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{{about|the geometric figure}}{{redirect|Elliptical|the syntactic omission of words|Ellipsis (linguistics)|the punctuation mark|Ellipsis}}{{short description|Plane curve: conic section}}File:Ellipse-conic.svg|thumb|An ellipse (red) obtained as the intersection of a conecone(File:Ellipse-def0.svg|300px|thumb|Ellipse: notations)(File:Ellipse-var.svg|thumb|Ellipses: examples with increasing eccentricity)In mathematics, an ellipse is a plane curve 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, 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 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).Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is:frac{x^2}{a^2}+frac{y^2}{b^2}= 1 .Assuming a ≥ b, the foci are (±c, 0) for c = sqrt{a^2-b^2}. The standard parametric equation is:(x,y)=(acos(t),bsin(t)) quad text{for} quad 0leq tleq 2pi.Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane (see figure). Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. An angled cross section of a cylinder is also an ellipse.An ellipse may also be defined in terms of one focus point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant. This constant ratio is the above-mentioned eccentricity e=tfrac ca=sqrt{1-tfrac{b^2}{a^2}}.Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in the solar system is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the barycenter of the Sun{{ndash}}planet 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 ellipsoids. A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under parallel or perspective projection. The ellipse is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical polarization of light in optics.The name, ({{transl|grc|élleipsis}}, "omission"), was given by Apollonius of Perga in his Conics.

Definition as locus of points

(File:Ellipse-def-e.svg|thumb|Ellipse: Definition by sum of distances to foci)(File:Ellipse-def-dc.svg|thumb|Ellipse: Definition by focus and circular directrix)An ellipse can be defined geometrically as a set or locus of points 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 = {Pin mathbb{R}^2 ,mid, |PF_2| +|PF_1 | = 2a } .
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 the vertex points 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{c}{a} 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 midpoint 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.{{citation|first1=Tom M.|last1=Apostol|first2=Mamikon A.|last2=Mnatsakanian|title=New Horizons in Geometry|year=2012|publisher=The Mathematical Association of America|series=The Dolciani Mathematical Expositions #47|isbn=978-0-88385-354-2|page=251}}The German term for this circle is Leitkreis which can be translated as "Director circle", but that term has a different meaning in the English literature (see Director circle). This property should not be confused with the definition of an ellipse using a directrix line below.Using 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

(File:Ellipse-param.svg|thumb|shape parameters:a: semi-major axis, b: semi-minor axisc: linear eccentricity, p: semi-latus rectum (usually ell).)

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{ (x-c)^2 + y^2 } and to the other focus sqrt{ (x+c)^2 + y^2 }. Hence the point (x,y) is on the ellipse whenever:
sqrt{(x-c)^2 + y^2} + sqrt{(x+c)^2 + y^2} = 2a .
Removing the radicals by suitable squarings and using b^2 = a^2-c^2 produces the standard equation of the ellipse:frac{x^2}{a^2}+frac{y^2}{b^2}= 1 ,or, solved for y:
y=pmfrac{b}{a}sqrt{a^2-x^2}

pm sqrt{(a^2{-}x^2)(1 {-} e^2)}.

The width and height parameters a,; b are called the semi-major and semi-minor axes. 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 - e x.It follows from the equation that the ellipse is symmetric with respect to the coordinate axes and hence with respect to the origin.

Parameters

Semi-major and semi-minor axes a ≥ b

Throughout this article a is the semi-major axis, i.e. age b>0 . In general the canonical ellipse equation tfrac{x^2}{a^2}+tfrac{y^2}{b^2}= 1 may have a < b (and hence the ellipse would be taller than it is wide); in this form the semi-major axis would be b. 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 c

This is the distance from the center to a focus: c = sqrt{a^2-b^2}.

Eccentricity e

The eccentricity can be expressed as: e=frac ca=sqrt{1-left(frac{b}{a}right)^2}, assuming a>b. An ellipse with equal axes (a=b) has zero eccentricity, and is a circle.

Semi-latus rectum l

The length of the chord through one of the foci, perpendicular to the major axis, is called the latus rectum. One half of it is the semi-latus rectum ell. A calculation shows: ell ,=, frac{b^2}a ,=, a (1{-}e^2).The semi-latus rectum ell is equal to the radius of curvature of the osculating circles at the vertices.

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{x^2}{a^2}+tfrac{y^2}{b^2}= 1; has the coordinate equation:
frac{x_1}{a^2}x+frac{y_1}{b^2}y = 1; . A vector parametric equation of the tangent is: vec x=begin{pmatrix}x_1 y_1 end{pmatrix}+s; begin{pmatrix}!-y_1a^2! x_1b^2! end{pmatrix}quad with quad sinmathbb{R} .
Proof:Let be (x_1,y_1) an ellipse point and ;vec x=begin{pmatrix}x_1 y_1end{pmatrix}+s; begin{pmatrix}u v end{pmatrix}; the equation of any line g containing (x_1,y_1). Inserting the line's equation into the ellipse equation and respecting ; tfrac{x_1^2}{a^2}+tfrac{y_1^2}{b^2}=1; yields:
frac{(x_1+su)^2}{a^2}+frac{(y_1+sv)^2}{b^2}= 1 quad Longrightarrowquad 2sleft(frac{x_1u}{a^2}+frac{y_1v}{b^2}right) + s^2left(frac{u^2}{a^2}+frac{v^2}{b^2}right)= 0 .
  • Case (1): tfrac{x_1}{a^2}u+tfrac{y_1}{b^2}v=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 (tfrac{x_1}{a^2},tfrac{y_1}{b^2}), so the tangent line has equation ;tfrac{x_1}{a^2}x+tfrac{y_1}{b^2}y = k; for some k. Because (x_1,y_1) is on the tangent and the ellipse, one obtains k=1.
  • Case (2): tfrac{x_1}{a^2}u+tfrac{y_1}{b^2}vne0. Then line g has a second point in common with the ellipse, and is a secant.
Using (1) one finds that (-y_1a^2,x_1b^2) 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
tfrac{x_1u}{a^2}+tfrac{y_1v}{b^2}=0;, then the points lie on two conjugate diameters (see below). (If a=b, the ellipse is a circle and "conjugate" means "orthogonal".)

Shifted ellipse

If the standard ellipse is shifted to have center (x_circ,y_circ), its equation isfrac{(x{-}x_circ)^2}{a^2}+frac{(y{-}y_circ)^2}{b^2}= 1 .The axes are still parallel to the x- and y-axes.(File:Elliko-sk.svg|thumb|The construction of points based on the parametric equation and the interpretation of parameter t, which is due to de la Hire)

General ellipse

In analytic geometry, the ellipse is defined as a quadric: the set of points (X,Y) of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equationBOOK,weblink Precalculus with Limits, Larson, Ron, Hostetler, Robert P., Falvo, David C., Cengage Learning, 2006, 978-0-618-66089-6, 767, Chapter 10,weblink BOOK,weblink Precalculus, Young, Cynthia Y., John Wiley and Sons, 2010, 978-0-471-75684-2, 831, Chapter 9,weblink
~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
Delta = begin{vmatrix} A & B/2 & D/2B/2 & C & E/2D/2 & E/2 & F end{vmatrix}
= left( AC - frac{B^2}{4} right) F + frac{BED}{4} - frac{CD^2}{4} - frac{AE^2}{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.Lawrence, J. Dennis, A Catalog of Special Plane Curves, Dover Publ., 1972.{{rp|p.63}}The general equation's coefficients can be obtained from known semi-major axis a, semi-minor axis b, center coordinates (x_circ, y_circ), and rotation angle Theta (the angle from the positive horizontal axis to the ellipse's major axis) using the formulae:
begin{align}A &= a^2 (sinTheta)^2 + b^2 (cosTheta)^2
B &= 2 (b^2-a^2) sinTheta cosThetaC &= a^2 (cosTheta)^2 + b^2 (sinTheta)^2D &= -2 A x_circ - B y_circE &= -B x_circ - 2 C y_circF &= A x_circ^2 + B x_circ y_circ + C y_circ^2 - a^2 b^2.end{align}These expressions can be derived from the canonical equation tfrac{x^2}{a^2} + tfrac{y^2}{b^2} = 1 by an affine transformation of the coordinates (x,y):
x = (X{-}x_circ) cosTheta + (Y{-}y_circ) sinTheta y = -(X{-}x_circ) sinTheta + (Y{-}y_circ) cosTheta.
Conversely, the canonical form parameters can be obtained from the general form coefficients by the equations:
begin{align}
a, b &= frac{sqrt{2 Big(A E^2 + C D^2 - B D E + (B^2 - 4 A C) FBig)left((A + C) pm sqrt{(A - C)^2 + B^2}right)}}{B^2 - 4 A C} x_circ &= frac{2 C D - B E}{B^2 - 4 A C} y_circ &= frac{2 A E - B D}{B^2 - 4 A C} Theta &= begin{cases}
0 & text{for } B = 0, A < C
90^circ & text{for } B = 0, A > C
arctanfrac{C - A - sqrt{(A - C)^2 + B^2}}{B} & text{for } B ne 0.
end{cases}end{align}

Parametric representation

(File:Ellipse-ratpar.svg|thumb|Ellipse points calculated by the rational representation with equal spaced parameters (Delta u=0.2))

Standard parametric representation

Using trigonometric functions, a parametric representation of the standard ellipse tfrac{x^2}{a^2}+tfrac{y^2}{b^2}= 1 is:(x,y)=(a cos t, b sin t), 0le t

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