Ellipse

Ellipse

Overview
{{Redirect|Elliptical|the exercise machine|Elliptical trainer}} {{about|the geometric figure}} {{distinguish|ellipsis}} In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, an ellipse (from Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

 ἔλλειψις elleipsis, a "falling short") is a plane curve
Plane curve
In mathematics, a plane curve is a curve in a Euclidean plane . The most frequently studied cases are smooth plane curves , and algebraic plane curves....

 that results from the intersection of a cone
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

 by a plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

 in a way that produces a closed curve. Circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

s are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis.
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Encyclopedia
{{Redirect|Elliptical|the exercise machine|Elliptical trainer}} {{about|the geometric figure}} {{distinguish|ellipsis}}
In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, an ellipse (from Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

 ἔλλειψις elleipsis, a "falling short") is a plane curve
Plane curve
In mathematics, a plane curve is a curve in a Euclidean plane . The most frequently studied cases are smooth plane curves , and algebraic plane curves....

 that results from the intersection of a cone
Cone (geometry)
A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...

 by a plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

 in a way that produces a closed curve. Circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

s are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis. An ellipse is also the locus
Locus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....

 of all points of the plane whose distances to two fixed points add to the same constant. Ellipses are closed curves and are the bounded
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...

 case of the conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

s, the curves that result from the intersection of a circular cone and a plane that does not pass through its apex
Apex (geometry)
In geometry, an apex is the vertex which is in some sense the highest of the figure to which it belongs.*In an isosceles triangle, the apex is the vertex where the two sides of equal length meet, opposite the unequal third side....

; the other two (open and unbounded) cases are parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

s and hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

s. Ellipses arise from the intersection of a right circular cylinder with a plane that is not parallel to the cylinder's main axis of symmetry. Ellipses also arise as images of a circle under parallel projection
Parallel projection
Parallel projections have lines of projection that are parallel both in reality and in the projection plane.Parallel projection corresponds to a perspective projection with an infinite focal length , or "zoom".Within parallel projection there is an ancillary category known as "pictorials"...

 and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection. It is also the simplest Lissajous figure, formed when the horizontal and vertical motions are sinusoids with the same frequency.

Elements of an ellipse

An ellipse is a smooth closed curve which is symmetric
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

 about its horizontal and vertical axes. The distance between antipodal
Antipodal point
In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite to it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter....

 points on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximum along the major axis or transverse diameter, and a minimum along the perpendicular minor axis or conjugate diameter. The semi-major axis
Semi-major axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

(denoted by a in the figure) and the semi-minor axis
Semi-minor axis
In geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis...

(denoted by b in the figure) are one half of the major and minor diameters, respectively. These are sometimes called (especially in technical fields) the major and minor semi-axes, the major and minor semiaxes, or major radius and minor radius. The foci
Focus (geometry)
In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...

 of the ellipse are two special points F1 and F2 on the ellipse's major axis and are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major diameter ( PF1 + PF2 = 2a ). Each of these two points is called a focus
Focus (geometry)
In geometry, the foci are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in defining conic sections, the four types of which are the circle, ellipse, parabola, and hyperbola...

of the ellipse. Refer to the lower Directrix section of this article for a second equivalent construction of an ellipse. The eccentricity
Eccentricity (mathematics)
In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...

of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to the length of the major axis or e = 2f/2a = f/a. For an ellipse the eccentricity is between 0 and 1 (0<e<1). When the eccentricity is 0 the foci coincide with the center point and the figure is a circle. As the eccentricity tends toward
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

 1, the ellipse gets a more elongated shape. It tends towards a line segment (see below) if the two foci remain a finite distance apart and a parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...

 if one focus is kept fixed as the other is allowed to move arbitrarily far away.
The distance ae from a focal point to the centre is called the linear eccentricity of the ellipse (f = ae).

The pins-and-string method

Video for this method
An ellipse can be drawn using two drawing pins, a length of string, and a pencil: NEWLINE
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Push the pins into the paper at two points, which will become the ellipse's foci. Tie the string into a loose loop around the two pins. Pull the loop taut with the pen's tip, so as to form a triangle
Triangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

. Move the pen around, while keeping the string taut, and its tip will trace out an ellipse. Using two pegs and a rope, this procedure is traditionally used by gardeners to outline an elliptical flower bed; thus it is called the gardener's ellipse.
NEWLINE If the ellipse is to be inscribed within a specified rectangle
Rectangle
In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

,[diagram needed] tangent to its four sides at their midpoints, one must first determine the position of the foci and the length of the string loop: NEWLINE
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Let A,B,C,D be the corners of the rectangle, in clockwise order, with A-B being one of the long sides. Draw a circle centered on A, whose radius is the short side A-D. From corner B draw a tangent to the circle. The length L of this tangent is the distance between the foci. This length L can be calculated with the Pythagorean theorem. As the tangent is at a right angles to the radius at the intersect of the tangent with the circle L equals square root ((A-B)squared - (A-D)squared) i.e. square root of the long side of the rectangle squared minus short side squared. Draw a horizontal line through the center of the rectangle. This will be the major axis of the ellipse. Place the foci on the major axis, at distance L/2 from the center.
NEWLINE To adjust the length of the string loop, insert a pin at one focus, and the second pin at the opposite side of the rectangle on the major axis. Loop the string around the two pins and tie it taut. Move the second pin to the other focus. Then draw the ellipse as above; it should fit snugly in the original rectangle. Unfortunately strings tend to be elastic so if you push harder on the pencil stretching the string more you will get a bigger ellipse, pushing less it will be smaller. It may take a few tries to push just hard enough to make the ellipse fit the rectangle. The string method was developed by James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...

, the discoverer of the electromagnetic nature of light, at age 12.

Other methods

An ellipse can also be drawn using a ruler
Ruler
A ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing and engineering/building to measure distances and/or to rule straight lines...

, a set square
Set square
A set square or triangle is an object used in engineering and technical drawing, with the aim of providing a straightedge at a right angle or other particular planar angle to a baseline....

, and a pencil: NEWLINE
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Draw two perpendicular lines M,N on the paper; these will be the major and minor axes of the ellipse. Mark three points A, B, C on the ruler. A->C being the length of the major axis and B->C the length of the minor axis. With one hand, move the ruler on the paper, turning and sliding it so as to keep point A always on line N, and B on line M. With the other hand, keep the pencil's tip on the paper, following point C of the ruler. The tip will trace out an ellipse.
NEWLINE The trammel of Archimedes
Trammel of Archimedes
A trammel of Archimedes is a mechanism that traces out an ellipse. It consists of two shuttles which are confined to perpendicular channels or rails, and a rod which is attached to the shuttles by pivots at fixed positions along the rod. As the shuttles move back and forth, each along its channel,...

 or ellipsograph is a mechanical device that implements this principle. The ruler is replaced by a rod with a pencil holder (point C) at one end, and two adjustable side pins (points A and B) that slide into two perpendicular slots cut into a metal plate. The mechanism can be used with a router to cut ellipses from board material. The mechanism is also used in a toy called the "nothing grinder".

Approximations to ellipses

An ellipse of low eccentricity can be represented reasonably accurately by a circle with its centre offset. With the exception of Mercury, all the planets have an orbit whose minor axis differs from the major axis by less than half of one percent. To draw the orbit with a pair of compasses the centre of the circle should be offset from the focus by an amount equal to the eccentricity multiplied by the radius.

Definition

In Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, an ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points is constant. The equivalence of these two definitions can be proved using the Dandelin spheres
Dandelin spheres
In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin...

.

Equations

The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.

Focus

The distance from the center C to either focus is f = ae, which can be expressed in terms of the major and minor radii:f = \sqrt{a^2-b^2}.

Eccentricity

The eccentricity
Eccentricity (mathematics)
In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...

 of the ellipse (commonly denoted as either e or \epsilon) isNEWLINE
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e=\varepsilon=\sqrt{\frac{a^2-b^2}{a^2}}
NEWLINE =\sqrt{1-\left(\frac{b}{a}\right)^2} =f/a (where again a and b are one-half of the ellipse's major and minor axes respectively, and f is the focal distance) or, as expressed in terms using the flattening
Flattening
The flattening, ellipticity, or oblateness of an oblate spheroid is a measure of the "squashing" of the spheroid's pole, towards its equator...

 factor g=1-\frac {b}{a}=1-\sqrt{1-e^2},e=\sqrt{g(2-g)}.

Directrix

Each focus F of the ellipse is associated with a line parallel to the minor axis called a directrix. Refer to the illustration on the right. The distance from any point P on the ellipse to the focus F is a constant fraction of that point's perpendicular distance to the directrix resulting in the equality, e=PF/PD. The ratio of these two distances is the eccentricity of the ellipse. This property (which can be proved using the Dandelin spheres
Dandelin spheres
In geometry, the Dandelin spheres are one or two spheres that are tangent both to a plane and to a cone that intersects the plane. The intersection of the cone and the plane is a conic section, and the point at which either sphere touches the plane is a focus of the conic section, so the Dandelin...

) can be taken as another definition of the ellipse.
Besides the well known ratio e=f/a, it is also true that e=a/d.

Circular directrix

The ellipse can also be defined as the set of points that are equidistant from one focus and a particular circle, the directrix circle, that is centered on the other focus. The radius of the directrix circle is greater than the distance between the center of this circle and the focus; thus, the focus is inside the directrix circle, as is the entire ellipse.

Ellipse as hypotrochoid

The ellipse is a special case of the hypotrochoid
Hypotrochoid
A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle....

 when R = 2r.

Area

The area enclosed by an ellipse is πab, where (as before) a and b are one-half of the ellipse's major and minor axes respectively. If the ellipse is given by the implicit equation A x^2+ B x y + C y^2 = 1 , then the area is \frac{2\pi}{\sqrt{ 4 A C - B^2 }}.

Circumference

The circumference
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....

 C of an ellipse is: C = 4 a E(e) where again e is the eccentricity and where the function E is the complete elliptic integral of the second kind. The exact infinite series is: C = 2\pi a \left[{1 - \left({1\over 2}\right)^2e^2 - \left({1\cdot 3\over 2\cdot 4}\right)^2{e^4\over 3} - \left({1\cdot 3\cdot 5\over 2\cdot 4\cdot 6}\right)^2{e^6\over5} - \cdots}\right]\,\! or C = - 2\pi a \sum_{n=0}^\infty {e^{2n}\over 2n - 1} \prod_{m=1}^n \left({ 2m-1 \over 2m}\right)^2 \,\! For computational purposes a much faster series where the denominators vanish at a rate \tfrac{27}{1024} \left (\tfrac{a-b}{a+b} \right )^{8} is given by: C = \frac{8\pi}{Q^{5/4}}\sum_{n=0}^\infty \frac{(\tfrac{1}{12})_{n}(\tfrac{5}{12})_{n}(v_{1}+nv_{2})r^{n}}{(n!)^{2}} NEWLINE
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NEWLINE r = \frac{432(a^{2}-b^{2})^{2}(a-b)^{6}ba}{Q^3} NEWLINE
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NEWLINE Q = b^{4}+60ab^{3}+134a^{2}b^{2}+60a^{3}b+a^{4}\, NEWLINE
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NEWLINE v_{1} = ba(15b^{4}+68ab^{3}+90a^{2}b^{2}+68a^{3}b+15a^{4})\, NEWLINE
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NEWLINE v_{2} = -a^{6}-b^{6}+126ab^{5}+1041a^{2}b^{4}+1764a^{3}b^{3}+1041a^{4}b^{2}+126a^{5}b\, A good approximation
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...

 is Ramanujan
Srinivasa Ramanujan
Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...

's: C \approx \pi \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]= \pi \left[3(a+b)-\sqrt{10ab+3(a^2+b^2)}\right] and a better approximation
Approximation
An approximation is a representation of something that is not exact, but still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.Approximations may be used because...

 is C\approx\pi\left(a+b\right)\left(1+\frac{3\left(\frac{a-b}{a+b}\right)^2}{10+\sqrt{4-3\left(\frac{a-b}{a+b}\right)^2}}\right).\!\, For the special case where the minor axis is half the major axis, these become: C \approx \frac{\pi a (9 - \sqrt{35})}{2} or, as an estimate of the better approximation, C \approx \frac{a}{2} \sqrt{93 + \frac{1}{2} \sqrt{3}} More generally, the arc length
Arc length
Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods were used for specific curves...

 of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral
Elliptic integral
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...

. {{See also|Meridian arc#Meridian distance on the ellipsoid}} The inverse function
Inverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...

, the angle subtended as a function of the arc length, is given by the elliptic functions.{{Citation needed|date=October 2010}}

Chords

The midpoints of a set of parallel chord
Chord (geometry)
A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...

s of an ellipse are collinear.{{rp|p.147}}

In projective geometry

In projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...

, an ellipse can be defined as the set of all points of intersection between corresponding lines of two pencils of lines
Pencil (mathematics)
A pencil in projective geometry is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a projective plane....

 which are related by a projective map. By projective duality, an ellipse can be defined also as the envelope
Envelope (mathematics)
In geometry, an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point. Classically, a point on the envelope can be thought of as the intersection of two "adjacent" curves, meaning the limit of intersections of nearby curves...

 of all lines that connect corresponding points of two lines which are related by a projective map. This definition also generates hyperbolae and parabolae. However, in projective geometry every conic section is equivalent to an ellipse. A parabola is an ellipse that is tangent to the line at infinity Ω, and the hyperbola is an ellipse that crosses Ω. An ellipse is also the result of projecting
Oblique projection
Oblique projection is a simple type of graphical projection used for producing pictorial, two-dimensional images of three-dimensional objects.- Overview :Oblique projection is a type of parallel projection:...

 a circle, sphere, or ellipse in three dimensions onto a plane, by parallel
Parallel (geometry)
Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...

 lines. It is also the result of conical (perspective) projection of any of those geometric objects from a point O onto a plane P, provided that the plane Q that goes through O and is parallel to P does not cut the object. The image of an ellipse by any affine map is an ellipse, and so is the image of an ellipse by any projective map M such that the line M−1(Ω) does not touch or cross the ellipse.

General ellipse

In analytic geometry
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...

, the ellipse is defined as the set of points (X,Y) of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation ~A X^2 + B X Y + C Y^2 + D X + E Y + F = 0 provided B^2 - 4AC < 0. To distinguish the degenerate cases from the non-degenerate case, let be the determinant of the 3×3 matrix [A, B/2, D/2 ; B/2, C, E/2 ; D/2, E/2, F ]: that is, = (AC - B2/4)F + BED/4 - CD2/4 - AE2/4. 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.{{rp|p.63}}

Canonical form

Let a>b. By a proper choice of coordinate system, the ellipse can be described by the canonical implicit equation
Canonical form
Generally, in mathematics, a canonical form of an object is a standard way of presenting that object....

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 Here (x,y) are the point coordinates in the canonical system, whose origin is the center (X_c,Y_c) of the ellipse, whose x-axis is the unit vector (X_a,Y_a) coinciding with the major axis, and whose y-axis is the perpendicular vector (-Y_a,X_a) coinciding with the minor axis. That is, x = X_a(X - X_c) + Y_a(Y - Y_c) and y = -Y_a(X - X_c) + X_a(Y - Y_c). In this system, the center is the origin (0,0) and the foci are (-e a, 0) and (+e a, 0). Any ellipse can be obtained by rotation and translation
Translation (geometry)
In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...

 of a canonical ellipse with the proper semi-diameters. Translation of an ellipse centered at (X_c,Y_c) is expressed as\frac{(x - X_c)^2}{a^2}+\frac{(y - Y_c)^2}{b^2}=1 Moreover, any canonical ellipse can be obtained by scaling the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...

 of \reals^2, defined by the equation X^2+Y^2=1\, by factors a and b along the two axes. For an ellipse in canonical form, we have Y = \pm b\sqrt{1 - (X/a)^2} = \pm \sqrt{(a^2-X^2)(1 - e^2)} The distances from a point (X,Y) on the ellipse to the left and right foci are a + e X and a - e X, respectively.

General parametric form

An ellipse in general position can be expressed parametrically
Parametric equation
In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....

 as the path of a point (X(t),Y(t)), whereX(t)=X_c + a\,\cos t\,\cos \varphi - b\,\sin t\,\sin\varphiY(t)=Y_c + a\,\cos t\,\sin \varphi + b\,\sin t\,\cos\varphi as the parameter t varies from 0 to 2π. Here (X_c,Y_c) is the center of the ellipse, and \varphi is the angle between the X-axis and the major axis of the ellipse.

Parametric form in canonical position

For an ellipse in canonical position (center at origin, major axis along the X-axis), the equation simplifies toX(t)=a\,\cos tY(t)=b\,\sin t Note that the parameter t (called the eccentric anomaly
Eccentric anomaly
In celestial mechanics, the eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit.For the point P orbiting around an ellipse, the eccentric anomaly is the angle E in the figure...

in astronomy) is not the angle of (X(t),Y(t)) with the X-axis. Formulae connecting a tangential angle
Tangential angle
In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent line to the curve at the given point and the x-axis. In geometry, the tangential angle of a curve in the Cartesian plane, at a specific point, is the angle between the tangent...

 \phi, the angle anchored at the ellipse's center \phi^\prime (called also the polar angle from the ellipse center), and the parametric angle
Parametric equation
In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion....

 t are: \tan \phi=\frac {a}{b} \tan t=\frac {\tan \phi'}{(1-g)^2}=\frac {\tan \phi'}{1-e^2}\tan \phi^\prime=(1-f) \tan t\tan t=\frac {b}{a} \tan \phi=\sqrt{(1-e^2)} \tan \phi=(1-g) \tan \phi=\frac {\tan \phi'}{\sqrt{(1-e^2)}}=\frac {a}{b} \tan \phi'

Polar form relative to center

In 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 isr(\theta)=\frac{ab}{\sqrt{(b \cos \theta)^2 + (a\sin \theta)^2}}

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 isr(\theta)=\frac{a (1-e^{2})}{1 \pm e\cos\theta} 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 isr=\frac{a (1-e^{2})}{1 - e\cos(\theta - \phi)}. The angle \theta in these formulas is called the true anomaly
True anomaly
In celestial mechanics, the true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse .The true anomaly is usually...

of the point. The numerator a (1-e^{2}) of these formulas is the semi-latus rectum of the ellipse, usually denoted l. It is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular
Perpendicular
In geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...

 to the major axis.

General polar form

The following equation on the polar coordinates (rθ) describes a general ellipse with semidiameters a and b, centered at a point (r0θ0), with the a axis rotated by φ relative to the polar axis: r(\theta )=\frac{P(\theta )+Q(\theta )}{R(\theta )} where P(\theta )=r_0 \left[\left(b^2-a^2\right) \cos \left(\theta +\theta _0-2 \varphi \right)+\left(a^2+b^2\right) \cos \left(\theta -\theta_0\right)\right] Q(\theta )=\sqrt{2} a b \sqrt{R(\theta )-2 r_0^2 \sin ^2\left(\theta -\theta_0\right)} R(\theta )=\left(b^2-a^2\right) \cos (2 \theta -2 \varphi )+a^2+b^2

Angular eccentricity

The angular eccentricity
Angular eccentricity
In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and eccentricity....

 \alpha is the angle whose sine is the eccentricity e; that is, \alpha=\sin^{-1}(e)=\cos^{-1}\left(\frac{b}{a}\right)=2\tan^{-1}\left(\sqrt{\frac{a-b}{a+b}}\right);\,\!

Degrees of freedom

An ellipse in the plane has five degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

 (the same as a general conic section), defining its position, orientation, shape, and scale. In comparison, circles have only three degrees of freedom (position and scale), while parabolae have four. Said another way, the set of all ellipses in the plane, with any natural metric (such as the Hausdorff distance) is a five-dimensional manifold. These degrees can be identified with, for example, the coefficients A,B,C,D,E of the implicit equation, or with the coefficients Xc, Yc, φ, a, b of the general parametric form.

Elliptical reflectors and acoustics

If the water's surface is disturbed at one focus of an elliptical water tank, the circular waves created by that disturbance, after being reflected
Reflection (physics)
Reflection is the change in direction of a wavefront at an interface between two differentmedia so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves...

 by the walls, will 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
Mirror
A mirror is an object that reflects light or sound in a way that preserves much of its original quality prior to its contact with the mirror. Some mirrors also filter out some wavelengths, while preserving other wavelengths in the reflection...

, 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
Prolate spheroid
A prolate spheroid is a spheroid in which the polar axis is greater than the equatorial diameter. Prolate spheroids stand in contrast to oblate spheroids...

), this property will hold 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
Fluorescent lamp
A fluorescent lamp or fluorescent tube is a gas-discharge lamp that uses electricity to excite mercury vapor. The excited mercury atoms produce short-wave ultraviolet light that then causes a phosphor to fluoresce, producing visible light. A fluorescent lamp converts electrical power into useful...

 along a line of the paper; such mirrors are used in some document scanner
Image scanner
In computing, an image scanner—often abbreviated to just scanner—is a device that optically scans images, printed text, handwriting, or an object, and converts it to a digital image. Common examples found in offices are variations of the desktop scanner where the document is placed on a glass...

s. 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 vaulted roof
Cupola
In architecture, a cupola is a small, most-often dome-like, structure on top of a building. Often used to provide a lookout or to admit light and air, it usually crowns a larger roof or dome....

 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
National Statuary Hall
National Statuary Hall is a chamber in the United States Capitol devoted to sculptures of prominent Americans. The hall, also known as the Old Hall of the House, is a large, two-story, semicircular room with a second story gallery along the curved perimeter. It is located immediately south of the...

 at the United States Capitol
United States Capitol
The United States Capitol is the meeting place of the United States Congress, the legislature of the federal government of the United States. Located in Washington, D.C., it sits atop Capitol Hill at the eastern end of the National Mall...

 (where John Quincy Adams
John Quincy Adams
John Quincy Adams was the sixth President of the United States . He served as an American diplomat, Senator, and Congressional representative. He was a member of the Federalist, Democratic-Republican, National Republican, and later Anti-Masonic and Whig parties. Adams was the son of former...

 is said to have used this property for eavesdropping on political matters), at an exhibit on sound at the Museum of Science and Industry
Museum of Science and Industry (Chicago)
The Museum of Science and Industry is located in Chicago, Illinois, USA in Jackson Park, in the Hyde Park neighborhood adjacent to Lake Michigan. It is housed in the former Palace of Fine Arts from the 1893 World's Columbian Exposition...

 in Chicago
Chicago
Chicago is the largest city in the US state of Illinois. With nearly 2.7 million residents, it is the most populous city in the Midwestern United States and the third most populous in the US, after New York City and Los Angeles...

, in front of the University of Illinois at Urbana-Champaign
University of Illinois at Urbana-Champaign
The University of Illinois at Urbana–Champaign is a large public research-intensive university in the state of Illinois, United States. It is the flagship campus of the University of Illinois system...

 Foellinger Auditorium, and also at a side chamber of the Palace of Charles V, in the Alhambra
Alhambra
The Alhambra , the complete form of which was Calat Alhambra , is a palace and fortress complex located in the Granada, Andalusia, Spain...

.

Planetary orbits

{{Main|Elliptic orbit}} In the 17th century, Johannes Kepler
Johannes Kepler
Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...

 discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun at one focus, in his first law of planetary motion
Kepler's laws of planetary motion
In astronomy, Kepler's laws give a description of the motion of planets around the Sun.Kepler's laws are:#The orbit of every planet is an ellipse with the Sun at one of the two foci....

. Later, Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 explained this as a corollary of his law of universal gravitation
Newton's law of universal gravitation
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...

. More generally, in the gravitational two-body problem
Two-body problem
In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic nucleus In...

, if the two bodies are bound to each other (i.e., the total energy is negative), their orbits are similar
Similarity (geometry)
Two geometrical objects are called similar if they both have the same shape. More precisely, either one is congruent to the result of a uniform scaling of the other...

 ellipses with the common barycenter being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one 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
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

 and quantum effects
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

 which become significant when the particles are moving at high speed.) For elliptical orbits, useful relations involving the eccentricity, e are: e={{r_\mathrm{ap}-r_\mathrm{per}}\over{r_\mathrm{ap}+r_\mathrm{per}}}={{r_\mathrm{ap}-r_\mathrm{per}}\over{2a}} r_\mathrm{ap}=(1+e)a\!\,             r_\mathrm{per}=(1-e)a\!\, where:NEWLINE
    NEWLINE
  • r_\mathrm{ap}\,\! is radius at apoapsis (i.e., the farthest distance).
  • NEWLINE
  • r_\mathrm{per}\,\! is radius at periapsis (the closest distance).
  • NEWLINE
  • a\,\! is the length of the semi-major axis
    Semi-major axis
    The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

    .
NEWLINE In addition, the semimajor axis a\,\! is the arithmetic mean
Arithmetic mean
In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

 of r_\text{ap}\,\! and r_\mathrm{per}\,\!, or a=\frac{{r_\text{ap}} + {r_\text{per}}}{2} , and the semiminor axis b\,\! is the geometric mean
Geometric mean
The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root of the resulting product is taken.For instance, the...

 of r_\text{ap}\,\! and r_\mathrm{per}\,\!, or b=\, ^2 \! \! \! \! \sqrt{r_\mathrm{ap} \times r_\mathrm{per}} .
Also the semi-latus rectum (the distance from a focus to a point on the ellipse along a line parallel to the minor axis) is the harmonic mean
Harmonic mean
In mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....

 of r_\text{ap}\,\! and r_\mathrm{per}\,\!, or \text {semi-latus rectum} = \frac{2}{\frac{1}{r_\text{ap}} + \frac{1}{r_\text{per}}} .

Harmonic oscillators

The general solution for a harmonic oscillator
Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....

 in two or more dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

s 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; 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
Electronics
Electronics is the branch of science, engineering and technology that deals with electrical circuits involving active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies...

, the relative phase of two sinusoidal signals can be compared by feeding them to the vertical and horizontal inputs of an oscilloscope
Oscilloscope
An oscilloscope is a type of electronic test instrument that allows observation of constantly varying signal voltages, usually as a two-dimensional graph of one or more electrical potential differences using the vertical or 'Y' axis, plotted as a function of time,...

. If the display is an ellipse, rather than a straight line, the two signals are out of phase.

Elliptical gears

Two non-circular gear
Non-circular gear
A non-circular gear is a special gear design with special characteristics and purpose. While a regular gear is optimized to transmit torque to another engaged member with minimum noise and wear and with maximum efficiency, a non-circular gear's main objective might be ratio variations, axle...

s with the same elliptical outline, each pivoting around one focus and positioned at the proper angle, will turn smoothly while maintaining contact at all times. Alternatively, they can be connected by a link chain or timing belt
Timing belt
A timing belt, or cam belt , is a part of an internal combustion engine that controls the timing of the engine's valves. Some engines, such as the flat-4 Volkswagen air-cooled engine, and the straight-6 Toyota F engine use timing gears...

, or in the case of a bicycle the main 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
Torque
Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

 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
Mechanical advantage
Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. Ideally, the device preserves the input power and simply trades off forces against movement to obtain a desired amplification in the output force...

. 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
Bobbin
A bobbin is a spindle or cylinder, with or without flanges, on which wire, yarn, thread or film is wound. Bobbins are typically found in sewing machines, cameras, and within electronic equipment....

 on a spinning
Spinning (textiles)
Spinning is a major industry. It is part of the textile manufacturing process where three types of fibre are converted into yarn, then fabric, then textiles. The textiles are then fabricated into clothes or other artifacts. There are three industrial processes available to spin yarn, and a...

 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
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

 depends on the direction of the light. The dependency can be described by an index ellipsoid
Index ellipsoid
In optics, an index ellipsoid is a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive indices in a crystal....

. (If the material is optically isotropic, this ellipsoid is a sphere.)

Ellipses in statistics and finance

In statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, a random vector (X, Y) is 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
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...

 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.

Ellipses in computer graphics

Drawing an ellipse as a graphics primitive is common in standard display libraries, such as the MacIntosh QuickDraw
QuickDraw
QuickDraw is the 2D graphics library and associated Application Programming Interface which is a core part of the classic Apple Macintosh operating system. It was initially written by Bill Atkinson and Andy Hertzfeld. QuickDraw still exists as part of the libraries of Mac OS X, but has been...

 API, and Direct2D
Direct2D
Direct2D is a 2D and vector graphics application programming interface designed by Microsoft and implemented in Windows 7 and Windows Server 2008 R2, and also Windows Vista and Windows Server 2008 and Platform Update Supplement for Windows Vista and for Windows Server 2008 & KB2505189 update...

 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 (IEEE CG&A, Sept. 1984). 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 Bezier spline paths

Multiple Bezier splines may also be used to draw an ellipse to sufficient accuracy, since any ellipse may be construed as an affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...

 of a circle. The spline methods used to draw a circle may be used to draw an ellipse, since the constituent Bezier curve
Bézier curve
A Bézier curve is a parametric curve frequently used in computer graphics and related fields. Generalizations of Bézier curves to higher dimensions are called Bézier surfaces, of which the Bézier triangle is a special case....

s will behave appropriately under such transformations.

Line segment as a type of degenerate ellipse

A line segment is a degenerate ellipse with semi-minor axis = 0 and eccentricity = 1, and with the focal points at the ends. Although the eccentricity is 1 this is not a parabola. A radial elliptic trajectory is a non-trivial special case of an elliptic orbit, where the ellipse is a line segment.

Ellipses in Optimization Theory

It is sometimes useful to find the minimum bounding ellipse on a set of points. The Ellipsoid method
Ellipsoid method
In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm, which finds an optimal solution in a finite number of steps.The...

is quite useful for attacking this problem

External links

{{Commons category|Ellipses}}NEWLINENEWLINE