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Ellipse

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From Wikipedia, the free encyclopedia

Plane curve

This article is about the geometric figure. For other uses, seeEllipse (disambiguation).

Not to be confused withEllipsis,Eclipse, orEcliptic.

An ellipse (red) obtained as the intersection of a cone with an inclined plane.

Ellipses: examples with increasing eccentricity

Inmathematics, anellipse is aplane curve surrounding twofocal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes acircle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by itseccentricity $e {\displaystyle e}$ , a number ranging from $e=0$ (thelimiting case of a circle) to $e=1$ (the limiting case of infinite elongation, no longer an ellipse but aparabola).

An ellipse has a simplealgebraic solution for its area, but forits perimeter (also known ascircumference),integration is required to obtain an exact solution.

The largest and smallestdiameters of an ellipse, also known as its width and height, are typically denoted2a and2b. An ellipse has fourextreme points: twovertices at the endpoints of themajor axis and twoco-vertices at the endpoints of the minor axis.

Notable points and line segments in an ellipse.

Analytically, the equation of a standard ellipse centered at the origin is: ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.$ Assuming $a\geq b$ , the foci are $(\pm c,0)$ where ${\textstyle c={\sqrt {a^{2}-b^{2}}}}$ , calledlinear eccentricity, is the distance from the center to a focus. The standardparametric equation is: $(x,y)=(a\cos(t),b\sin(t))\quad {\text{for}}\quad 0\leq t\leq 2\pi .$

Ellipses are theclosed type ofconic section: a plane curve tracing the intersection of acone with aplane (see figure). Ellipses have many similarities with the other two forms of conic sections, parabolas andhyperbolas, both of which areopen andunbounded. An angledcross section of a right circularcylinder is also an ellipse.

An ellipse may also be defined in terms of one focal point and a line outside the ellipse called thedirectrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant, called theeccentricity: $e={\frac {c}{a}}={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}.$

Ellipses are common inphysics,astronomy andengineering. For example, theorbit of each planet in theSolar System is approximately an ellipse with the Sun at one focus point (more precisely, the focus is thebarycenter of the Sun–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 byellipsoids. A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle underparallel orperspective projection. The ellipse is also the simplestLissajous figure formed when the horizontal and vertical motions aresinusoids with the same frequency: a similar effect leads toelliptical polarization of light inoptics.

The name,ἔλλειψις (élleipsis, "omission"), was given byApollonius of Perga in hisConics.

Definition as locus of points

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Ellipse: definition by sum of distances to foci

Ellipse: definition by focus and circular directrix

An ellipse can be defined geometrically as a set orlocus 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 {\displaystyle P}

such that the sum of the distances

|PF_{1}|,\ |PF_{2}|

is equal to

2a

E=\left\{P\in \mathbb {R} ^{2}\,\mid \,\left|PF_{2}\right|+\left|PF_{1}\right|=2a\right\}.

The midpoint $C {\displaystyle C}$ of the line segment joining the foci is called thecenter of the ellipse. The line through the foci is called themajor axis, and the line perpendicular to it through the center is theminor axis.The major axis intersects the ellipse at twovertices $V_{1},V_{2}$ , which have distance $a {\displaystyle a}$ to the center. The distance $c {\displaystyle c}$ of the foci to the center is called thefocal distance or linear eccentricity. The quotient $e={\tfrac {c}{a}}$ is defined as theeccentricity.

The case $F_{1}=F_{2}$ yields a circle and is included as a special type of ellipse.

The equation $\left|PF_{2}\right|+\left|PF_{1}\right|=2a$ can be viewed in a different way (see figure):

c_{2}

is the circle with center

F_{2}

and radius

2a

, then the distance of a point

P {\displaystyle P}

to the circle

c_{2}

equals the distance to the focus

F_{1}

\left|PF_{1}\right|=\left|Pc_{2}\right|.

$c_{2}$ is called thecircular directrix (related to focus $F_{2}$ ) of the ellipse.^[1]^[2] This property should not be confused with the definition of an ellipse using a directrix line below.

UsingDandelin spheres, one can prove that any 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

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Shape parameters:
a: semi-major axis,
b: semi-minor axis,
c: linear eccentricity,
p: semi-latus rectum (usually $\ell$ ).

{\displaystyle \ell } — Shape parameters:
a: semi-major axis,
b: semi-minor axis,
c: linear eccentricity,
p: semi-latus rectum (usually $\ell$ ).

Standard equation

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The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, thex-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 ${\textstyle {\sqrt {(x-c)^{2}+y^{2}}}}$ and to the other focus ${\textstyle {\sqrt {(x+c)^{2}+y^{2}}}}$ . Hence the point $(x,\,y)$ is on the ellipse whenever: ${\sqrt {(x-c)^{2}+y^{2}}}+{\sqrt {(x+c)^{2}+y^{2}}}=2a\ .$

Removing theradicals by suitable squarings and using $b^{2}=a^{2}-c^{2}$ (see diagram) produces the standard equation of the ellipse:^[3] ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,$ or, solved fory: $y=\pm {\frac {b}{a}}{\sqrt {a^{2}-x^{2}}}=\pm {\sqrt {\left(a^{2}-x^{2}\right)\left(1-e^{2}\right)}}.$

The width and height parameters $a,\;b$ are called thesemi-major and semi-minor axes. The top and bottom points $V_{3}=(0,\,b),\;V_{4}=(0,\,-b)$ are theco-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 issymmetric with respect to the coordinate axes and hence with respect to the origin.

Parameters

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Principal axes

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Throughout this article, thesemi-major and semi-minor axes are denoted $a {\displaystyle a}$ and $b {\displaystyle b}$ , respectively, i.e. $a\geq b>0\ .$

In principle, 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). This form can be converted to the standard form by transposing the variable names $x {\displaystyle x}$ and $y {\displaystyle y}$ and the parameter names $a {\displaystyle a}$ and $b . {\displaystyle b.}$

Linear eccentricity

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

Eccentricity

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The eccentricity can be expressed as: $e={\frac {c}{a}}={\sqrt {1-\left({\frac {b}{a}}\right)^{2}}},$

assuming $a>b.$ An ellipse with equal axes ( $a=b$ ) has zero eccentricity, and is a circle.

Semi-latus rectum

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The length of the chord through one focus, perpendicular to the major axis, is called thelatus rectum. One half of it is thesemi-latus rectum $\ell$ . A calculation shows:^[4] $\ell ={\frac {b^{2}}{a}}=a\left(1-e^{2}\right).$

The semi-latus rectum $\ell$ is equal to theradius of curvature at the vertices (see sectioncurvature).

Tangent

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An arbitrary line $g {\displaystyle g}$ intersects an ellipse at 0, 1, or 2 points, respectively called anexterior line,tangent andsecant. 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 vectorparametric equation of the tangent is: ${\vec {x}}={\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}+s\left({\begin{array}{r}-y_{1}a^{2}\\x_{1}b^{2}\end{array}}\right),\quad s\in \mathbb {R} .$

Proof:Let $(x_{1},\,y_{1})$ be a point on an ellipse and ${\textstyle {\vec {x}}={\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}+s{\begin{pmatrix}u\\v\end{pmatrix}}}$ be the equation of any line $g {\displaystyle g}$ containing $(x_{1},\,y_{1})$ . Inserting the line's equation into the ellipse equation and respecting ${\textstyle {\frac {x_{1}^{2}}{a^{2}}}+{\frac {y_{1}^{2}}{b^{2}}}=1}$ yields: ${\frac {\left(x_{1}+su\right)^{2}}{a^{2}}}+{\frac {\left(y_{1}+sv\right)^{2}}{b^{2}}}=1\ \quad \Longrightarrow \quad 2s\left({\frac {x_{1}u}{a^{2}}}+{\frac {y_{1}v}{b^{2}}}\right)+s^{2}\left({\frac {u^{2}}{a^{2}}}+{\frac {v^{2}}{b^{2}}}\right)=0\ .$ There are then cases:

${\frac {x_{1}}{a^{2}}}u+{\frac {y_{1}}{b^{2}}}v=0.$ Then line $g {\displaystyle g}$ and the ellipse have only point $(x_{1},\,y_{1})$ in common, and $g {\displaystyle g}$ is a tangent. The tangent direction hasperpendicular vector ${\begin{pmatrix}{\frac {x_{1}}{a^{2}}}&{\frac {y_{1}}{b^{2}}}\end{pmatrix}}$ , so the tangent line has equation ${\textstyle {\frac {x_{1}}{a^{2}}}x+{\tfrac {y_{1}}{b^{2}}}y=k}$ for some $k {\displaystyle k}$ . Because $(x_{1},\,y_{1})$ is on the tangent and the ellipse, one obtains $k=1$ .
${\frac {x_{1}}{a^{2}}}u+{\frac {y_{1}}{b^{2}}}v\neq 0.$ Then line $g {\displaystyle g}$ has a second point in common with the ellipse, and is a secant.

Using (1) one finds that ${\begin{pmatrix}-y_{1}a^{2}&x_{1}b^{2}\end{pmatrix}}$ 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 ${\textstyle {\frac {x_{1}u}{a^{2}}}+{\tfrac {y_{1}v}{b^{2}}}=0}$ , then the points lie on twoconjugate diameters (seebelow). (If $a=b$ , the ellipse is a circle and "conjugate" means "orthogonal".)

Shifted ellipse

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If the standard ellipse is shifted to have center $\left(x_{\circ },\,y_{\circ }\right)$ , its equation is ${\frac {\left(x-x_{\circ }\right)^{2}}{a^{2}}}+{\frac {\left(y-y_{\circ }\right)^{2}}{b^{2}}}=1\ .$

The axes are still parallel to thex- andy-axes.

General ellipse

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Main article:Matrix representation of conic sections

A general ellipse in the plane can be uniquely described as a bivariate quadratic equation of Cartesian coordinates, or using center, semi-major and semi-minor axes, and angle

Inanalytic geometry, the ellipse is defined as aquadric: the set of points $(x,\,y)$ of theCartesian plane that, in non-degenerate cases, satisfy theimplicit equation^[5]^[6] $Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$ provided $B^{2}-4AC<0.$

To distinguish thedegenerate cases from the non-degenerate case, let∆ be thedeterminant $\Delta ={\begin{vmatrix}A&{\frac {1}{2}}B&{\frac {1}{2}}D\\{\frac {1}{2}}B&C&{\frac {1}{2}}E\\{\frac {1}{2}}D&{\frac {1}{2}}E&F\end{vmatrix}}=ACF+{\tfrac {1}{4}}BDE-{\tfrac {1}{4}}(AE^{2}+CD^{2}+FB^{2}).$

Then the ellipse is a non-degenerate real ellipse if and only ifC∆ < 0. IfC∆ > 0, we have an imaginary ellipse, and if∆ = 0, we have a point ellipse.^[7]^: 63

The general equation's coefficients can be obtained from known semi-major axis $a {\displaystyle a}$ , semi-minor axis $b {\displaystyle b}$ , 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: ${\begin{aligned}A&=a^{2}\sin ^{2}\theta +b^{2}\cos ^{2}\theta &B&=2\left(b^{2}-a^{2}\right)\sin \theta \cos \theta \\[1ex]C&=a^{2}\cos ^{2}\theta +b^{2}\sin ^{2}\theta &D&=-2Ax_{\circ }-By_{\circ }\\[1ex]E&=-Bx_{\circ }-2Cy_{\circ }&F&=Ax_{\circ }^{2}+Bx_{\circ }y_{\circ }+Cy_{\circ }^{2}-a^{2}b^{2}.\end{aligned}}$

These expressions can be derived from the canonical equation ${\frac {X^{2}}{a^{2}}}+{\frac {Y^{2}}{b^{2}}}=1$ by a Euclidean transformation of the coordinates $(X,\,Y)$ : ${\begin{aligned}X&=\left(x-x_{\circ }\right)\cos \theta +\left(y-y_{\circ }\right)\sin \theta ,\\Y&=-\left(x-x_{\circ }\right)\sin \theta +\left(y-y_{\circ }\right)\cos \theta .\end{aligned}}$

Conversely, the canonical form parameters can be obtained from the general-form coefficients by the equations:^[3]

${\begin{aligned}a,b&={\frac {-{\sqrt {2{\big (}AE^{2}+CD^{2}-BDE+(B^{2}-4AC)F{\big )}{\big (}(A+C)\pm {\sqrt {(A-C)^{2}+B^{2}}}{\big )}}}}{B^{2}-4AC}},\\x_{\circ }&={\frac {2CD-BE}{B^{2}-4AC}},\\[5mu]y_{\circ }&={\frac {2AE-BD}{B^{2}-4AC}},\\[5mu]\theta &={\tfrac {1}{2}}\operatorname {atan2} (-B,\,C-A),\end{aligned}}$

whereatan2 is the 2-argument arctangent function.

Parametric representation

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The construction of points based on the parametric equation and the interpretation of parametert, which is due to de la Hire

Ellipse points calculated by the rational representation with equally spaced parameters ( $\Delta u=0.2$ ).

{\displaystyle \Delta u=0.2} — Ellipse points calculated by the rational representation with equally spaced parameters ( $\Delta u=0.2$ ).

Standard parametric representation

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Usingtrigonometric functions, a parametric representation of the standard ellipse ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1$ is: $(x,\,y)=(a\cos t,\,b\sin t),\ 0\leq t<2\pi \,.$

The parametert (called theeccentric anomaly in astronomy) is not the angle of $(x(t),y(t))$ with thex-axis, but has a geometric meaning due toPhilippe de La Hire (see§ Drawing ellipses below).^[8]

Rational representation

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With the substitution ${\textstyle u=\tan \left({\frac {t}{2}}\right)}$ and trigonometric formulae one obtains $\cos t={\frac {1-u^{2}}{1+u^{2}}}\ ,\quad \sin t={\frac {2u}{1+u^{2}}}$

and therational parametric equation of an ellipse ${\begin{cases}x(u)=a\,{\dfrac {1-u^{2}}{1+u^{2}}}\\[10mu]y(u)=b\,{\dfrac {2u}{1+u^{2}}}\\[10mu]-\infty<u<\infty \end{cases}}$

which covers any point of the ellipse ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1$ except the left vertex $(-a,\,0)$ .

For $u\in [0,\,1],$ this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing $u . {\displaystyle u.}$ The left vertex is the limit ${\textstyle \lim _{u\to \pm \infty }(x(u),\,y(u))=(-a,\,0)\;.}$

Alternately, if the parameter $[u:v]$ is considered to be a point on thereal projective line ${\textstyle \mathbf {P} (\mathbf {R} )}$ , then the corresponding rational parametrization is $[u:v]\mapsto \left(a{\frac {v^{2}-u^{2}}{v^{2}+u^{2}}},b{\frac {2uv}{v^{2}+u^{2}}}\right).$

Then ${\textstyle [1:0]\mapsto (-a,\,0).}$

Rational representations of conic sections are commonly used incomputer-aided design (seeBézier curve).

Tangent slope as parameter

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A parametric representation, which uses the slope $m {\displaystyle m}$ of the tangent at a point of the ellipsecan be obtained from the derivative of the standard representation ${\vec {x}}(t)=(a\cos t,\,b\sin t)^{\mathsf {T}}$ : ${\vec {x}}'(t)=(-a\sin t,\,b\cos t)^{\mathsf {T}}\quad \rightarrow \quad m=-{\frac {b}{a}}\cot t\quad \rightarrow \quad \cot t=-{\frac {ma}{b}}.$

With help oftrigonometric formulae one obtains: $\cos t={\frac {\cot t}{\pm {\sqrt {1+\cot ^{2}t}}}}={\frac {-ma}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}\ ,\quad \quad \sin t={\frac {1}{\pm {\sqrt {1+\cot ^{2}t}}}}={\frac {b}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}.$

Replacing $\cos t$ and $\sin t$ of the standard representation yields: ${\vec {c}}_{\pm }(m)=\left(-{\frac {ma^{2}}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}},\;{\frac {b^{2}}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}\right),\,m\in \mathbb {R} .$

Here $m {\displaystyle m}$ 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 $n {\displaystyle n}$ can be determined by inserting the coordinates of the corresponding ellipse point ${\vec {c}}_{\pm }(m)$ : $y=mx\pm {\sqrt {m^{2}a^{2}+b^{2}}}\,.$

This description of the tangents of an ellipse is an essential tool for the determination of theorthoptic of an ellipse. The orthoptic article contains another proof, without differential calculus and trigonometric formulae.

General ellipse

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Ellipse as an affine image of the unit circle

Another definition of an ellipse usesaffine transformations:

Anyellipse 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 $A {\displaystyle A}$ is a regularmatrix (with non-zerodeterminant) and ${\vec {f}}\!_{0}$ is an arbitrary vector. If ${\vec {f}}\!_{1},{\vec {f}}\!_{2}$ are the column vectors of the matrix $A {\displaystyle A}$ , 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 twoconjugate diameters, in general not perpendicular.

Vertices

The four vertices of the ellipse are ${\vec {p}}(t_{0}),\;{\vec {p}}\left(t_{0}\pm {\tfrac {\pi }{2}}\right),\;{\vec {p}}\left(t_{0}+\pi \right)$ , for a parameter $t=t_{0}$ defined by: $\cot(2t_{0})={\frac {{\vec {f}}\!_{1}^{\,2}-{\vec {f}}\!_{2}^{\,2}}{2{\vec {f}}\!_{1}\cdot {\vec {f}}\!_{2}}}.$

(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 $A=\pi \left|\det({\vec {f}}_{1},{\vec {f}}_{2})\right|.$

Semiaxes

With the abbreviations $\;M={\vec {f}}_{1}^{2}+{\vec {f}}_{2}^{2},\ N=\left|\det({\vec {f}}_{1},{\vec {f}}_{2})\right|$ the statements of Apollonios's theorem can be written as: $a^{2}+b^{2}=M,\quad ab=N\ .$ Solving this nonlinear system for $a, b {\displaystyle a,b}$ yields the semiaxes: ${\begin{aligned}a&={\frac {1}{2}}({\sqrt {M+2N}}+{\sqrt {M-2N}})\\[1ex]b&={\frac {1}{2}}({\sqrt {M+2N}}-{\sqrt {M-2N}})\,.\end{aligned}}$

Implicit representation

Solving the parametric representation for $\;\cos t,\sin t\;$ byCramer's rule and using $\;\cos ^{2}t+\sin ^{2}t-1=0\;$ , one obtains the implicit representation $\det {\left({\vec {x}}\!-\!{\vec {f}}\!_{0},{\vec {f}}\!_{2}\right)^{2}}+\det {\left({\vec {f}}\!_{1},{\vec {x}}\!-\!{\vec {f}}\!_{0}\right)^{2}}-\det {\left({\vec {f}}\!_{1},{\vec {f}}\!_{2}\right)^{2}}=0.$

Conversely: If theequation

x^{2}+2cxy+d^{2}y^{2}-e^{2}=0\ ,

with

\;d^{2}-c^{2}>0\;,

of an ellipse centered at the origin is given, then the two vectors ${\vec {f}}_{1}={e \choose 0},\quad {\vec {f}}_{2}={\frac {e}{\sqrt {d^{2}-c^{2}}}}{-c \choose 1}$ 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 ${\vec {f}}_{1}={1 \choose 0},\quad {\vec {f}}_{2}={\frac {1}{\sqrt {2}}}{-1 \choose 1}.$

Whirls: nested, scaled and rotated ellipses. The spiral is not drawn: we see it as thelocus of points where the ellipses are especially close to each other.

Rotated standard ellipse

For ${\vec {f}}_{0}={0 \choose 0},\;{\vec {f}}_{1}=a{\cos \theta \choose \sin \theta },\;{\vec {f}}_{2}=b{-\sin \theta \choose \;\cos \theta }$ one obtains a parametric representation of the standard ellipserotated by angle $\theta$ : ${\begin{aligned}x&=x_{\theta }(t)=a\cos \theta \cos t-b\sin \theta \sin t\,,\\y&=y_{\theta }(t)=a\sin \theta \cos t+b\cos \theta \sin t\,.\end{aligned}}$

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

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Polar form relative to center

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Polar coordinates centered at the center.

Inpolar 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^[7]^: 75 $r(\theta )={\frac {ab}{\sqrt {(b\cos \theta )^{2}+(a\sin \theta )^{2}}}}={\frac {b}{\sqrt {1-(e\cos \theta )^{2}}}}$ where $e {\displaystyle e}$ is the eccentricity (notEuler's number).

Polar form relative to focus

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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 $r(\theta )={\frac {a(1-e^{2})}{1\pm e\cos \theta }}$

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.

The angle $\theta$ is called thetrue anomaly of the point. The numerator $\ell =a(1-e^{2})$ is thesemi-latus rectum.

Eccentricity and the directrix property

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Each of the two lines parallel to the minor axis, and at a distance of ${\textstyle d={\frac {a^{2}}{c}}={\frac {a}{e}}}$ from it, is called adirectrix of the ellipse (see diagram).

For an arbitrary point

P {\displaystyle 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 {\left|PF_{1}\right|}{\left|Pl_{1}\right|}}={\frac {\left|PF_{2}\right|}{\left|Pl_{2}\right|}}=e={\frac {c}{a}}\ .

The proof for the pair $F_{1},l_{1}$ follows from the fact that ${\textstyle \left|PF_{1}\right|^{2}=(x-c)^{2}+y^{2},\ \left|Pl_{1}\right|^{2}=\left(x-{\tfrac {a^{2}}{c}}\right)^{2}}$ and $y^{2}=b^{2}-{\tfrac {b^{2}}{a^{2}}}x^{2}$ satisfy the equation $\left|PF_{1}\right|^{2}-{\frac {c^{2}}{a^{2}}}\left|Pl_{1}\right|^{2}=0\,.$

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 {\displaystyle F}

(focus), any line

l {\displaystyle l}

(directrix) not through

F {\displaystyle F}

, and any real number

e {\displaystyle 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, {\displaystyle e,}

that is:

E=\left\{P\ \left|\ {\frac {|PF|}{|Pl|}}=e\right.\right\}.

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 theline at infinity in theprojective plane.

(The choice $e=1$ yields a parabola, and if $e>1$ , a hyperbola.)

Pencil of conics with a common vertex and common semi-latus rectum

Proof

Let $F=(f,\,0),\ e>0$ , and assume $(0,\,0)$ is a point on the curve. The directrix $l {\displaystyle l}$ has equation $x=-{\tfrac {f}{e}}$ . 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 {f}{e}}\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 anellipse ( $e<1$ ), or aparabola ( $e=1$ ), or ahyperbola ( $e>1$ ). 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 $1-e^{2}={\tfrac {b^{2}}{a^{2}}},{\text{ and }}\ p={\tfrac {b^{2}}{a}}$ , and then the equation above becomes ${\frac {(x-a)^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1\,,$

which is the equation of an ellipse with center $(a,\,0)$ , thex-axis as major axis, andthe major/minor semi axis $a,\,b$ .

Construction of a directrix

Because of $c\cdot {\tfrac {a^{2}}{c}}=a^{2}$ point $L_{1}$ of directrix $l_{1}$ (see diagram) and focus $F_{1}$ are inverse with respect to thecircle inversion 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 $\left(x-f_{1}\right)^{2}+\left(y-f_{2}\right)^{2}=e^{2}{\frac {\left(ux+vy+w\right)^{2}}{u^{2}+v^{2}}}\ .$

(The right side of the equation uses theHesse normal form of a line to calculate the distance $|Pl|$ .)

Focus-to-focus reflection property

[edit]

Ellipse: the tangent bisects the supplementary angle of the angle between the lines to the foci.

Rays from one focus reflect off the ellipse to pass through the other focus.

An ellipse possesses the following property:

The normal at a point

P {\displaystyle P}

bisects the angle between the lines

{\overline {PF_{1}}},\,{\overline {PF_{2}}}

Proof

Because the tangent line is perpendicular to the normal, an equivalent statement is that the tangent is the external angle bisector of the lines to the foci (see diagram).Let $L {\displaystyle L}$ be the point on the line ${\overline {PF_{2}}}$ with distance $2a$ to the focus $F_{2}$ , where $a {\displaystyle a}$ is the semi-major axis of the ellipse. Let line $w {\displaystyle w}$ be the external angle bisector of the lines ${\overline {PF_{1}}}$ and ${\overline {PF_{2}}}.$ Take any other point $Q {\displaystyle Q}$ on $w . {\displaystyle w.}$ By thetriangle inequality and theangle bisector theorem, $2a=\left|LF_{2}\right|<{}$ $\left|QF_{2}\right|+\left|QL\right|={}$ $\left|QF_{2}\right|+\left|QF_{1}\right|,$ so $Q {\displaystyle Q}$ must be outside the ellipse. As this is true for every choice of $Q, {\displaystyle Q,}$ $w {\displaystyle w}$ only intersects the ellipse at the single point $P {\displaystyle P}$ so must be the tangent line.

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 (seewhispering gallery).

Additionally, because of the focus-to-focus reflection property of ellipses, if the rays are allowed to continue propagating, reflected rays will eventually align closely with the major axis.

Conjugate diameters

[edit]

Definition of conjugate diameters

[edit]

Orthogonal diameters of a circle with a square of tangents, midpoints of parallel chords and an affine image, which is an ellipse with conjugate diameters, a parallelogram of tangents and midpoints of chords.

Main article: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 areconjugate if the midpoints of chords parallel to $d_{1}$ lie on $d_{2}\ .$

From the diagram one finds:

Two diameters

{\overline {P_{1}Q_{1}}},\,{\overline {P_{2}Q_{2}}}

of an ellipse are conjugate whenever the tangents at

P_{1}

and

Q_{1}

are parallel to

{\overline {P_{2}Q_{2}}}

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 ${\vec {p}}\left(t+{\tfrac {\pi }{2}}\right),\ {\vec {p}}\left(t-{\tfrac {\pi }{2}}\right)$ belong to its conjugate diameter.

For the common parametric representation $(a\cos t,b\sin t)$ of the ellipse with equation ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1$ one gets: The points

(x_{1},y_{1})=(\pm a\cos t,\pm b\sin t)\quad

(signs: (+,+) or (−,−) )

(x_{2},y_{2})=({\color {red}{\mp }}a\sin t,\pm b\cos t)\quad

(signs: (−,+) or (+,−) )

are conjugate and

{\frac {x_{1}x_{2}}{a^{2}}}+{\frac {y_{1}y_{2}}{b^{2}}}=0\ .

In case of a circle the last equation collapses to $x_{1}x_{2}+y_{1}y_{2}=0\ .$

Theorem of Apollonios on conjugate diameters

[edit]

For an ellipse with semi-axes $a,\,b$ the following is true:^[9]^[10]

Let

c_{1}

and

c_{2}

be halves of two conjugate diameters (see diagram) then

$c_{1}^{2}+c_{2}^{2}=a^{2}+b^{2}$ .
Thetriangle $O,P_{1},P_{2}$ with sides $c_{1},\,c_{2}$ (see diagram) has the constant area ${\textstyle A_{\Delta }={\frac {1}{2}}ab}$ , which can be expressed by $A_{\Delta }={\tfrac {1}{2}}c_{2}d_{1}={\tfrac {1}{2}}c_{1}c_{2}\sin \alpha$ , 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 sectionmetric properties) can be written as $A_{el}=\pi ab=\pi c_{2}d_{1}=\pi c_{1}c_{2}\sin \alpha$ .
The parallelogram of tangents adjacent to the given conjugate diameters has the ${\text{Area}}_{12}=4ab\ .$

Proof

Let the ellipse be in the canonical form with parametric equation ${\vec {p}}(t)=(a\cos t,\,b\sin t).$

The two points ${\textstyle {\vec {c}}_{1}={\vec {p}}(t),\ {\vec {c}}_{2}={\vec {p}}\left(t+{\frac {\pi }{2}}\right)}$ are on conjugate diameters (see previous section). From trigonometric formulae one obtains ${\vec {c}}_{2}=(-a\sin t,\,b\cos t)^{\mathsf {T}}$ and $\left|{\vec {c}}_{1}\right|^{2}+\left|{\vec {c}}_{2}\right|^{2}=\cdots =a^{2}+b^{2}\,.$

The area of the triangle generated by ${\vec {c}}_{1},\,{\vec {c}}_{2}$ is $A_{\Delta }={\tfrac {1}{2}}\det \left({\vec {c}}_{1},\,{\vec {c}}_{2}\right)=\cdots ={\tfrac {1}{2}}ab$

and from the diagram it can be seen that the area of the parallelogram is 8 times that of $A_{\Delta }$ . Hence ${\text{Area}}_{12}=4ab\,.$

Orthogonal tangents

[edit]

Main article:Orthoptic (geometry)

For the ellipse ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1$ the intersection points oforthogonal tangents lie on the circle $x^{2}+y^{2}=a^{2}+b^{2}$ .

This circle is calledorthoptic ordirector circle of the ellipse (not to be confused with the circular directrix defined above).

Drawing ellipses

[edit]

Ellipses appear indescriptive 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 was known to the 5th century mathematicianProclus, and the tool now known as anelliptical trammel was invented byLeonardo da Vinci.^[11]

If there is no ellipsograph available, one can draw an ellipse using anapproximation 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 ofRytz's construction the axes and semi-axes can be retrieved.

de La Hire's point construction

[edit]

The following construction of single points of an ellipse is due tode La Hire.^[12] It is based on thestandard parametric representation $(a\cos t,\,b\sin t)$ of an ellipse:

Draw the twocircles centered at the center of the ellipse with radii $a, b {\displaystyle a,b}$ and the axes of the ellipse.
Draw aline through the center, which intersects the two circles at point $A {\displaystyle A}$ and $B {\displaystyle B}$ , respectively.
Draw aline through $A {\displaystyle A}$ that is parallel to the minor axis and aline through $B {\displaystyle B}$ that is parallel to the major axis. These lines meet at an ellipse point $P {\displaystyle P}$ (see diagram).
Repeat steps (2) and (3) with different lines through the center.

de La Hire's method
Animation of the method

Pins-and-string method

[edit]

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 twodrawing pins, 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 thegardener's ellipse. The Byzantine architectAnthemius of Tralles (c. 600) described how this method could be used to construct an elliptical reflector,[13] and it was elaborated in a now-lost 9th-century treatise byAl-Ḥasan ibn Mūsā.^[14]

A similar method for drawingconfocal ellipses with aclosed string is due to the Irish bishopCharles Graves.

Paper strip methods

[edit]

The two following methods rely on the parametric representation (see§ Standard parametric representation, 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 $P {\displaystyle P}$ . If the strip slides with both ends on the axes of the desired ellipse, then point $P {\displaystyle P}$ traces the ellipse. For the proof one shows that point $P {\displaystyle P}$ has the parametric representation $(a\cos t,\,b\sin t)$ , where parameter $t {\displaystyle t}$ is the angle of the slope of the paper strip.

A technical realization of the motion of the paper strip can be achieved by aTusi couple (see animation). The device is able to draw any ellipse with afixed 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.

Ellipse construction: paper strip method 1
Ellipses with Tusi couple. Two examples: red and cyan.

A variation of the paper strip method 1 uses the observation that the midpoint $N {\displaystyle N}$ of the paper strip is moving on the circle with center $M {\displaystyle M}$ (of the ellipse) and radius ${\tfrac {a+b}{2}}$ . Hence, the paperstrip can be cut at point $N {\displaystyle N}$ into halves, connected again by a joint at $N {\displaystyle N}$ and the sliding end $K {\displaystyle K}$ fixed at the center $M {\displaystyle M}$ (see diagram). After this operation the movement of the unchanged half of the paperstrip is unchanged.^[15] This variation requires only one sliding shoe.

Variation of the paper strip method 1
Animation of the variation of the paper strip method 1

Ellipse construction: paper strip method 2

Method 2

The second method starts with

a strip of paper of length

a {\displaystyle a}

One marks the point, which divides the strip into two substrips of length $b {\displaystyle b}$ 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 $t {\displaystyle t}$ is the angle of slope of the paper strip.

This method is the base for severalellipsographs (see section below).

Similar to the variation of the paper strip method 1 avariation of the paper strip method 2 can be established (see diagram) by cutting the part between the axes into halves.

Elliptical trammel (principle)
Ellipsograph due toBenjamin Bramer
Variation of the paper strip method 2

Most ellipsographdrafting instruments are based on the second paperstrip method.

Approximation of an ellipse with osculating circles

Approximation by osculating circles

[edit]

FromMetric properties below, one obtains:

The radius of curvature at the vertices $V_{1},\,V_{2}$ is: ${\tfrac {b^{2}}{a}}$
The radius of curvature at the co-vertices $V_{3},\,V_{4}$ is: ${\tfrac {a^{2}}{b}}\ .$

The diagram shows an easy way to find the centers of curvature $C_{1}=\left(a-{\tfrac {b^{2}}{a}},0\right),\,C_{3}=\left(0,b-{\tfrac {a^{2}}{b}}\right)$ 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 $H {\displaystyle H}$ , 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 aFrench curve one draws a curve, which has smooth contact to theosculating circles.

Steiner generation

[edit]

The following method to construct single points of an ellipse relies on theSteiner generation of a conic section:

Given twopencils

B(U),\,B(V)

of lines at two points

U,\,V

(all lines containing

U {\displaystyle U}

and

V {\displaystyle V}

, respectively) and a projective but not perspective mapping

\pi

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 ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1$ one uses the pencils at the vertices $V_{1},\,V_{2}$ . Let $P=(0,\,b)$ be an upper co-vertex of the ellipse and $A=(-a,\,2b),\,B=(a,\,2b)$ .

$P {\displaystyle P}$ is the center of the rectangle $V_{1},\,V_{2},\,B,\,A$ . The side ${\overline {AB}}$ 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 ${\overline {V_{1}B}}$ 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 aparallelogram method because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.

As hypotrochoid

[edit]

An ellipse (in red) as a special case of thehypotrochoid with R = 2r

The ellipse is a special case of thehypotrochoid when $R=2r$ , as shown in the adjacent image. The special case of a moving circle with radius $r {\displaystyle r}$ inside a circle with radius $R=2r$ is called aTusi couple.

Inscribed angles and three-point form

[edit]

Circles

[edit]

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 theinscribed 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 radianθ, but here the following measurement is more convenient:

In order to measure the angle between two lines with equations

y=m_{1}x+d_{1},\ y=m_{2}x+d_{2},\ m_{1}\neq m_{2},

one uses the quotient:

{\frac {1+m_{1}m_{2}}{m_{2}-m_{1}}}=\cot \theta \ .

Inscribed angle theorem for circles

[edit]

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 {(x_{4}-x_{1})(x_{4}-x_{2})+(y_{4}-y_{1})(y_{4}-y_{2})}{(y_{4}-y_{1})(x_{4}-x_{2})-(y_{4}-y_{2})(x_{4}-x_{1})}}={\frac {(x_{3}-x_{1})(x_{3}-x_{2})+(y_{3}-y_{1})(y_{3}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}.

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

[edit]

As a consequence, one obtains an equation for the circle determined by three non-collinear points

P_{i}=\left(x_{i},\,y_{i}\right)

{\frac {({\color {red}x}-x_{1})({\color {red}x}-x_{2})+({\color {red}y}-y_{1})({\color {red}y}-y_{2})}{({\color {red}y}-y_{1})({\color {red}x}-x_{2})-({\color {red}y}-y_{2})({\color {red}x}-x_{1})}}={\frac {(x_{3}-x_{1})(x_{3}-x_{2})+(y_{3}-y_{1})(y_{3}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}.

For example, for $P_{1}=(2,\,0),\;P_{2}=(0,\,1),\;P_{3}=(0,\,0)$ the three-point equation is:

{\frac {(x-2)x+y(y-1)}{yx-(y-1)(x-2)}}=0

, which can be rearranged to

(x-1)^{2}+\left(y-{\tfrac {1}{2}}\right)^{2}={\tfrac {5}{4}}\ .

Using vectors,dot products anddeterminants this formula can be arranged more clearly, letting ${\vec {x}}=(x,\,y)$ : ${\frac {\left({\color {red}{\vec {x}}}-{\vec {x}}_{1}\right)\cdot \left({\color {red}{\vec {x}}}-{\vec {x}}_{2}\right)}{\det \left({\color {red}{\vec {x}}}-{\vec {x}}_{1},{\color {red}{\vec {x}}}-{\vec {x}}_{2}\right)}}={\frac {\left({\vec {x}}_{3}-{\vec {x}}_{1}\right)\cdot \left({\vec {x}}_{3}-{\vec {x}}_{2}\right)}{\det \left({\vec {x}}_{3}-{\vec {x}}_{1},{\vec {x}}_{3}-{\vec {x}}_{2}\right)}}.$

The center of the circle $\left(x_{\circ },\,y_{\circ }\right)$ satisfies: ${\begin{bmatrix}1&{\dfrac {y_{1}-y_{2}}{x_{1}-x_{2}}}\\[2ex]{\dfrac {x_{1}-x_{3}}{y_{1}-y_{3}}}&1\end{bmatrix}}{\begin{bmatrix}x_{\circ }\\[1ex]y_{\circ }\end{bmatrix}}={\begin{bmatrix}{\dfrac {x_{1}^{2}-x_{2}^{2}+y_{1}^{2}-y_{2}^{2}}{2(x_{1}-x_{2})}}\\[2ex]{\dfrac {y_{1}^{2}-y_{3}^{2}+x_{1}^{2}-x_{3}^{2}}{2(y_{1}-y_{3})}}\end{bmatrix}}.$

The radius is the distance between any of the three points and the center. $r={\sqrt {\left(x_{1}-x_{\circ }\right)^{2}+\left(y_{1}-y_{\circ }\right)^{2}}}={\sqrt {\left(x_{2}-x_{\circ }\right)^{2}+\left(y_{2}-y_{\circ }\right)^{2}}}={\sqrt {\left(x_{3}-x_{\circ }\right)^{2}+\left(y_{3}-y_{\circ }\right)^{2}}}.$

Ellipses

[edit]

This section considers the family of ellipses defined by equations ${\tfrac {\left(x-x_{\circ }\right)^{2}}{a^{2}}}+{\tfrac {\left(y-y_{\circ }\right)^{2}}{b^{2}}}=1$ with afixed eccentricity $e {\displaystyle e}$ . It is convenient to use the parameter: ${\color {blue}q}={\frac {a^{2}}{b^{2}}}={\frac {1}{1-e^{2}}},$

and to write the ellipse equation as: $\left(x-x_{\circ }\right)^{2}+{\color {blue}q}\,\left(y-y_{\circ }\right)^{2}=a^{2},$

whereq 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 thex-axis; if $q>1$ , it is parallel to they-axis.)

Like a circle, such an ellipse is determined by three points not on a line.

For this family of ellipses, one introduces the followingq-analog angle measure, which isnot a function of the usual angle measureθ:^[16]^[17]

In order to measure an angle between two lines with equations

y=m_{1}x+d_{1},\ y=m_{2}x+d_{2},\ m_{1}\neq m_{2}

one uses the quotient:

{\frac {1+{\color {blue}q}\;m_{1}m_{2}}{m_{2}-m_{1}}}\ .

Inscribed angle theorem for ellipses

[edit]

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}+{\color {blue}q}\,(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 {(x_{4}-x_{1})(x_{4}-x_{2})+{\color {blue}q}\;(y_{4}-y_{1})(y_{4}-y_{2})}{(y_{4}-y_{1})(x_{4}-x_{2})-(y_{4}-y_{2})(x_{4}-x_{1})}}={\frac {(x_{3}-x_{1})(x_{3}-x_{2})+{\color {blue}q}\;(y_{3}-y_{1})(y_{3}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}\ .

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

[edit]

A consequence, one obtains an equation for the ellipse determined by three non-collinear points

P_{i}=\left(x_{i},\,y_{i}\right)

{\frac {({\color {red}x}-x_{1})({\color {red}x}-x_{2})+{\color {blue}q}\;({\color {red}y}-y_{1})({\color {red}y}-y_{2})}{({\color {red}y}-y_{1})({\color {red}x}-x_{2})-({\color {red}y}-y_{2})({\color {red}x}-x_{1})}}={\frac {(x_{3}-x_{1})(x_{3}-x_{2})+{\color {blue}q}\;(y_{3}-y_{1})(y_{3}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}\ .

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 {(x-2)x+4y(y-1)}{yx-(y-1)(x-2)}}=0

and after conversion

{\frac {(x-1)^{2}}{2}}+{\frac {\left(y-{\frac {1}{2}}\right)^{2}}{\frac {1}{2}}}=1.

Analogously to the circle case, the equation can be written more clearly using vectors: ${\frac {\left({\color {red}{\vec {x}}}-{\vec {x}}_{1}\right)*\left({\color {red}{\vec {x}}}-{\vec {x}}_{2}\right)}{\det \left({\color {red}{\vec {x}}}-{\vec {x}}_{1},{\color {red}{\vec {x}}}-{\vec {x}}_{2}\right)}}={\frac {\left({\vec {x}}_{3}-{\vec {x}}_{1}\right)*\left({\vec {x}}_{3}-{\vec {x}}_{2}\right)}{\det \left({\vec {x}}_{3}-{\vec {x}}_{1},{\vec {x}}_{3}-{\vec {x}}_{2}\right)}},$

where $*$ is the modifieddot product ${\vec {u}}*{\vec {v}}=u_{x}v_{x}+{\color {blue}q}\,u_{y}v_{y}.$

Pole-polar relation

[edit]

Any ellipse can be described in a suitable coordinate system by an equation ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1$ . The equation of the tangent at a point $P_{1}=\left(x_{1},\,y_{1}\right)$ of the ellipse is ${\tfrac {x_{1}x}{a^{2}}}+{\tfrac {y_{1}y}{b^{2}}}=1.$ 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 {x_{1}x}{a^{2}}}+{\tfrac {y_{1}y}{b^{2}}}=1

, not through the center of the ellipse.

This relation between points and lines is abijection.

Theinverse function maps

line $y=mx+d,\ d\neq 0$ onto the point $\left(-{\tfrac {ma^{2}}{d}},\,{\tfrac {b^{2}}{d}}\right)$ and
line $x=c,\ c\neq 0$ onto the point $\left({\tfrac {a^{2}}{c}},\,0\right).$

Such a relation between points and lines generated by a conic is calledpole-polar relation orpolarity. 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 $P {\displaystyle P}$ outside the ellipse, the intersection points of its polar with the ellipse are the tangency points of the two tangents passing $P {\displaystyle P}$ (see diagram: $P_{2},\,p_{2}$ ).
For a pointwithin 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 $x={\tfrac {a^{2}}{c}}$ and $x=-{\tfrac {a^{2}}{c}}$ , 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 (seeInversive geometry).

Pole-polar relations exist for hyperbolas and parabolas as well.

Metric properties

[edit]

All metric properties given below refer to an ellipse with equation

{\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1

except for the section on the area enclosed by a tilted ellipse, where the generalized form of Eq.(1) will be given.

Area

[edit]

Thearea $A_{\text{ellipse}}$ enclosed by an ellipse is:

A_{\text{ellipse}}=\pi ab

where $a {\displaystyle a}$ and $b {\displaystyle b}$ are the lengths of the semi-major and semi-minor axes, respectively. The area formula $\pi ab$ is intuitive: start with a circle of radius $b {\displaystyle b}$ (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: $\pi b^{2}(a/b)=\pi ab.$ ^[18] However, using the same approach for the circumference would be fallacious – compare theintegrals ${\textstyle \int f(x)\,dx}$ and ${\textstyle \int {\sqrt {1+f'^{2}(x)}}\,dx}$ . It is also easy to rigorously prove the area formula using integration as follows. Equation (1) can be rewritten as ${\textstyle y(x)=b{\sqrt {1-x^{2}/a^{2}}}.}$ For $x\in [-a,a],$ this curve is the top half of the ellipse. So twice the integral of $y(x)$ over the interval $[-a,a]$ will be the area of the ellipse: ${\begin{aligned}A_{\text{ellipse}}&=\int _{-a}^{a}2b{\sqrt {1-{\frac {x^{2}}{a^{2}}}}}\,dx\\&={\frac {b}{a}}\int _{-a}^{a}2{\sqrt {a^{2}-x^{2}}}\,dx.\end{aligned}}$

The second integral is the area of a circle of radius $a, {\displaystyle a,}$ that is, $\pi a^{2}.$ So $A_{\text{ellipse}}={\frac {b}{a}}\pi a^{2}=\pi ab.$

An ellipse defined implicitly by $Ax^{2}+Bxy+Cy^{2}=1$ has area $2\pi /{\sqrt {4AC-B^{2}}}.$

The area can also be expressed in terms of eccentricity and the length of the semi-major axis as $a^{2}\pi {\sqrt {1-e^{2}}}$ (obtained by solving forflattening, then computing the semi-minor axis).

The area enclosed by a tilted ellipse is $\pi \;y_{\text{int}}\,x_{\text{max}}$ .

{\displaystyle \pi \;y_{\text{int}}\,x_{\text{max}}} — The area enclosed by a tilted ellipse is $\pi \;y_{\text{int}}\,x_{\text{max}}$ .

So far we have dealt witherect ellipses, whose major and minor axes are parallel to the $x {\displaystyle x}$ and $y {\displaystyle y}$ axes. However, some applications requiretilted ellipses. In charged-particle beam optics, for instance, the enclosed area of an erect or tilted ellipse is an important property of the beam, itsemittance. In this case a simple formula still applies, namely

A_{\text{ellipse}}=\pi \;y_{\text{int}}\,x_{\text{max}}=\pi \;x_{\text{int}}\,y_{\text{max}}

where $y_{\text{int}}$ , $x_{\text{int}}$ are intercepts and $x_{\text{max}}$ , $y_{\text{max}}$ are maximum values. It follows directly fromApollonios's theorem.

Circumference

[edit]

Main article:Perimeter of an ellipse

Further information:Meridian arc § Quarter meridian

The circumference $C {\displaystyle C}$ of an ellipse is: $C\,=\,4a\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta \,=\,4a\,E(e)$

where again $a {\displaystyle a}$ is the length of the semi-major axis, ${\textstyle e={\sqrt {1-b^{2}/a^{2}}}}$ is the eccentricity, and the function $E {\displaystyle E}$ is thecomplete elliptic integral of the second kind, $E(e)\,=\,\int _{0}^{\pi /2}{\sqrt {1-e^{2}\sin ^{2}\theta }}\ d\theta$ which is in general not anelementary function.

The circumference of the ellipse may be evaluated in terms of $E(e)$ usingGauss's arithmetic-geometric mean;^[19] this is a quadratically converging iterative method (seehere for details).

The exactinfinite series is: ${\begin{aligned}{\frac {C}{2\pi a}}&=1-\left({\frac {1}{2}}\right)^{2}e^{2}-\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}{\frac {e^{4}}{3}}-\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right)^{2}{\frac {e^{6}}{5}}-\cdots \\&=1-\sum _{n=1}^{\infty }\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {e^{2n}}{2n-1}}\\&=-\sum _{n=0}^{\infty }\left({\frac {(2n-1)!!}{(2n)!!}}\right)^{2}{\frac {e^{2n}}{2n-1}},\end{aligned}}$ where $n!! {\displaystyle n!!}$ is thedouble factorial (extended to negative odd integers in the usual way, giving $(-1)!!=1$ and $(-3)!!=-1$ ).

This series converges, but by expanding in terms of $h=(a-b)^{2}/(a+b)^{2},$ James Ivory,^[20]Bessel^[21] andKummer^[22] derived a series that converges much more rapidly. It is most concisely written in terms of thebinomial coefficient with $n=1/2$ : ${\begin{aligned}{\frac {C}{\pi (a+b)}}&=\sum _{n=0}^{\infty }{{\frac {1}{2}} \choose n}^{2}h^{n}\\&=\sum _{n=0}^{\infty }\left({\frac {(2n-3)!!}{(2n)!!}}\right)^{2}h^{n}\\&=\sum _{n=0}^{\infty }\left({\frac {(2n-3)!!}{2^{n}n!}}\right)^{2}h^{n}\\&=\sum _{n=0}^{\infty }\left({\frac {1}{(2n-1)4^{n}}}{\binom {2n}{n}}\right)^{2}h^{n}\\&=1+{\frac {h}{4}}+{\frac {h^{2}}{64}}+{\frac {h^{3}}{256}}+{\frac {25\,h^{4}}{16384}}+{\frac {49\,h^{5}}{65536}}+{\frac {441\,h^{6}}{2^{20}}}+{\frac {1089\,h^{7}}{2^{22}}}+\cdots .\end{aligned}}$ The coefficients are slightly smaller (by a factor of $2n-1$ ), but also $e^{4}/16\leq h\leq e^{4}$ is numerically much smaller than $e {\displaystyle e}$ except at $h=e=0$ and $h=e=1$ . For eccentricities less than 0.5( $h<0.005$ ), the error is at the limits ofdouble-precision floating-point after the $h^{4}$ term.^[23]

Srinivasa Ramanujan gave two closeapproximations for the circumference in §16 of "Modular Equations and Approximations to $\pi$ ";^[24] they are ${\frac {C}{\pi }}\approx 3(a+b)-{\sqrt {(3a+b)(a+3b)}}=3(a+b)-{\sqrt {3(a+b)^{2}+4ab}}$ and ${\frac {C}{\pi (a+b)}}\approx 1+{\frac {3h}{10+{\sqrt {4-3h}}}},$ where $h {\displaystyle h}$ takes on the same meaning as above. The errors in these approximations, which were obtained empirically, are of order $h^{3}$ and $h^{5},$ respectively.^[25]^[26] This is because the second formula's infinite series expansion matches Ivory's formula up to the $h^{4}$ term.^[25]^: 3

Arc length

[edit]

Further information:Meridian arc § Calculation

More generally, thearc length of a portion of the circumference, as a function of the angle subtended (orx coordinates of any two points on the upper half of the ellipse), is given by an incompleteelliptic integral. The upper half of an ellipse is parameterized by $y=b\ {\sqrt {1-{\frac {x^{2}}{a^{2}}}\ }}~.$

Then the arc length $s {\displaystyle s}$ from $\ x_{1}\$ to $\ x_{2}\$ is: $s=-b\int _{\arccos {\frac {x_{1}}{a}}}^{\arccos {\frac {x_{2}}{a}}}{\sqrt {\ 1+\left({\tfrac {a^{2}}{b^{2}}}-1\right)\ \sin ^{2}z~}}\;dz~.$

This is equivalent to $s=b\ \left[\;E\left(z\;{\Biggl |}\;1-{\frac {a^{2}}{b^{2}}}\right)\;\right]_{z\ =\ \arccos {\frac {x_{2}}{a}}}^{\arccos {\frac {x_{1}}{a}}}$

where $E(z\mid m)$ is the incomplete elliptic integral of the second kind with parameter $m=k^{2}.$

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^[27] ${\begin{aligned}2\pi b&\leq C\leq 2\pi a\ ,\\\pi (a+b)&\leq C\leq 4(a+b)\ ,\\4{\sqrt {a^{2}+b^{2}\ }}&\leq C\leq {\sqrt {2\ }}\pi {\sqrt {a^{2}+b^{2}\ }}~.\end{aligned}}$

Here the upper bound $\ 2\pi a\$ is the circumference of acircumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound $4{\sqrt {a^{2}+b^{2}}}$ is the perimeter of aninscribed rhombus withvertices at the endpoints of the major and the minor axes.

Given an ellipse whose axes are drawn, we can construct the endpoints of a particular elliptic arc whose length is one eighth of the ellipse's circumference using onlystraightedge and compass in a finite number of steps; for some specific shapes of ellipses, such as when the axes have a length ratio of⁠ ${\sqrt {2}}:1$ ⁠, it is additionally possible to construct the endpoints of a particular arc whose length is one twelfth of the circumference.^[28] (The vertices and co-vertices are already endpoints of arcs whose length is one half or one quarter of the ellipse's circumference.) However, the general theory of straightedge-and-compass elliptic division appears to be unknown, unlike inthe case of the circle andthe lemniscate. The division in special cases has been investigated byLegendre in his classical treatise.^[29]

Curvature

[edit]

Thecurvature is given by:

$\kappa ={\frac {1}{a^{2}b^{2}}}\left({\frac {x^{2}}{a^{4}}}+{\frac {y^{2}}{b^{4}}}\right)^{-{\frac {3}{2}}}\ ,$

and theradius of curvature, ρ = 1/κ, at point $(x,y)$ : $\rho =a^{2}b^{2}\left({\frac {x^{2}}{a^{4}}}+{\frac {y^{2}}{b^{4}}}\right)^{\frac {3}{2}}={\frac {1}{a^{4}b^{4}}}{\sqrt {\left(a^{4}y^{2}+b^{4}x^{2}\right)^{3}}}\ .$ The radius of curvature of an ellipse, as a function of angleθ from the center, is: $R(\theta )={\frac {a^{2}}{b}}{\biggl (}{\frac {1-e^{2}(2-e^{2})(\cos \theta )^{2})}{1-e^{2}(\cos \theta )^{2}}}{\biggr )}^{3/2}\,,$ where e is the eccentricity.

Radius of curvature at the twovertices $(\pm a,0)$ and the centers of curvature: $\rho _{0}={\frac {b^{2}}{a}}=p\ ,\qquad \left(\pm {\frac {c^{2}}{a}}\,{\bigg |}\,0\right)\ .$

Radius of curvature at the twoco-vertices $(0,\pm b)$ and the centers of curvature: $\rho _{1}={\frac {a^{2}}{b}}\ ,\qquad \left(0\,{\bigg |}\,\pm {\frac {c^{2}}{b}}\right)\ .$ The locus of all the centers of curvature is called anevolute. In the case of an ellipse, the evolute is anastroid.

In triangle geometry

[edit]

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 theSteiner inellipse and theMandart inellipse.

As plane sections of quadrics

[edit]

Ellipses appear as plane sections of the followingquadrics:

Ellipsoid
Elliptic cone
Elliptic cylinder
Hyperboloid of one sheet
Hyperboloid of two sheets

Applications

[edit]

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 hisfirst law of planetary motion. Later,Isaac Newton explained this as a corollary of hislaw of universal gravitation.

More generally, in the gravitationaltwo-body problem, if the two bodies are bound to each other (that is, the total energy is negative), their orbits aresimilar ellipses with the commonbarycenter 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 toelectromagnetic radiation andquantum effects, which become significant when the particles are moving at high speed.)

Forelliptical orbits, useful relations involving the eccentricity $e {\displaystyle e}$ are: ${\begin{aligned}e&={\frac {r_{a}-r_{p}}{r_{a}+r_{p}}}={\frac {r_{a}-r_{p}}{2a}}\\r_{a}&=(1+e)a\\r_{p}&=(1-e)a\end{aligned}}$

where

$r_{a}$ is the radius atapoapsis, i.e., the farthest distance of the orbit to thebarycenter of the system, which is afocus of the ellipse
$r_{p}$ is the radius atperiapsis, the closest distance
$a {\displaystyle a}$ is the length of thesemi-major axis

Also, in terms of $r_{a}$ and $r_{p}$ , the semi-major axis $a {\displaystyle a}$ is theirarithmetic mean, the semi-minor axis $b {\displaystyle b}$ is theirgeometric mean, and thesemi-latus rectum $\ell$ is theirharmonic mean. In other words, ${\begin{aligned}a&={\frac {r_{a}+r_{p}}{2}}\\[2pt]b&={\sqrt {r_{a}r_{p}}}\\[2pt]\ell &={\frac {2}{{\frac {1}{r_{a}}}+{\frac {1}{r_{p}}}}}={\frac {2r_{a}r_{p}}{r_{a}+r_{p}}}.\end{aligned}}$

Harmonic oscillators

[edit]

The general solution for aharmonic oscillator in two or moredimensions 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 elasticspring; 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

[edit]

Inelectronics, the relative phase of two sinusoidal signals can be compared by feeding them to the vertical and horizontal inputs of anoscilloscope. If theLissajous figure display is an ellipse, rather than a straight line, the two signals are out of phase.

Elliptical gears

[edit]

Twonon-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 alink chain ortiming belt, or in the case of a bicycle the mainchainring may be elliptical, or anovoid similar to an ellipse in form. Such elliptical gears may be used in mechanical equipment to produce variableangular speed ortorque from a constant rotation of the driving axle, or in the case of a bicycle to allow a varying crank rotation speed with inversely varyingmechanical advantage.

Elliptical bicycle gears make it easier for the chain to slide off the cog when changing gears.^[30]

An example gear application would be a device that winds thread onto a conicalbobbin on aspinning machine. The bobbin would need to wind faster when the thread is near the apex than when it is near the base.^[31]

Optics

[edit]

In a material that is opticallyanisotropic (birefringent), therefractive index depends on the direction of the light. The dependency can be described by anindex ellipsoid. (If the material is opticallyisotropic, this ellipsoid is a sphere.)
In lamp-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).^[32]
In laser-plasma producedEUV light sources used in microchiplithography, 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.^[33]

Statistics and finance

[edit]

Instatistics, a bivariaterandom vector $(X,Y)$ isjointly 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 themultivariate normal distribution. The elliptical distributions are important in the financial field 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.^[34]^[35]

Computer graphics

[edit]

Drawing an ellipse as agraphics primitive is common in standard display libraries, such as the MacIntoshQuickDraw API, andDirect2D 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.^[36] Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.^[37]

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.^[38] 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 anaffine transformation of a circle. The spline methods used to draw a circle may be used to draw an ellipse, since the constituentBézier curves behave appropriately under such transformations.

Optimization theory

[edit]

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

Notes

[edit]

^Apostol, Tom M.; Mnatsakanian, Mamikon A. (2012),New Horizons in Geometry, The Dolciani Mathematical Expositions #47, The Mathematical Association of America, p. 251,ISBN 978-0-88385-354-2
^The German term for this circle isLeitkreis which can be translated as "Director circle", but that term has a different meaning in the English literature (seeDirector circle).
^^a ^b"Ellipse - from Wolfram MathWorld". Mathworld.wolfram.com. 2020-09-10. Retrieved2020-09-10.
^Protter & Morrey (1970, pp. 304, APP-28)
^Larson, Ron; Hostetler, Robert P.; Falvo, David C. (2006)."Chapter 10".Precalculus with Limits. Cengage Learning. p. 767.ISBN 978-0-618-66089-6.
^Young, Cynthia Y. (2010)."Chapter 9".Precalculus. John Wiley and Sons. p. 831.ISBN 978-0-471-75684-2.
^^a ^bLawrence, J. Dennis,A Catalog of Special Plane Curves, Dover Publ., 1972.
^Strubecker, K. (1967).Vorlesungen über Darstellende Geometrie. Göttingen: Vandenhoeck & Ruprecht. p. 26.OCLC 4886184.
^Bronstein&Semendjajew:Taschenbuch der Mathematik, Verlag Harri Deutsch, 1979,ISBN 3871444928, p. 274.
^Encyclopedia of Mathematics, Springer, URL:http://encyclopediaofmath.org/index.php?title=Apollonius_theorem&oldid=17516 .
^Blake, E. M. (1900). "The Ellipsograph of Proclus".American Journal of Mathematics.22 (2):146–153.doi:10.2307/2369752.JSTOR 2369752.
^K. Strubecker:Vorlesungen über Darstellende Geometrie. Vandenhoeck & Ruprecht, Göttingen 1967, S. 26.
^FromΠερί παραδόξων μηχανημάτων [Concerning Wondrous Machines]: "If, then, we stretch a string surrounding the points A, B tightly around the first point from which the rays are to be reflected, the line will be drawn which is part of the so-called ellipse, with respect to which the surface of the mirror must be situated."
Huxley, G. L. (1959).Anthemius of Tralles: A Study in Later Greek Geometry. Cambridge, MA. pp. 8–9.LCCN 59-14700.{{cite book}}: CS1 maint: location missing publisher (link)
^Al-Ḥasan's work was titledKitāb al-shakl al-mudawwar al-mustaṭīl [The Book of the Elongated Circular Figure].
Rashed, Roshdi (2014).Classical Mathematics from Al-Khwarizmi to Descartes. Translated by Shank, Michael H. New York: Routledge. p. 559.ISBN 978-13176-2-239-0.
^J. van Mannen:Seventeenth century instruments for drawing conic sections. In:The Mathematical Gazette. Vol. 76, 1992, p. 222–230.
^E. Hartmann: Lecture Note 'Planar Circle Geometries', an Introduction to Möbius-, Laguerre- and Minkowski Planes, p. 55
^W. Benz,Vorlesungen über Geomerie der Algebren,Springer (1973)
^Archimedes. (1897).The works of Archimedes. Heath, Thomas Little, Sir, 1861-1940. Mineola, N.Y.: Dover Publications. p. 115.ISBN 0-486-42084-1.OCLC 48876646.{{cite book}}:ISBN / Date incompatibility (help)
^Carlson, B. C. (2010),"Elliptic Integrals", inOlver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),NIST Handbook of Mathematical Functions, Cambridge University Press,ISBN 978-0-521-19225-5,MR 2723248.
^Ivory, J. (1798)."A new series for the rectification of the ellipsis".Transactions of the Royal Society of Edinburgh.4 (2):177–190.doi:10.1017/s0080456800030817.S2CID 251572677.
^Bessel, F. W. (2010). "The calculation of longitude and latitude from geodesic measurements (1825)".Astron. Nachr.331 (8):852–861.arXiv:0908.1824.Bibcode:2010AN....331..852K.doi:10.1002/asna.201011352.S2CID 118760590. English translation ofBessel, F. W. (1825). "Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermesssungen".Astron. Nachr. (in German).4 (16):241–254.arXiv:0908.1823.Bibcode:1825AN......4..241B.doi:10.1002/asna.18260041601.S2CID 118630614.
^Linderholm, Carl E.; Segal, Arthur C. (June 1995). "An Overlooked Series for the Elliptic Perimeter".Mathematics Magazine.68 (3):216–220.doi:10.1080/0025570X.1995.11996318. which cites toKummer, Ernst Eduard (1836)."Uber die Hypergeometrische Reihe" [About the hypergeometric series].Journal für die Reine und Angewandte Mathematik (in German).15 (1, 2):39–83,127–172.doi:10.1515/crll.1836.15.39.
^Cook, John D. (28 May 2023)."Comparing approximations for ellipse perimeter".John D. Cook Consulting blog. Retrieved2024-09-16.
^Ramanujan, Srinivasa (1914)."Modular Equations and Approximations toπ"(PDF).Quart. J. Pure App. Math.45:350–372.ISBN 978-0-8218-2076-6.{{cite journal}}:ISBN / Date incompatibility (help)
^^a ^bVillarino, Mark B. (20 June 2005). "Ramanujan's Perimeter of an Ellipse".arXiv:math.CA/0506384.We present a detailed analysis of Ramanujan's most accurate approximation to the perimeter of an ellipse. In particular, the second equation underestimates the circumference by $\pi (a+b)h^{5}\theta (h),$ where $22.888\cdot 10^{-6}<3\cdot 2^{-17}<\theta (h)\leq 4\left(1-{\frac {7\pi }{22}}\right)<1.60935\cdot 10^{-3}$ is an increasing function of $0\leq h\leq 1.$
^Cook, John D. (22 September 2024)."Error in Ramanujan's approximation for ellipse perimeter".John D. Cook Consulting blog. Retrieved2024-12-01.the relative error whenb = 1 anda varies ... is bound by4/π − 14/11 = 0.00051227….
^Jameson, G.J.O. (2014). "Inequalities for the perimeter of an ellipse".Mathematical Gazette.98 (542):227–234.doi:10.1017/S002555720000125X.S2CID 125063457.
^Prasolov, V.; Solovyev, Y. (1997).Elliptic Functions and Elliptic Integrals. American Mathematical Society. pp. 58–60.ISBN 0-8218-0587-8.
^Legendre'sTraité des fonctions elliptiques et des intégrales eulériennes
^David Drew."Elliptical Gears".[1]
^Grant, George B. (1906).A treatise on gear wheels. Philadelphia Gear Works. p. 72.
^Encyclopedia of Laser Physics and Technology - lamp-pumped lasers, arc lamps, flash lamps, high-power, Nd:YAG laser
^"Cymer - EUV Plasma Chamber Detail Category Home Page". Archived fromthe original on 2013-05-17. Retrieved2013-06-20.
^Chamberlain, G. (February 1983). "A characterization of the distributions that imply mean—Variance utility functions".Journal of Economic Theory.29 (1):185–201.doi:10.1016/0022-0531(83)90129-1.
^Owen, J.; Rabinovitch, R. (June 1983). "On the class of elliptical distributions and their applications to the theory of portfolio choice".Journal of Finance.38 (3):745–752.doi:10.1111/j.1540-6261.1983.tb02499.x.JSTOR 2328079.
^Pitteway, M.L.V. (1967)."Algorithm for drawing ellipses or hyperbolae with a digital plotter".The Computer Journal.10 (3):282–9.doi:10.1093/comjnl/10.3.282.
^Van Aken, J.R. (September 1984). "An Efficient Ellipse-Drawing Algorithm".IEEE Computer Graphics and Applications.4 (9):24–35.doi:10.1109/MCG.1984.275994.S2CID 18995215.
^Smith, L.B. (1971)."Drawing ellipses, hyperbolae or parabolae with a fixed number of points".The Computer Journal.14 (1):81–86.doi:10.1093/comjnl/14.1.81.

References

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Besant, W.H. (1907)."Chapter III. The Ellipse".Conic Sections. London: George Bell and Sons. p. 50.
Coxeter, H.S.M. (1969).Introduction to Geometry (2nd ed.). New York: Wiley. pp. 115–9.
Meserve, Bruce E. (1983) [1959],Fundamental Concepts of Geometry, Dover Publications,ISBN 978-0-486-63415-9
Miller, Charles D.; Lial, Margaret L.; Schneider, David I. (1990).Fundamentals of College Algebra (3rd ed.). Scott Foresman/Little. p. 381.ISBN 978-0-673-38638-0.
Protter, Murray H.; Morrey, Charles B. Jr. (1970),College Calculus with Analytic Geometry (2nd ed.), Reading:Addison-Wesley,LCCN 76087042