g containing (x_1,\, y_1). Inserting the line's equation into the ellipse equation and respecting \frac + \frac = 1 yields: : \frac + \frac = 1\ \quad\Longrightarrow\quad 2s\left(\frac + \frac\right) + s^2\left(\frac + \frac\right) = 0\ . : There are then cases: :# \fracu + \fracv = 0. Then line g and the ellipse have only point (x_1,\, y_1) in common, and g is a tangent. The tangent direction has perpendicular vector \begin\frac & \frac\end, so the tangent line has equation \fracx + \tfracy = k for some k. Because (x_1,\, y_1) is on the tangent and the ellipse, one obtains k = 1. :# \fracu + \fracv \ne 0. Then line g has a second point in common with the ellipse, and is a secant. Using (1) one finds that \begin -y_1a^2 & x_1b^2 \end is a tangent vector at point (x_1,\, y_1), which proves the vector equation. If (x_1, y_1) and (u, v) are two points of the ellipse such that \frac + \tfrac = 0, then the points lie on two ''conjugate diameters'' (see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
). (If a = b, the ellipse is a circle and "conjugate" means "orthogonal".)


Shifted ellipse

If the standard ellipse is shifted to have center \left(x_\circ,\, y_\circ\right), its equation is : \frac + \frac = 1 \ . The axes are still parallel to the x- and y-axes.


General ellipse

In
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches ...
, the ellipse is defined as a
quadric In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimension thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics Physics is the natural science that s ...
: the set of points (X,\, Y) of the
Cartesian plane A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...

Cartesian plane
that, in non-degenerate cases, satisfy the implicit equation : AX^2 + B X Y + C Y^2 + D X + E Y + F = 0 provided B^2 - 4AC < 0. To distinguish the
degenerate cases Degeneracy may refer to: Science * Codon degeneracy * Degeneracy (biology), the ability of elements that are structurally different to perform the same function or yield the same output * Degeneration (medical) ** Degenerative diseaseDegener ...
from the non-degenerate case, let ''∆'' be the
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

determinant
:\Delta = \begin A & \fracB & \fracD \\ \fracB & C & \fracE \\ \fracD & \fracE & F \end = \left(AC - \frac\right) F + \frac - \frac - \frac. Then the ellipse is a non-degenerate real ellipse if and only if ''C∆'' < 0. If ''C∆'' > 0, we have an imaginary ellipse, and if ''∆'' = 0, we have a point ellipse.Lawrence, J. Dennis, ''A Catalog of Special Plane Curves'', Dover Publ., 1972. The general equation's coefficients can be obtained from known semi-major axis a, semi-minor axis b, center coordinates \left(x_\circ,\, y_\circ\right), and rotation angle \theta (the angle from the positive horizontal axis to the ellipse's major axis) using the formulae: :\begin A &= a^2 \sin^2\theta + b^2 \cos^2\theta \\ B &= 2\left(b^2 - a^2\right) \sin\theta \cos\theta \\ C &= a^2 \cos^2\theta + b^2 \sin^2\theta \\ D &= -2A x_\circ - B y_\circ \\ E &= - B x_\circ - 2C y_\circ \\ F &= A x_\circ^2 + B x_\circ y_\circ + C y_\circ^2 - a^2 b^2. \end These expressions can be derived from the canonical equation \tfrac + \tfrac = 1 by an affine transformation of the coordinates (x,\, y): :\begin x &= \left(X - x_\circ\right) \cos\theta + \left(Y - y_\circ\right) \sin\theta \\ y &= -\left(X - x_\circ\right) \sin\theta + \left(Y - y_\circ\right) \cos\theta. \end Conversely, the canonical form parameters can be obtained from the general form coefficients by the equations: :\begin a, b &= \frac \\ x_\circ &= \frac \\
pt
pt
y_\circ &= \frac \\
pt
pt
\theta &= \begin \arctan\left(\frac\left(C - A - \sqrt\right)\right) & \text B \ne 0 \\ 0 & \text B = 0,\ A < C \\ 90^\circ & \text B = 0,\ A > C \\ \end \end


Parametric representation


Standard parametric representation

Using
trigonometric function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

trigonometric function
s, a parametric representation of the standard ellipse \tfrac+\tfrac = 1 is: : (x,\, y) = (a \cos t,\, b \sin t),\ 0 \le t < 2\pi\ . The parameter ''t'' (called the ''
eccentric anomalyIn orbital mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit In celestial mechanics, a Kepler orbit (or Keplerian orbit, named after the German astronomer ...
'' in astronomy) is not the angle of (x(t),y(t)) with the ''x''-axis, but has a geometric meaning due to Philippe de La Hire (see '' Drawing ellipses'' below).


Rational representation

With the substitution u = \tan\left(\frac\right) and trigonometric formulae one obtains :\cos t = \frac\ ,\quad \sin t = \frac and the ''rational'' parametric equation of an ellipse : \begin x(u) &= a\frac \\ y(u) &= \frac \end\;,\quad -\infty < u < \infty\;, which covers any point of the ellipse \tfrac + \tfrac = 1 except the left vertex (-a,\, 0). For u \in ,\, 1 this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing u. The left vertex is the limit \lim_ (x(u),\, y(u)) = (-a,\, 0)\;. Rational representations of conic sections are commonly used in
computer-aided design Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve com ...
(see Bezier curve).


Tangent slope as parameter

A parametric representation, which uses the slope m of the tangent at a point of the ellipse can be obtained from the derivative of the standard representation \vec x(t) = (a \cos t,\, b \sin t)^\mathsf: :\vec x'(t) = (-a\sin t,\, b\cos t)^\mathsf \quad \rightarrow \quad m = -\frac\cot t\quad \rightarrow \quad \cot t = -\frac. With help of trigonometric formulae one obtains: :\cos t = \frac = \frac\ ,\quad\quad \sin t = \frac = \frac. Replacing \cos t and \sin t of the standard representation yields: : \vec c_\pm(m) = \left(-\frac,\;\frac\right),\, m \in \R. Here m is the slope of the tangent at the corresponding ellipse point, \vec c_+ is the upper and \vec c_- the lower half of the ellipse. The vertices(\pm a,\, 0), having vertical tangents, are not covered by the representation. The equation of the tangent at point \vec c_\pm(m) has the form y = mx + n. The still unknown n can be determined by inserting the coordinates of the corresponding ellipse point \vec c_\pm(m): : y = mx \pm\sqrt\; . This description of the tangents of an ellipse is an essential tool for the determination of the orthoptic of an ellipse. The orthoptic article contains another proof, without differential calculus and trigonometric formulae.


General ellipse

Another definition of an ellipse uses
affine transformation In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...
s: : Any ''ellipse'' is an affine image of the unit circle with equation x^2 + y^2 = 1. ;Parametric representation An affine transformation of the Euclidean plane has the form \vec x \mapsto \vec f\!_0 + A\vec x, where A is a regular
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
(with non-zero
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

determinant
) and \vec f\!_0 is an arbitrary vector. If \vec f\!_1, \vec f\!_2 are the column vectors of the matrix A, the unit circle (\cos(t), \sin(t)), 0 \leq t \leq 2\pi, is mapped onto the ellipse: : \vec x = \vec p(t) = \vec f\!_0 + \vec f\!_1 \cos t + \vec f\!_2 \sin t \ . Here \vec f\!_0 is the center and \vec f\!_1,\; \vec f\!_2 are the directions of two
conjugate diameterIn geometry, two diameters of a conic section are said to be conjugate if each chord (geometry), chord parallel (geometry), parallel to one diameter is bisection, bisected by the other diameter. For example, two diameters of a circle are conjugate i ...
s, in general not perpendicular. ;Vertices The four vertices of the ellipse are \vec p(t_0),\;\vec p\left(t_0 \pm \tfrac\right),\; \vec p\left(t_0 + \pi\right), for a parameter t = t_0 defined by: : \cot (2t_0) = \frac. (If \vec f\!_1 \cdot \vec f\!_2 = 0, then t_0 = 0.) This is derived as follows. The tangent vector at point \vec p(t) is: : \vec p\,'(t) = -\vec f\!_1\sin t + \vec f\!_2\cos t \ . At a vertex parameter t = t_0, the tangent is perpendicular to the major/minor axes, so: : 0 = \vec p'(t) \cdot \left(\vec p(t) -\vec f\!_0\right) = \left(-\vec f\!_1\sin t + \vec f\!_2\cos t\right) \cdot \left(\vec f\!_1 \cos t + \vec f\!_2 \sin t\right). Expanding and applying the identities \; \cos^2 t -\sin^2 t=\cos 2t,\ \ 2\sin t \cos t = \sin 2t\; gives the equation for t = t_0\; . ;Area From Apollonios theorem (see below) one obtains:
The area of an ellipse \;\vec x = \vec f_0 +\vec f_1 \cos t +\vec f_2 \sin t\; is :A=\pi, \det(\vec f_1, \vec f_2), \ . ;Semiaxes With the abbreviations \; M=\vec f_1^2+\vec f_2^2, \ N = \left, \det(\vec f_1,\vec f_2)\ the statements of Apollonios's theorem can be written as: :a^2+b^2=M, \quad ab=N \ . Solving this nonlinear system for a,b yields the semiaxes: :a=\frac(\sqrt+\sqrt) :b=\frac(\sqrt-\sqrt)\ . ;Implicit representation Solving the parametric representation for \; \cos t,\sin t\; by
Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical s ...
and using \;\cos^2t+\sin^2t -1=0\; , one obtains the implicit representation :\det(\vec x\!-\!\vec f\!_0,\vec f\!_2)^2+\det(\vec f\!_1,\vec x\!-\!\vec f\!_0)^2-\det(\vec f\!_1,\vec f\!_2)^2=0. Conversely: If the
equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
:x^2+2cxy+d^2y^2-e^2=0\ , with \; d^2-c^2 >0 \; , of an ellipse centered at the origin is given, then the two vectors :\vec f_1=,\quad \vec f_2=\frac\ point to two conjugate points and the tools developed above are applicable. ''Example'': For the ellipse with equation \;x^2+2xy+3y^2-1=0\; the vectors are :\vec f_1=,\quad \vec f_2=\frac . ;Rotated Standard ellipse For \vec f_0= ,\;\vec f_1= a ,\;\vec f_2= b one obtains a parametric representation of the standard ellipse rotated by angle \theta: :x=x_\theta(t)=a\cos\theta\cos t-b\sin\theta\sin t\ , :y=y_\theta(t)=a\sin\theta\cos t+b\cos\theta\sin t\ . ;Ellipse in space The definition of an ellipse in this section gives a parametric representation of an arbitrary ellipse, even in space, if one allows \vec f\!_0, \vec f\!_1, \vec f\!_2 to be vectors in space.


Polar forms


Polar form relative to center

In
polar coordinates In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

polar coordinates
, with the origin at the center of the ellipse and with the angular coordinate \theta measured from the major axis, the ellipse's equation is : r(\theta) = \frac=\frac


Polar form relative to focus

If instead we use polar coordinates with the origin at one focus, with the angular coordinate \theta = 0 still measured from the major axis, the ellipse's equation is : r(\theta)=\frac where the sign in the denominator is negative if the reference direction \theta = 0 points towards the center (as illustrated on the right), and positive if that direction points away from the center. In the slightly more general case of an ellipse with one focus at the origin and the other focus at angular coordinate \phi, the polar form is :r(\theta)=\frac. The angle \theta in these formulas is called the
true anomaly In celestial mechanics Celestial mechanics is the branch of astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, ...
of the point. The numerator of these formulas is the
semi-latus rectum In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
\ell=a (1-e^2).


Eccentricity and the directrix property

Each of the two lines parallel to the minor axis, and at a distance of d = \frac = \frac from it, is called a ''directrix'' of the ellipse (see diagram). : For an arbitrary point P of the ellipse, the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity: :: \frac = \frac = e = \frac\ . The proof for the pair F_1, l_1 follows from the fact that \left, PF_1\^2 = (x - c)^2 + y^2,\ \left, Pl_1\^2 = \left(x - \tfrac\right)^2 and y^2 = b^2 - \tfracx^2 satisfy the equation :\left, PF_1\^2 - \frac\left, Pl_1\^2 = 0\ . The second case is proven analogously. The converse is also true and can be used to define an ellipse (in a manner similar to the definition of a parabola): : For any point F (focus), any line l (directrix) not through F, and any real number e with 0 < e < 1, the ellipse is the locus of points for which the quotient of the distances to the point and to the line is e, that is: :: E = \left\. The extension to e = 0, which is the eccentricity of a circle, is not allowed in this context in the Euclidean plane. However, one may consider the directrix of a circle to be the
line at infinity In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

line at infinity
(with \left, PF\ being the radius of the circle) in the
projective plane In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. (The choice e = 1 yields a parabola, and if e > 1, a hyperbola.) ;Proof Let F = (f,\, 0),\ e > 0, and assume (0,\, 0) is a point on the curve. The directrix l has equation x = -\tfrac. With P = (x,\, y), the relation , PF, ^2 = e^2, Pl, ^2 produces the equations :(x - f)^2 + y^2 = e^2\left(x + \frac\right)^2 = (ex + f)^2 and x^2\left(e^2 - 1\right) + 2xf(1 + e) - y^2 = 0. The substitution p = f(1 + e) yields : x^2\left(e^2 - 1\right) + 2px - y^2 = 0. This is the equation of an ''ellipse'' (e < 1), or a ''parabola'' (e = 1), or a ''hyperbola'' (e > 1). All of these non-degenerate conics have, in common, the origin as a vertex (see diagram). If e < 1, introduce new parameters a,\, b so that 1 - e^2 = \tfrac, \text\ p = \tfrac, and then the equation above becomes :\frac + \frac = 1\ , which is the equation of an ellipse with center (a,\, 0), the ''x''-axis as major axis, and the major/minor semi axis a,\, b. ;Construction of a directrix Because of c\cdot\tfrac=a^2 point L_1 of directrix l_1 (see diagram) and focus F_1 are inverse with respect to the
circle inversion In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...
at circle x^2+y^2=a^2 (in diagram green). Hence L_1 can be constructed as shown in the diagram. Directrix l_1 is the perpendicular to the main axis at point L_1. ;General ellipse If the focus is F = \left(f_1,\, f_2\right) and the directrix ux + vy + w = 0, one obtains the equation :\left(x - f_1\right)^2 + \left(y - f_2\right)^2 = e^2 \frac\ . (The right side of the equation uses the
Hesse normal form The Hesse normal form named after Otto Hesse, is an equation used in analytic geometry, and describes a line (geometry), line in \mathbb^2 or a plane (geometry), plane in Euclidean space \mathbb^3 or a hyperplane in higher dimensions.John Vince: '' ...

Hesse normal form
of a line to calculate the distance , Pl, .)


Focus-to-focus reflection property

An ellipse possesses the following property: : The normal at a point P bisects the angle between the lines \overline,\, \overline. ; Proof Because the tangent is perpendicular to the normal, the statement is true for the tangent and the supplementary angle of the angle between the lines to the foci (see diagram), too. Let L be the point on the line \overline with the distance 2a to the focus F_2, a is the semi-major axis of the ellipse. Let line w be the bisector of the supplementary angle to the angle between the lines \overline,\, \overline. In order to prove that w is the tangent line at point P, one checks that any point Q on line w which is different from P cannot be on the ellipse. Hence w has only point P in common with the ellipse and is, therefore, the tangent at point P. From the diagram and the
triangle inequality In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

triangle inequality
one recognizes that 2a = \left, LF_2\ < \left, QF_2\ + \left, QL\ = \left, QF_2\ + \left, QF_1\ holds, which means: \left, QF_2\ + \left, QF_1\ > 2a . The equality \left, QL\ = \left, QF_1\ is true from the
Angle bisector theorem In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that Bisection, bisects the opposite angle. It equates their relative lengths to the relative lengt ...

Angle bisector theorem
because \frac=\frac and \left, PL\=\left, PF_1\ . But if Q is a point of the ellipse, the sum should be 2a. ; Application The rays from one focus are reflected by the ellipse to the second focus. This property has optical and acoustic applications similar to the reflective property of a parabola (see
whispering gallery The Whispering Gallery of St Paul's Cathedral, London A whispering gallery is usually a circular, hemispherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-di ...
).


Conjugate diameters


Definition of conjugate diameters

A circle has the following property: : The midpoints of parallel chords lie on a diameter. An affine transformation preserves parallelism and midpoints of line segments, so this property is true for any ellipse. (Note that the parallel chords and the diameter are no longer orthogonal.) ; Definition: Two diameters d_1,\, d_2 of an ellipse are ''conjugate'' if the midpoints of chords parallel to d_1 lie on d_2\ . From the diagram one finds: : Two diameters \overline,\, \overline of an ellipse are conjugate whenever the tangents at P_1 and Q_1 are parallel to \overline. Conjugate diameters in an ellipse generalize orthogonal diameters in a circle. In the parametric equation for a general ellipse given above, : \vec x = \vec p(t) = \vec f\!_0 +\vec f\!_1 \cos t + \vec f\!_2 \sin t, any pair of points \vec p(t),\ \vec p(t + \pi) belong to a diameter, and the pair \vec p\left(t + \tfrac\right),\ \vec p\left(t - \tfrac\right) belong to its conjugate diameter.


Theorem of Apollonios on conjugate diameters

For an ellipse with semi-axes a,\, b the following is true: : Let c_1 and c_2 be halves of two conjugate diameters (see diagram) then :# c_1^2 + c_2^2 = a^2 + b^2. :# The ''triangle'' O,P_1,P_2 with sides c_1,\, c_2 (see diagram) has the constant area A_\Delta = \fracab, which can be expressed by A_\Delta=\tfrac 1 2 c_2d_1=\tfrac 1 2 c_1c_2\sin\alpha, too. d_1 is the altitude of point P_1 and \alpha the angle between the half diameters. Hence the area of the ellipse (see section metric properties) can be written as A_=\pi ab=\pi c_2d_1=\pi c_1c_2\sin\alpha. :# The parallelogram of tangents adjacent to the given conjugate diameters has the \text_ = 4ab\ . ; Proof: Let the ellipse be in the canonical form with parametric equation : \vec p(t) = (a\cos t,\, b\sin t). The two points \vec c_1 = \vec p(t),\ \vec c_2 = \vec p\left(t + \frac\right) are on conjugate diameters (see previous section). From trigonometric formulae one obtains \vec c_2 = (-a\sin t,\, b\cos t)^\mathsf and : \left, \vec c_1\^2 + \left, \vec c_2\^2 = \cdots = a^2 + b^2\ . The area of the triangle generated by \vec c_1,\, \vec c_2 is : A_\Delta = \frac\det\left(\vec c_1,\, \vec c_2\right) = \cdots = \fracab and from the diagram it can be seen that the area of the parallelogram is 8 times that of A_\Delta. Hence : \text_ = 4ab\ .


Orthogonal tangents

For the ellipse \tfrac+\tfrac=1 the intersection points of ''orthogonal'' tangents lie on the circle x^2+y^2=a^2+b^2. This circle is called ''orthoptic'' or
director circle upright=1.35, An ellipse, its minimum bounding box, and its director circle. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the ...
of the ellipse (not to be confused with the circular directrix defined above).


Drawing ellipses

Ellipses appear in descriptive geometry as images (parallel or central projection) of circles. There exist various tools to draw an ellipse. Computers provide the fastest and most accurate method for drawing an ellipse. However, technical tools ('' ellipsographs'') to draw an ellipse without a computer exist. The principle of ellipsographs were known to Greek mathematicians such as
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

Archimedes
and Proklos. If there is no ellipsograph available, one can draw an ellipse using an approximation by the four osculating circles at the vertices. For any method described below, knowledge of the axes and the semi-axes is necessary (or equivalently: the foci and the semi-major axis). If this presumption is not fulfilled one has to know at least two conjugate diameters. With help of
Rytz's construction The Rytz’s axis construction is a basic method of descriptive Geometry to find the axes, the semi-major axis and semi-minor axis and the vertices of an ellipse, starting from two conjugate diameters, conjugated half-diameters. If the center and t ...
the axes and semi-axes can be retrieved.


de La Hire's point construction

The following construction of single points of an ellipse is due to de La Hire. It is based on the standard parametric representation (a\cos t,\, b\sin t) of an ellipse: # Draw the two ''circles'' centered at the center of the ellipse with radii a,b and the axes of the ellipse. # Draw a ''line through the center'', which intersects the two circles at point A and B, respectively. # Draw a ''line'' through A that is parallel to the minor axis and a ''line'' through B that is parallel to the major axis. These lines meet at an ellipse point (see diagram). # Repeat steps (2) and (3) with different lines through the center. Elliko-sk.svg, de La Hire's method Parametric ellipse.gif, Animation of the method


Pins-and-string method

The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two
drawing pin A drawing pin (British English) or thumb tack (North American English) is a short nail Nail or Nails may refer to: In biology * Nail (anatomy), toughened protective protein-keratin (known as alpha-keratin, also found in hair) at the end of ...

drawing pin
s, a length of string, and a pencil. In this method, pins are pushed into the paper at two points, which become the ellipse's foci. A string is tied at each end to the two pins; its length after tying is 2a. The tip of the pencil then traces an ellipse if it is moved while keeping the string taut. Using two pegs and a rope, gardeners use this procedure to outline an elliptical flower bed—thus it is called the ''gardener's ellipse''. A similar method for drawing confocal ellipses with a ''closed'' string is due to the Irish bishop .


Paper strip methods

The two following methods rely on the parametric representation (see section ''
parametric representation In mathematics, a parametric equation defines a group of quantities as Function (mathematics), functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that m ...
'', above): : (a\cos t,\, b\sin t) This representation can be modeled technically by two simple methods. In both cases center, the axes and semi axes a,\, b have to be known. ;Method 1 The first method starts with : a strip of paper of length a + b. The point, where the semi axes meet is marked by P. If the strip slides with both ends on the axes of the desired ellipse, then point P traces the ellipse. For the proof one shows that point P has the parametric representation (a\cos t,\, b\sin t), where parameter t is the angle of the slope of the paper strip. A technical realization of the motion of the paper strip can be achieved by a
Tusi couple The Tusi couple is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. I ...

Tusi couple
(see animation). The device is able to draw any ellipse with a ''fixed'' sum a + b, which is the radius of the large circle. This restriction may be a disadvantage in real life. More flexible is the second paper strip method. Elliko-pap1.svg, Ellipse construction: paper strip method 1 Tusi couple vs Paper strip plus Ellipses horizontal.gif, Ellipses with Tusi couple. Two examples: red and cyan. A variation of the paper strip method 1 uses the observation that the midpoint N of the paper strip is moving on the circle with center M (of the ellipse) and radius \tfrac. Hence, the paperstrip can be cut at point N into halves, connected again by a joint at N and the sliding end K fixed at the center M (see diagram). After this operation the movement of the unchanged half of the paperstrip is unchanged. This variation requires only one sliding shoe. Ellipse-papsm-1a.svg, Variation of the paper strip method 1 Ellipses with SliderCrank inner Ellipses.gif, Animation of the variation of the paper strip method 1 ; Method 2: The second method starts with : a strip of paper of length a. One marks the point, which divides the strip into two substrips of length b and a - b. The strip is positioned onto the axes as described in the diagram. Then the free end of the strip traces an ellipse, while the strip is moved. For the proof, one recognizes that the tracing point can be described parametrically by (a\cos t,\, b\sin t), where parameter t is the angle of slope of the paper strip. This method is the base for several ''ellipsographs'' (see section below). Similar to the variation of the paper strip method 1 a ''variation of the paper strip method 2'' can be established (see diagram) by cutting the part between the axes into halves. File:Archimedes Trammel.gif, (principle) File:L-Ellipsenzirkel.png, Ellipsograph due to
Benjamin Bramer Benjamin Bramer (15 February 1588 – 17 March 1652) was a German mathematician, architect, inventor, and adviser. Early life Bramer was born on 15 February 1588 in Felsberg, Germany to a Protestant minister father. The minister later died when Br ...
File:Ellipses with SliderCrank Ellipses at Slider Side.gif, Variation of the paper strip method 2
Most ellipsograph
drafting Drafting or draughting may refer to: * Campdrafting, an Australian equestrian sport * Drafting (aerodynamics), slipstreaming * Drafting (writing), writing something that is likely to be amended * Technical drawing, the act and discipline of compos ...

drafting
instruments are based on the second paperstrip method.


Approximation by osculating circles

From ''Metric properties'' below, one obtains: * The radius of curvature at the vertices V_1,\, V_2 is: \tfrac * The radius of curvature at the co-vertices V_3,\, V_4 is: \tfrac\ . The diagram shows an easy way to find the centers of curvature C_1 = \left(a - \tfrac, 0\right),\, C_3 = \left(0, b - \tfrac\right) at vertex V_1 and co-vertex V_3, respectively: # mark the auxiliary point H = (a,\, b) and draw the line segment V_1 V_3\ , # draw the line through H, which is perpendicular to the line V_1 V_3\ , # the intersection points of this line with the axes are the centers of the osculating circles. (proof: simple calculation.) The centers for the remaining vertices are found by symmetry. With help of a
French curve A French curve is a template usually made from metal A metal (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in S ...

French curve
one draws a curve, which has smooth contact to the
osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve in ...

osculating circle
s.


Steiner generation

The following method to construct single points of an ellipse relies on the Steiner generation of a conic section: : Given two
pencils A pencil is a Writing implement, writing or drawing implement with a solid pigment core encased in a sleeve, barrel, or shaft that prevents breaking the core or marking a user's hand. Pencils create marks by physical abrasion (mechanical), ...
B(U),\, B(V) of lines at two points U,\, V (all lines containing U and V, respectively) and a projective but not perspective mapping \pi of B(U) onto B(V), then the intersection points of corresponding lines form a non-degenerate projective conic section. For the generation of points of the ellipse \tfrac + \tfrac = 1 one uses the pencils at the vertices V_1,\, V_2. Let P = (0,\, b) be an upper co-vertex of the ellipse and A = (-a,\, 2b),\, B = (a,\,2b). P is the center of the rectangle V_1,\, V_2,\, B,\, A. The side \overline of the rectangle is divided into n equal spaced line segments and this division is projected parallel with the diagonal AV_2 as direction onto the line segment \overline and assign the division as shown in the diagram. The parallel projection together with the reverse of the orientation is part of the projective mapping between the pencils at V_1 and V_2 needed. The intersection points of any two related lines V_1 B_i and V_2 A_i are points of the uniquely defined ellipse. With help of the points C_1,\, \dotsc the points of the second quarter of the ellipse can be determined. Analogously one obtains the points of the lower half of the ellipse. Steiner generation can also be defined for hyperbolas and parabolas. It is sometimes called a ''parallelogram method'' because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.


As hypotrochoid

The ellipse is a special case of the
hypotrochoid A hypotrochoid is a roulette Roulette is a casino game named after the French word meaning ''little wheel''. In the game, players may choose to place bets on either a single number, various groupings of numbers, the colors red or black, wheth ...

hypotrochoid
when R = 2r, as shown in the adjacent image. The special case of a moving circle with radius r inside a circle with radius R = 2r is called a
Tusi couple The Tusi couple is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. I ...

Tusi couple
.


Inscribed angles and three-point form


Circles

A circle with equation \left(x - x_\circ\right)^2 + \left(y - y_\circ\right)^2 = r^2 is uniquely determined by three points \left(x_1, y_1\right),\; \left(x_2,\,y_2\right),\; \left(x_3,\, y_3\right) not on a line. A simple way to determine the parameters x_\circ,y_\circ,r uses the ''
inscribed angle theorem In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space tha ...

inscribed angle theorem
'' for circles: : For four points P_i = \left(x_i,\, y_i\right),\ i = 1,\, 2,\, 3,\, 4,\, (see diagram) the following statement is true: : The four points are on a circle if and only if the angles at P_3 and P_4 are equal. Usually one measures inscribed angles by a degree or radian ''θ,'' but here the following measurement is more convenient: : In order to measure the angle between two lines with equations y = m_1x + d_1,\ y = m_2x + d_2,\ m_1 \ne m_2, one uses the quotient: :: \frac = \cot\theta\ .


Inscribed angle theorem for circles

For four points P_i = \left(x_i,\, y_i\right),\ i = 1,\, 2,\, 3,\, 4,\, no three of them on a line, we have the following (see diagram): : The four points are on a circle, if and only if the angles at P_3 and P_4 are equal. In terms of the angle measurement above, this means: :: \frac = \frac . At first the measure is available only for chords not parallel to the y-axis, but the final formula works for any chord.


Three-point form of circle equation

: As a consequence, one obtains an equation for the circle determined by three non-colinear points P_i = \left(x_i,\, y_i\right): :: \frac = \frac . For example, for P_1 = (2,\, 0),\; P_2 = (0,\, 1),\; P_3 = (0,\,0) the three-point equation is: : \frac = 0, which can be rearranged to (x - 1)^2 + \left(y - \tfrac\right)^2 = \tfrac\ . Using vectors,
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
s and
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

determinant
s this formula can be arranged more clearly, letting \vec x = (x,\, y): : \frac = \frac . The center of the circle \left(x_\circ,\, y_\circ\right) satisfies: : \begin 1 & \frac \\ \frac & 1 \end \begin x_\circ \\ y_\circ \end = \begin \frac \\ \frac \end. The radius is the distance between any of the three points and the center. : r = \sqrt = \sqrt = \sqrt.


Ellipses

This section, we consider the family of ellipses defined by equations \tfrac + \tfrac = 1 with a ''fixed'' eccentricity e. It is convenient to use the parameter: : = \frac = \frac, and to write the ellipse equation as: : \left(x - x_\circ\right)^2 + \, \left(y - y_\circ\right)^2 = a^2, where ''q'' is fixed and x_\circ,\, y_\circ,\, a vary over the real numbers. (Such ellipses have their axes parallel to the coordinate axes: if q < 1, the major axis is parallel to the ''x''-axis; if q > 1, it is parallel to the ''y''-axis.) Like a circle, such an ellipse is determined by three points not on a line. For this family of ellipses, one introduces the following
q-analog In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
angle measure, which is ''not'' a function of the usual angle measure ''θ'': : In order to measure an angle between two lines with equations y = m_1x + d_1,\ y = m_2x + d_2,\ m_1 \ne m_2 one uses the quotient: :: \frac\ .


Inscribed angle theorem for ellipses

: Given four points P_i = \left(x_i,\, y_i\right),\ i = 1,\, 2,\, 3,\, 4, no three of them on a line (see diagram). : The four points are on an ellipse with equation (x - x_\circ)^2 + \, (y - y_\circ)^2 = a^2 if and only if the angles at P_3 and P_4 are equal in the sense of the measurement above—that is, if :: \frac = \frac \ . At first the measure is available only for chords which are not parallel to the y-axis. But the final formula works for any chord. The proof follows from a straightforward calculation. For the direction of proof given that the points are on an ellipse, one can assume that the center of the ellipse is the origin.


Three-point form of ellipse equation

: A consequence, one obtains an equation for the ellipse determined by three non-colinear points P_i = \left(x_i,\, y_i\right): :: \frac = \frac \ . For example, for P_1 = (2,\, 0),\; P_2 = (0,\,1),\; P_3 = (0,\, 0) and q = 4 one obtains the three-point form : \frac = 0 and after conversion \frac + \frac = 1. Analogously to the circle case, the equation can be written more clearly using vectors: : \frac = \frac , where * is the modified
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
\vec u*\vec v = u_x v_x + \,u_y v_y.


Pole-polar relation

Any ellipse can be described in a suitable coordinate system by an equation \tfrac + \tfrac = 1. The equation of the tangent at a point P_1 = \left(x_1,\, y_1\right) of the ellipse is \tfrac + \tfrac = 1. If one allows point P_1 = \left(x_1,\, y_1\right) to be an arbitrary point different from the origin, then : point P_1 = \left(x_1,\, y_1\right) \neq (0,\, 0) is mapped onto the line \tfrac + \tfrac = 1, not through the center of the ellipse. This relation between points and lines is a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijection
. The
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
maps * line y = mx + d,\ d \ne 0 onto the point \left(-\tfrac,\, \tfrac\right) and * line x = c,\ c \ne 0 onto the point \left(\tfrac,\, 0\right). Such a relation between points and lines generated by a conic is called '''' or ''polarity''. The pole is the point; the polar the line. By calculation one can confirm the following properties of the pole-polar relation of the ellipse: * For a point (pole) ''on'' the ellipse, the polar is the tangent at this point (see diagram: P_1,\, p_1). * For a pole P ''outside'' the ellipse, the intersection points of its polar with the ellipse are the tangency points of the two tangents passing P (see diagram: P_2,\, p_2). * For a point ''within'' the ellipse, the polar has no point with the ellipse in common (see diagram: F_1,\, l_1). # The intersection point of two polars is the pole of the line through their poles. # The foci (c,\, 0) and (-c,\, 0), respectively, and the directrices x = \tfrac and x = -\tfrac, respectively, belong to pairs of pole and polar. Because they are even polar pairs with respect to the circle x^2+y^2=a^2, the directrices can be constructed by compass and straightedge (see
Inversive geometry In geometry, inversive geometry is the study of ''Inversion in a sphere, inversion'', a transformation of the Euclidean plane that maps circles or Line (geometry), lines to other circles or lines and that preserves the angles between crossing curv ...
). Pole-polar relations exist for hyperbolas and parabolas as well.


Metric properties

All metric properties given below refer to an ellipse with equation except for the section on the area enclosed by a tilted ellipse, where the generalized form of Eq.() will be given.


Area

The
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

area
A_\text enclosed by an ellipse is: where a and b are the lengths of the semi-major and semi-minor axes, respectively. The area formula \pi a b is intuitive: start with a circle of radius b (so its area is \pi b^2) and stretch it by a factor a/b to make an ellipse. This scales the area by the same factor: \pi b^2(a/b) = \pi a b. It is also easy to rigorously prove the area formula using integration as follows. Equation () can be rewritten as y(x)= b \sqrt. For x\in a,a this curve is the top half of the ellipse. So twice the integral of y(x) over the interval a,a/math> will be the area of the ellipse: : \begin A_\text &= \int_^a 2b\sqrt\,dx\\ &= \frac ba \int_^a 2\sqrt\,dx. \end The second integral is the area of a circle of radius a, that is, \pi a^2. So : A_\text = \frac\pi a^2 = \pi ab. An ellipse defined implicitly by Ax^2+ Bxy + Cy^2 = 1 has area 2\pi / \sqrt. The area can also be expressed in terms of eccentricity and the length of the semi-major axis as a^2\pi\sqrt (obtained by solving for
flattening Flattening is a measure of the compression of a circle A circle is a shape A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water drople ...
, then computing the semi-minor axis). So far we have dealt with ''erect'' ellipses, whose major and minor axes are parallel to the x and y axes. However, some applications require ''tilted'' ellipses. In charged-particle beam optics, for instance, the enclosed area of an erect or tilted ellipse is an important property of the beam, its ''emittance''. In this case a simple formula still applies, namely where y_, x_ are intercepts and x_, y_ are maximum values. It follows directly from Appolonio's theorem.


Circumference

The circumference C of an ellipse is: : C \,=\, 4a\int_0^\sqrt \ d\theta \,=\, 4 a \,E(e) where again a is the length of the semi-major axis, e=\sqrt is the eccentricity, and the function E is the complete elliptic integral of the second kind, : E(e) \,=\, \int_0^\sqrt \ d\theta which is in general not an
elementary function In mathematics, an elementary function is a function (mathematics), function of a single variable (mathematics), variable (typically Function of a real variable, real or Complex analysis#Complex functions, complex) that is defined as taking addit ...
. The circumference of the ellipse may be evaluated in terms of E(e) using Gauss's arithmetic-geometric mean; this is a quadratically converging iterative method. The exact
infinite series In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
is: :\begin C &= 2\pi a \left
right Rights are legal Law is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is desc ...

right
\\ &= 2\pi a \left - \sum_^\infty \left(\frac\right)^2 \frac\right\\ &= -2\pi a \sum_^\infty \left(\frac\right)^2 \frac, \end where n!! is the double factorial (extended to negative odd integers by the recurrence relation (2n-1)!! = (2n+1)!!/(2n+1), for n \le 0). This series converges, but by expanding in terms of h = (a-b)^2 / (a+b)^2, James Ivory (mathematician), James Ivory and Bessel derived an expression that converges much more rapidly: :\begin C &= \pi (a+b) \sum_^\infty \left(\frac\right)^2 h^n \\ &= \pi (a+b) \left[1 + \frac + \sum_^\infty \left(\frac\right)^2 h^n\right] \\ &= \pi (a+b) \left[1 + \sum_^\infty \left(\frac\right)^2 \frac\right]. \end Srinivasa Ramanujan gives two close approximations for the circumference in §16 of "Modular Equations and Approximations to \pi"; they are : C \approx \pi \biggl[3(a + b) - \sqrt \biggr] = \pi \biggl[3(a + b) - \sqrt\biggr] and : C\approx\pi\left(a+b\right)\left(1+\frac\right). The errors in these approximations, which were obtained empirically, are of order h^3 and h^5, respectively.


Arc length

More generally, the arc length of a portion of the circumference, as a function of the angle subtended (or -coordinates of any two points on the upper half of the ellipse), is given by an incomplete elliptic integral. The upper half of an ellipse is parameterized by : y=b\sqrt. Then the arc length s from x_ to x_ is: : s = -b\int_^ \sqrt \, dz. This is equivalent to : s = -b\left[E\left(z \;\Biggl, \; 1 - \frac\right)\right]^_ where E(z \mid m) is the incomplete elliptic integral of the second kind with parameter m=k^. The
inverse function In mathematics, the inverse function of a Function (mathematics), function (also called the inverse of ) is a function (mathematics), function that undoes the operation of . The inverse of exists if and only if is Bijection, bijective, and i ...
, the angle subtended as a function of the arc length, is given by a certain elliptic functions, elliptic function. Some lower and upper bounds on the circumference of the canonical ellipse x^2/a^2 + y^2/b^2 = 1 with a\geq b are : \begin 2\pi b &\le C \le 2\pi a, \\ \pi (a+b) &\le C \le 4(a+b), \\ 4\sqrt &\le C \le \sqrt \pi \sqrt . \end Here the upper bound 2\pi a is the circumference of a circumscribed circle, circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound 4\sqrt is the perimeter of an inscribed figure, inscribed rhombus with vertex (geometry), vertices at the endpoints of the major and the minor axes.


Curvature

The curvature is given by \kappa = \frac\left(\frac+\frac\right)^\ ,
radius of curvature In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a co ...

radius of curvature
at point (x,y): : \rho = a^2 b^2 \left(\frac + \frac\right)^\frac = \frac \sqrt \ . Radius of curvature at the two ''vertices'' (\pm a,0) and the centers of curvature: : \rho_0 = \frac=p\ , \qquad \left(\pm\frac\,\bigg, \,0\right)\ . Radius of curvature at the two ''co-vertices'' (0,\pm b) and the centers of curvature: : \rho_1 = \frac\ , \qquad \left(0\,\bigg, \,\pm\frac\right)\ .


In triangle geometry

Ellipses appear in triangle geometry as # Steiner ellipse: ellipse through the vertices of the triangle with center at the centroid, # inellipses: ellipses which touch the sides of a triangle. Special cases are the Steiner inellipse and the Mandart inellipse.


As plane sections of quadrics

Ellipses appear as plane sections of the following quadrics: * Ellipsoid * Elliptic cone * Elliptic cylinder * Hyperboloid of one sheet * Hyperboloid of two sheets Ellipsoid Quadric.png, Ellipsoid Quadric Cone.jpg, Elliptic cone Elliptic Cylinder Quadric.png, Elliptic cylinder Hyperboloid1.png, Hyperboloid of one sheet Hyperboloid2.png, Hyperboloid of two sheets


Applications


Physics


Elliptical reflectors and acoustics

If the water's surface is disturbed at one focus of an elliptical water tank, the circular waves of that disturbance, after reflection (physics), reflecting off the walls, converge simultaneously to a single point: the ''second focus''. This is a consequence of the total travel length being the same along any wall-bouncing path between the two foci. Similarly, if a light source is placed at one focus of an elliptic mirror, all light rays on the plane of the ellipse are reflected to the second focus. Since no other smooth curve has such a property, it can be used as an alternative definition of an ellipse. (In the special case of a circle with a source at its center all light would be reflected back to the center.) If the ellipse is rotated along its major axis to produce an ellipsoidal mirror (specifically, a prolate spheroid), this property holds for all rays out of the source. Alternatively, a cylindrical mirror with elliptical cross-section can be used to focus light from a linear fluorescent lamp along a line of the paper; such mirrors are used in some image scanner, document scanners. Sound waves are reflected in a similar way, so in a large elliptical room a person standing at one focus can hear a person standing at the other focus remarkably well. The effect is even more evident under a cupola, vaulted roof shaped as a section of a prolate spheroid. Such a room is called a ''whisper chamber''. The same effect can be demonstrated with two reflectors shaped like the end caps of such a spheroid, placed facing each other at the proper distance. Examples are the National Statuary Hall at the United States Capitol (where John Quincy Adams is said to have used this property for eavesdropping on political matters); the Mormon Tabernacle at Temple Square in Salt Lake City, Utah; at an exhibit on sound at the Museum of Science and Industry (Chicago), Museum of Science and Industry in Chicago; in front of the University of Illinois at Urbana–Champaign Foellinger Auditorium; and also at a side chamber of the Palace of Charles V, in the Alhambra.


Planetary orbits

In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun [approximately] at one focus, in his Kepler's laws of planetary motion, first law of planetary motion. Later, Isaac Newton explained this as a corollary of his Newton's law of universal gravitation, law of universal gravitation. More generally, in the gravitational two-body problem, if the two bodies are bound to each other (that is, the total energy is negative), their orbits are Similarity (geometry), similar ellipses with the common barycenter being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. The orbit of either body in the reference frame of the other is also an ellipse, with the other body at the same focus. Keplerian elliptical orbits are the result of any radially directed attraction force whose strength is inversely proportional to the square of the distance. Thus, in principle, the motion of two oppositely charged particles in empty space would also be an ellipse. (However, this conclusion ignores losses due to electromagnetic radiation and quantum mechanics, quantum effects, which become significant when the particles are moving at high speed.) For elliptical orbits, useful relations involving the eccentricity e are: : \begin e &= \frac = \frac \\ r_a &= (1 + e)a \\ r_p &= (1 - e)a \end where * r_a is the radius at apoapsis (the farthest distance) * r_p is the radius at periapsis (the closest distance) * a is the length of the semi-major axis Also, in terms of r_a and r_p, the semi-major axis a is their arithmetic mean, the semi-minor axis b is their geometric mean, and the conic section#Features, semi-latus rectum \ell is their harmonic mean. In other words, :\begin a &= \frac \\[2pt] b &= \sqrt \\[2pt] \ell &= \frac = \frac \end.


Harmonic oscillators

The general solution for a harmonic oscillator in two or more dimensions is also an ellipse. Such is the case, for instance, of a long pendulum that is free to move in two dimensions; of a mass attached to a fixed point by a perfectly elastic spring (mechanics), spring; or of any object that moves under influence of an attractive force that is directly proportional to its distance from a fixed attractor. Unlike Keplerian orbits, however, these "harmonic orbits" have the center of attraction at the geometric center of the ellipse, and have fairly simple equations of motion.


Phase visualization

In electronics, the relative phase of two sinusoidal signals can be compared by feeding them to the vertical and horizontal inputs of an oscilloscope. If the
Lissajous figure A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations : x=A\sin(at+\delta),\quad y=B\sin(bt), which describe complex harmonic motion. This family of curve In mathematics, a ...
display is an ellipse, rather than a straight line, the two signals are out of phase.


Elliptical gears

Two non-circular gears with the same elliptical outline, each pivoting around one focus and positioned at the proper angle, turn smoothly while maintaining contact at all times. Alternatively, they can be connected by a link chain or toothed belt, timing belt, or in the case of a bicycle the main chainwheel#ovoid chainwheels, chainring may be elliptical, or an ovoid similar to an ellipse in form. Such elliptical gears may be used in mechanical equipment to produce variable angular speed or torque from a constant rotation of the driving axle, or in the case of a bicycle to allow a varying crank rotation speed with inversely varying mechanical advantage. Elliptical bicycle gears make it easier for the chain to slide off the cog when changing gears. An example gear application would be a device that winds thread onto a conical bobbin on a Spinning (textiles), spinning machine. The bobbin would need to wind faster when the thread is near the apex than when it is near the base.


Optics

* In a material that is optically anisotropic (birefringent), the refractive index depends on the direction of the light. The dependency can be described by an index ellipsoid. (If the material is optically isotropic, this ellipsoid is a sphere.) * In lamp-laser pumping, pumped solid-state lasers, elliptical cylinder-shaped reflectors have been used to direct light from the pump lamp (coaxial with one ellipse focal axis) to the active medium rod (coaxial with the second focal axis). * In laser-plasma produced Extreme ultraviolet, EUV light sources used in microchip Extreme ultraviolet lithography, lithography, EUV light is generated by plasma positioned in the primary focus of an ellipsoid mirror and is collected in the secondary focus at the input of the lithography machine.


Statistics and finance

In statistics, a bivariate random vector (X, Y) is elliptical distribution, jointly elliptically distributed if its iso-density contours—loci of equal values of the density function—are ellipses. The concept extends to an arbitrary number of elements of the random vector, in which case in general the iso-density contours are ellipsoids. A special case is the multivariate normal distribution. The elliptical distributions are important in finance because if rates of return on assets are jointly elliptically distributed then all portfolios can be characterized completely by their mean and variance—that is, any two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return.


Computer graphics

Drawing an ellipse as a graphics primitive is common in standard display libraries, such as the MacIntosh QuickDraw API, and Direct2D on Windows. Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. M. L. V. Pitteway extended Bresenham's algorithm for lines to conics in 1967. Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken. In 1970 Danny Cohen presented at the "Computer Graphics 1970" conference in England a linear algorithm for drawing ellipses and circles. In 1971, L. B. Smith published similar algorithms for all conic sections and proved them to have good properties. These algorithms need only a few multiplications and additions to calculate each vector. It is beneficial to use a parametric formulation in computer graphics because the density of points is greatest where there is the most curvature. Thus, the change in slope between each successive point is small, reducing the apparent "jaggedness" of the approximation. ;Drawing with Bézier paths: Composite Bézier curves may also be used to draw an ellipse to sufficient accuracy, since any ellipse may be construed as an
affine transformation In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...
of a circle. The spline methods used to draw a circle may be used to draw an ellipse, since the constituent Bézier curves behave appropriately under such transformations.


Optimization theory

It is sometimes useful to find the minimum bounding ellipse on a set of points. The ellipsoid method is quite useful for solving this problem.


See also

* Cartesian oval, a generalization of the ellipse * Circumconic and inconic * Distance of closest approach of ellipses * Ellipse fitting * Elliptic coordinates, an orthogonal coordinate system based on families of ellipses and
hyperbola In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it ...

hyperbola
e * Elliptic partial differential equation * Elliptical distribution, in statistics * Elliptical dome * Geodesics on an ellipsoid * Great ellipse * Kepler's laws of planetary motion * n-ellipse, ''n''-ellipse, a generalization of the ellipse for ''n'' foci * Oval * Spheroid, the ellipsoid obtained by rotating an ellipse about its major or minor axis * Stadium (geometry), a two-dimensional geometric shape constructed of a rectangle with semicircles at a pair of opposite sides * Steiner circumellipse, the unique ellipse circumscribing a triangle and sharing its centroid * Superellipse, a generalization of an ellipse that can look more rectangular or more "pointy" * true anomaly, True, eccentric anomaly, eccentric, and mean anomaly


Notes


References

* * * * *


External links

* * * * *
Apollonius' Derivation of the Ellipse
at Convergence
''The Shape and History of The Ellipse in Washington, D.C.''
by Clark Kimberling
Ellipse circumference calculator


*
Trammel according Frans van Schooten
{{Authority control Conic sections Plane curves Elementary shapes Algebraic curves