Ellipse

This article is about the geometric figure of the ellipse, for other meanings see Ellipse (disambiguation). For the algebraic curve, see Elliptic curve.

Ellipses are special closed oval curves in geometry. They belong to the conic sections along with the parabolas and the hyperbolas. An illustrative definition uses the property that the sum of the distances of an ellipse point from two given points, the foci, is the same for all points. If the foci are identical, you get a circle.

Every ellipse can be $\;(a\cos t,b\sin t)\;$ described in a suitable coordinate system by an equation $\;{\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1\;$ or parameter representation From this it can be seen that an ellipse can be thought of as a unit circle stretched on the x-axis around and on the y-axis around

The ellipse (from Greek ἔλλειψις élleipsis 'defect') was introduced and named by Apollonios of Perge (c. 262-190 BC), the name referring to the eccentricity ε

Ellipses do not only occur as plane sections of a cone. There are also ellipses on cylinders, ellipsoids, hyperboloids and elliptic paraboloids.

In nature, ellipses occur in the form of undisturbed Keplerian planetary orbits around the sun. Ellipses are also often needed when drawing oblique images, since a circle is generally mapped onto an ellipse by a parallel projection (see Ellipse (Descriptive Geometry)).

Ellipse with center , foci $F_{1}$ and $F_{2}$ , vertices $S_{1},\dotsc ,S_{4}$ , major axis (red) and minor axis (green)

Ellipse as a conic section. The central axis of the cone is inclined so far that the ellipse appears in true size in the side view from the right.

Saturn's rings appear elliptical.

Definition of an ellipse as a geometric place

There are different ways to define ellipses. Besides the usual definition by certain distances of points, it is also possible to define an ellipse as an intersection curve between a corresponding inclined plane and a cone (see 1st picture) or as an affine image of the unit circle.

An ellipse is the geometric locus of all points the plane for which the sum of the distances to two given points $F_{1}$ and $F_{2}$ equal to a given constant. This constant is usually denoted by The points $F_{1}$ and $F_{2}$ called foci:

$E=\{P\mid |PF_{2}|+|PF_{1}|=2a\}$

To exclude a distance, assume that greater than the distance $|F_{1}F_{2}|$ of the foci. If the two foci coincide, a circle with radius . This simple case is often tacitly excluded in the following considerations, since most statements about ellipses become trivial in the circle case.
The center of the line ${\overline {F_{1}F_{2}}}$ is called the center of the ellipse. The straight line through the foci is the major axis and the orthogonal straight line through the minor axis. The two ellipse points $S_{1},\;S_{2}$ on the major axis are the major vertices. The distance of the principal vertices from the center is and is called the major semimajor axis. The two ellipse vertices $S_{3},\;S_{4}$ on the minor axis are the minor vertices, and their distance to the center is respectively the minor semi-axis . The distance the foci to the center is called the linear eccentricity and ε $\varepsilon=e/a$ the numerical eccentricity. Using the Pythagorean theorem, $a^{2}=e^{2}+b^{2}$ (see drawing).

The equation $|PF_{2}|+|PF_{1}|=2a$ can also be interpreted like this: If $c_{2}$ the circle around $F_{2}$ with radius , then the distance of the point to the circle $c_{2}$ equal to the distance of the point to the focal point $F_{1}$ :

$|PF_{1}|=|Pc_{2}|$

$c_{2}$ is called the guiding circle of the ellipse with respect to the focal point $F_{2}$ . This property should not be confused with the guiding property of an ellipse (see below).

With the help of Dandelin spheres one proves that holds:

Any intersection of a cone with a plane that does not contain the apex of the cone and whose slope is less than that of the cone's generatrices is an ellipse.

Due to the guide circle property, an ellipse is the equidistance curve to each of its foci and the guide circle with the other focal point as the center.

This graphic shows the designations used in the following text

Ellipse: definition with guide circle

Ellipse in Cartesian coordinates

Equation

A. Introduce Cartesian coordinates such that the center of the ellipse is at the origin, the axis is the major axis, and

the foci are the points $F_{1}=(e,0),\ F_{2}=(-e,0)$ ,

the main vertices $S_{1}=(a,0),\ S_{2}=(-a,0)$ are

so for any point (x,y) the distance to the focal point (e,0) is $\sqrt{ (x-e)^2 + y^2 }$ and to the second focal point $\sqrt{ (x+e)^2 + y^2 }$ . So the point lies on the ellipse (x,y) exactly when the following condition is satisfied:

${\sqrt {(x-e)^{2}+y^{2}}}+{\sqrt {(x+e)^{2}+y^{2}}}=2a$

After removing the roots by suitable squaring and using the relation $b^{2}=a^{2}-e^{2}$ (see above) we obtain the equation

${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1$ or resolved to

$y=\pm {\frac {b}{a}}{\sqrt {a^{2}-x^{2}}}.$

$S_{3}=(0,b),\;S_{4}=(0,-b)$ are the minor vertices. From the relation $b^{2}=a^{2}-e^{2}$ we get the equations

$e={\sqrt {a^{2}-b^{2}}}$ and ε $\varepsilon ={\frac {\sqrt {a^{2}-b^{2}}}{a}}\ .$

From this still result the relations

$b=a{\sqrt {1-\varepsilon ^{2}}}$

$p=a\cdot (1-\varepsilon ^{2})$

If a=b , then ε $\varepsilon =0$ and the ellipse is a circle.
If b=e , then ε $\varepsilon ={\tfrac {1}{\sqrt {2}}}$ , and the ellipse is called an equilateral ellipse or ellipse of most beautiful form.

B. The ellipse in A. can also be written using a bilinear form as the solution set of the equation ${\vec {x}}^{\mathrm {T} }M\,{\vec {x}}=1$ . Here, the vectors and $x^{\mathrm {T} }$ identified with the same point Introducing Cartesian coordinates, the matrix ${\begin{pmatrix}{\frac {1}{a^{2}}}&0\\0&{\frac {1}{b^{2}}}\end{pmatrix}}$ , ${\vec {x}}^{\mathrm {T} }=(x,y)$ a row vector and ${\vec {x}}={\begin{pmatrix}x\\y\end{pmatrix}}$ a column vector.

C. An ellipse with the center at the origin and the foci on the axis is also called in 1st principal position. When the above ellipse equation is mentioned here, it is always assumed that $a\geq b$ and thus the ellipse is in 1st principal position, but this need not be the case in "real life". It is also possible that a<b , which means that the ellipse is in 2nd main position (the foci are on the y {\displaystyle ).

Due to the definition of an ellipse holds:

An ellipse is symmetrical about its axes and therefore also about its center.

(The symmetry property can also be easily seen in the equation of an ellipse derived here).

Half parameter

Half the length of an ellipse passing through a focal point and perpendicular to the major axis is called the semi-parameter, sometimes just parameter or semi-latus rectum (half of latus rectum = $2\cdot p$ ) of the ellipse. With the help of the equation of an ellipse it is easy to calculate that

$p={\frac {b^{2}}{a}}$

is valid. The half parameter has the additional meaning (see below): The radius of curvature in the main vertices is .

Tangent

A. For the major vertex (a,0) or (-a,0) the tangent has the equation x=a or $x=-a$ . The simplest way to determine the equation of the tangent line at an ellipse point $(x_{0},y_{0}\neq 0)$ is, implicitly differentiate the equation ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1$ of the ellipse. Hereby results for the derivation

${\frac {2x}{a^{2}}}+{\frac {2yy'}{b^{2}}}=0\ \rightarrow \ y'=-{\frac {x}{y}}{\frac {b^{2}}{a^{2}}}$

and thus the point-slope form of the tangent line at the point $(x_{0},y_{0})$ :

$y=-{\frac {x_{0}}{y_{0}}}{\frac {b^{2}}{a^{2}}}(x-x_{0})+y_{0}$

Considering ${\tfrac {x_{0}^{2}}{a^{2}}}+{\tfrac {y_{0}^{2}}{b^{2}}}=1$ , then the equation of the tangent line at the point obtained $(x_{0},y_{0})$ :

${\frac {x_{0}}{a^{2}}}x+{\frac {y_{0}}{b^{2}}}y=1$

This form also includes the tangents through the main vertices. The latter also applies to the vector form

${\vec {x}}={\begin{pmatrix}x_{0}\\y_{0}\end{pmatrix}}+s\;{\begin{pmatrix}-ay_{0}/b\\bx_{0}/a\end{pmatrix}}\quad {\text{mit}}\quad s\in \mathbb {R}$ .

B. The tangent equation ${\tfrac {x_{0}}{a^{2}}}x+{\tfrac {y_{0}}{b^{2}}}y=1$ A. can also be introduced without differential calculus as a special case of a polar equation (see below derivation of the position relations of pole and polar, D.). It corresponds to a normal form with the normal vector ${\vec {n}}=\left({\tfrac {x_{0}}{a^{2}}},{\tfrac {y_{0}}{b^{2}}}\right)$ . From this, a direction vector ${\vec {u}}$ of perpendicular to it can be read. Since ${\vec {u}}$ is unique except for one scalar, it has the forms

${\vec {u}}={\begin{pmatrix}-y_{0}/b^{2}\\x_{0}/a^{2}\end{pmatrix}}=$ ${\frac {1}{ab}}{\begin{pmatrix}-ay_{0}/b\\bx_{0}/a\end{pmatrix}}=$ $-{\frac {y_{0}}{b^{2}}}{\begin{pmatrix}1\\{\frac {-x_{0}}{y_{0}}}{\frac {b^{2}}{a^{2}}}\end{pmatrix}}$ ;

this gives the direction vector of the vector shape given in A. and also the slope of the point slope shape given there.

A graphical determination of elliptic tangents can be found in the article Ellipse (Descriptive Geometry).

Equation of a displaced ellipse

If we move the above ellipse so that the center $(m_{1},m_{2})$ is the point , we get the center shape of an ellipse whose axes are parallel to the coordinate axes:

${\frac {(x-m_{1})^{2}}{a^{2}}}+{\frac {(y-m_{2})^{2}}{b^{2}}}=1$

Parameter representations

Standard display

The usual parameter representation of an ellipse uses the sine and cosine function. Because of $\cos ^{2}t+\sin ^{2}t=1$ describes

$(a\cos t,b\sin t),\ 0\leq t<2\pi$

the ellipse ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1.$

Different ways of interpreting the parameter geometrically are given in the section Drawing ellipses.

Rational parameter representation

With the substitution $u=\tan(t/2)$ and trigonometric formulas we get

$\cos t=(1-u^{2})/(1+u^{2})\ ,\quad \sin t=2u/(1+u^{2})$

and thus the rational parameter representation of an ellipse:

${\begin{array}{lcl}x(u)&=&a(1-u^{2})/(1+u^{2})\\y(u)&=&2bu/(1+u^{2})\end{array}},\quad -\infty <u<\infty \;.$

The rational parameter representation has the following properties (see figure):

For the main positive vertex is shown: $x(0)=a,y(0)=0$ ; for the minor positive vertex: $x(1)=0,y(1)=b$ .
Transition to the opposite number of the parameter reflects the represented point on the axis: $x(-u)=x(u),y(-u)=-y(u)$ ;
Transition to the inverse of the parameter reflects the represented point on the axis: $x\left({\tfrac {1}{u}}\right)=-x(u),y\left({\tfrac {1}{u}}\right)=y(u)$ .
The negative principal vertex cannot be represented with any real parameter The coordinates of the same are the limits of the parameter representation for infinite positive or negative : $\lim _{u\to \pm \infty }x(u)=-a,\lim _{u\to \pm \infty }y(u)=0$ .

Rational parameter representations of conic sections (ellipse, hyperbola, parabola) play an important role in CAD for quadratic rational Bezier curves.

Tangent slope as parameter

A parameter representation using the tangent slope at the particular ellipse point is obtained by differentiating the parameter representation ${\vec {x}}(t)=(a\cos t,b\sin t)^{\mathrm {T} }$ :

${\vec {x}}'(t)=(-a\sin t,b\cos t)^{\mathrm {T} }\quad \rightarrow \quad m=-{\frac {b}{a}}\cot t\quad \rightarrow \quad \cot t=-{\frac {ma}{b}}.$

With the help of trigonometric formulas we get

$\cos t={\frac {\cot t}{\pm {\sqrt {1+\cot ^{2}t}}}}={\frac {-ma}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}\ ,\quad \quad \sin t={\frac {1}{\pm {\sqrt {1+\cot ^{2}t}}}}={\frac {b}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}.$

Replacing $\cos t$ and standard representation. $\sin t$ one finally obtains

${\vec {c}}_{\pm }(m)=\left(-{\frac {ma^{2}}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}\;,\;{\frac {b^{2}}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}\right)^{\mathrm {T} },\,m\in \mathbb {R} .$

Here is the tangent slope at the respective ellipse point, ${\vec {c}}_{+}$ upper and ${\vec {c}}_{-}$ the lower half of the ellipse. The points with perpendicular tangents (vertex $(\pm a,0)$ ) are not captured by this parameter representation.
The equation of the tangent at the point ${\vec c}_{\pm }(m)$ has the form $y=mx+n$ . The -intercept obtained by substituting the coordinates of the corresponding ellipse point ${\vec c}_{\pm }(m)$ :

$y=mx\pm {\sqrt {m^{2}a^{2}+b^{2}}}$

This main form of the tangent equation is an essential tool in determining the orthoptic curve of an ellipse.

Remark. The main form of the tangent equation and the coordinates of ${\vec c}_{\pm }(m)$ can also be derived without differential calculus and without trigonometric formulas by taking the tangent as a special case of a polar (see below Derivation of the positional relations of pole and polar, D.).

Displaced ellipse

A displaced ellipse with center $(m_{1},m_{2})$ is given by

$(m_{1}+a\cos t\;,\;m_{2}+b\sin t),\ 0\leq t<2\pi$

described.

A parameter representation of an arbitrary ellipse is given in the section Ellipse as an affine image of the unit circle.

Points of an ellipse calculated using the rational parameter representation ( $\Delta u=0{,}2$ )

Properties

Focal Point Property

→ Main article: Focal point (geometry)

The connecting line between a focal point and a point of the ellipse is called focal line, guide ray or focal ray. Focal points and focal rays got their name because of the following property:

The angle between the two focal rays at one point of the ellipse is bisected by the normal at that point.

Applications

The angle of incidence formed by one focal ray with the tangent is equal to the angle of reflection formed by the tangent with the other focal ray. A light beam that originates from one focal point is thus reflected at the elliptical tangent in such a way that it hits the other focal point. In the case of an elliptical mirror, all light rays emanating from one focal point therefore meet at the other focal point.
Since all paths from one focal point to the other (along associated focal rays) are of equal length, sound, for example, is "amplified" by constructive interference.
The tangent at the ellipse point is the bisector of the exterior angle. Since angle bisectors are easy to construct, the focal point property provides a simple method to construct the tangent at an ellipse point (Another tangent construction is described in Ellipse (Descriptive Geometry)).

Two ellipses with the same foci $F_{1},\;F_{2}$ called confocal. Through each point that is not between the foci, there is exactly one ellipse with foci $F_{1},\;F_{2}$ . Two confocal ellipses have no intersection point (see definition of an ellipse).

Proof of the focal point property

Since the tangent is perpendicular to the normal, the above assertion is proven if the analogous statement holds for the tangent:

The exterior angle of the focal rays ${\overline {PF_{1}}},{\overline {PF_{2}}}$ in an ellipse point is bisected by the tangent line at this point (see figure).

Let be the point on the straight line $\overline{PF_2}$ with distance to the focal point $F_{2}$ ( is the major semi-axis of the ellipse). Let the straight be the bisector of the exterior angles of the focal rays ${\overline {PF_{1}}},{\overline {PF_{2}}}$ . To prove that is the tangent line, show that no other ellipse point can lie on From the drawing and the triangle inequality we can see that

$|QF_{2}|+|QF_{1}|=|QF_{2}|+|QL|>|LF_{2}|=2a$

holds. This means that But if were a point of the ellipse, the sum would have to be equal to

Remark: A proof by means of analytic geometry is in the proof archive.

Natural occurrence and application in technology:

The ceilings of some caves resemble an ellipse half. If you are - with your ears - in one focal point of this ellipse, you hear any sound whose origin lies in the second focal point amplified ("whispering vault"). This type of sound transmission even works from platform to platform in some stations of the Paris Métro. The same principle of sound focusing is used today to break up kidney stones with shock waves. A reflector in the shape of an ellipse is also used in the lamp-pumped Nd:YAG laser. The pump source - either a flash lamp or an arc lamp - is positioned in one focal point, and the doped crystal is placed in the other focal point.

Directrix

For a true ellipse, i.e. $e>0$ , a parallel to the minor axis at distance $a^{2}/e$ a directrix or directrix. For any point of the ellipse, the ratio of its distance from a focal point to the distance from the directrix on the corresponding side of the minor axis is equal to the numerical eccentricity:

$|PF_{1}|:|Pd_{1}|=|PF_{2}|:|Pd_{2}|=\varepsilon .$ It is ε $\varepsilon >0.$

Proof:
With ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1\Leftrightarrow y^{2}=b^{2}-{\tfrac {b^{2}x^{2}}{a^{2}}}$ and $e^{2}+b^{2}=a^{2}$ and the binomial formulas is

$|PF_{1,2}|^{2}=(x\mp e)^{2}+y^{2}=$

${(x\mp e)^{2}+b^{2}-{\frac {b^{2}x^{2}}{a^{2}}}={\frac {a^{2}x^{2}}{a^{2}}}\mp 2ex+e^{2}+b^{2}-{\frac {b^{2}x^{2}}{a^{2}}}={\frac {e^{2}x^{2}}{a^{2}}}\mp 2{\frac {ex}{a}}\cdot a+a^{2}=\left({\frac {ex}{a}}\mp {\frac {ea^{2}}{ae}}\right)^{2}=}$

$\left({\frac {e}{a}}\right)^{2}\left(x\mp {\frac {a^{2}}{e}}\right)^{2}=\varepsilon ^{2}|Pd_{1,2}|^{2}$ .

The inverse of this statement is also true and can be used to further define an ellipse (similar to a parabola):

For a point (focal point), a straight line (directrix) not through and a real number ε $\varepsilon$ with $0 < \varepsilon < 1$ , the set of points (geometric locus) for which the quotient of the distances to the point and the straight line equals ε $\varepsilon$ an ellipse:

$E=\{P\mid {\frac {|PF|}{|Pd|}}=\varepsilon \}$

The choice ε $\varepsilon=0$ , i.e. the eccentricity of a circle, is not allowed in this context. One can take the infinitely distant straight line as the directrix of a circle.

Proof:

$F=(f,0),\ \varepsilon >0$ 0 , be (0,0) point of the curve. The directrix has the equation $x=-\tfrac{f}{\varepsilon}$ . With P=(x,y) and the relation $|PF|^{2}=\varepsilon ^{2}|Pd|^{2}$ results in

$(x-f)^{2}+y^{2}=\varepsilon ^{2}(x+{\tfrac {f}{\varepsilon }})^{2}=(\varepsilon x+f)^{2}$ and $x^2(\varepsilon^2-1)+2xf(1+\varepsilon)-y^2=0.$

The substitution $p=f(1+\varepsilon)$ yields

$x^2(\varepsilon^2-1)+2px-y^2=0.$

This is the equation of an ellipse ( $\varepsilon<1$ a parabola ( ε = $\varepsilon =1$ a hyperbola ( ε > All these non-degenerate conic sections have the origin as vertex in common (see figure).

For ε $\varepsilon<1$ one introduces new parameters $a={\tfrac {p}{1-\varepsilon ^{2}}}$ and $b^{2}=ap\Rightarrow 1-\varepsilon ^{2}={\tfrac {b^{2}}{a^{2}}}$ }}}}; the above equation becomes then

${\frac {(x-a)^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1\ ,$

which is the equation of an ellipse with center (a,0) , the axis as major axis and the semi-axes a,b .

General case:

For the focal point $F=(f_{1},f_{2})$ and the directrix $ux+vy+w=0$ we get the equation

$\left(x-f_{1}\right)^{2}+\left(y-f_{2}\right)^{2}=\varepsilon ^{2}\cdot {\frac {\left(ux+vy+w\right)^{2}}{u^{2}+v^{2}}}.$

The right side of the equation uses the Hessian normal form of a straight line to calculate the distance of a point from a straight line.

Guideline construction:

Because of $e\cdot {\tfrac {a^{2}}{e}}=a^{2}$ point $D_{1}$ the directrix (see picture) and the focal point are inverse with $F_{1}$ respect to the reflection at the large vertex circle (green in the picture). Thus $D_{1}$ can be constructed from $F_{1}$ using the large vertex circle as shown in the image. Further justification for the construction is provided by the fact that the focal point $F_{1}$ and the directrix $d_{1}$ form a pole-polar pair (see below) with respect to both the ellipse and the large vertex circle.

Conjugate diameters

If for any ellipse diameter (of an ellipse chord through the ellipse center) consider ${\overline {PP'}}$ all parallel chords, their centers also lie on an ellipse diameter ${\overline {QQ'}}$ . One calls ${\overline {QQ'}}$ the ${\overline {PP'}}$ diameter conjugate to
If we form the conjugate diameter again to the conjugate diameter, we get the original one again. Thus, in the drawing, the diameter conjugate to ${\overline {QQ'}}$ coincides with the original diameter ${\overline {PP'}}$
The tangents in the endpoints of a diameter (say ${\overline {PP'}}$ ) are parallel to the conjugate diameter (in the example ${\overline {QQ'}}$ ).
Major and minor axes are the only pair of orthogonal conjugate diameters.
If the ellipse is a circle, exactly the orthogonal diameters are (also) conjugate.
If conjugate diameters are not orthogonal, the product of their slopes is ${\tfrac {-b^{2}}{a^{2}}}$ .
Let $d_{1}$ , $d_{2}$ conjugate diameters. Then $\left({\tfrac {d_{1}}{2}}\right)^{2}+\left({\tfrac {d_{2}}{2}}\right)^{2}=a^{2}+b^{2}$ . (Theorem of Apollonius)

Conjugate diameters (primarily of ellipses) are also treated on a separate Wikipedia page, as well as the theorem of Apollonius (including proof). A complete analytical proof of all statements listed here, which starts from the common bilinear form of two lines of origin, can be found in the proof archive. This proof needs neither trigonometric functions nor parameter representations nor an affine mapping.

One possible application in the field of technical drawing is the ability to find the highest point of an ellipse or elliptical arc of any position above a line - useful e.g. for correct 2D representations of non-orthogonal views of cylindrical bodies or rounded edges without using 3D programs. This is important for the clean connection of lines running tangentially away from the ellipse. For this purpose, two chords parallel to the desired tangent direction and the line of the corresponding conjugate diameter defined by the centers of the two chords must be drawn into the ellipse or the elliptical arc. The intersection of this line with the ellipse or the elliptical arc defines the connection point of the tangent (and normally the end point of the elliptical arc).

Orthogonal tangents

→ Main article: Orthoptic curve

For the ellipse $\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1$ the intersections of orthogonal tangents lie on the circle $x^{2}+y^{2}=a^{2}+b^{2}$ .

This circle is called the orthoptic curve of the given ellipse, it is the circumcircle of the rectangle circumscribing the ellipse.

Pole-Polar Relationship

If one introduces Cartesian coordinates in such a way that the center of the ellipse lies in the origin, then an arbitrary ellipse can be described with the equation ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1$ (see above section Equation). Further, for a given ellipse, a function assigns to each a point P_0=(x_0,y_0) the straight line ${\frac {x_{0}x}{a^{2}}}+{\frac {y_{0}y}{b^{2}}}=1$ With respect to $P_{0}$ called a pole, the associated straight line a polar. is a bijection; the inverse function maps one polar to each pole. The ellipse center (0,0) is not contained in any polar thus defined; correspondingly, (0,0) no polar exists to The given equation of the polar can be $\left({\tfrac {x_{0}}{a^{2}}},{\tfrac {y_{0}}{b^{2}}}\right)$ understood as a normal form with the corresponding normal vector

Such a relationship between points and lines mediated by a conic section is called a pole-polar relationship or simply polarity. Pole-polar relationships also exist for hyperbolas and parabolas, see also Pole and polar.

The following positional relationships apply to pole and polar:

The focal point and the directrix $x={\tfrac {a^{2}}{e}}$ are polar to each other. Since both are also polar with respect to the vertex circle $x^{2}+y^{2}=a^{2}$ , the directrix can also be constructed with the help of compass and ruler (see also circle mirroring). (1)
Exactly when the pole is outside the ellipse, the polar has two points in common with the ellipse (see figure: $P_2,\ p_2$ ). (2)
Exactly when the pole lies on the ellipse, the polar has exactly one point in common with the ellipse (= the polar is a tangent; see figure: $P_1,\ p_1$ ). (3)
Exactly when the pole is inside the ellipse, the polar has no point in common with the ellipse (see figure: $F_{1},\ l_{1}$ ). (4)
Each common point of a polar and an ellipse is a tangent point from the corresponding pole to the ellipse (see figure: $P_2,\ p_2$ ). (5)
The intersection of two polars is the pole of the straight line through the poles. (6)

Derivation of the positional relationships of pole and polar; alternative derivation of a tangent and an ellipse equation.

A. If a polar is parallel to the axis, it also has the form $0\neq c=x\Leftrightarrow {\frac {1}{c}}\cdot x+0\cdot y=1$ . With the associated normal vector $\left({\tfrac {x_{0}}{a^{2}}}={\tfrac {1}{c}},{\tfrac {y_{0}}{b^{2}}}=0\right)$ is the associated pole $\left(x_{0}={\tfrac {a^{2}}{c}},y_{0}=0\right).$ In particular, for $x_{0}={\tfrac {a^{2}}{c}}=e\Leftrightarrow c={\tfrac {a^{2}}{e}}$ follows the polarity (1) of focal point and directrix.

Inserting the considered polar curve into the center form of an ellipse results in the condition for the ordinate any intersection point ${\tfrac {c^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1$ ; the discriminant of this quadratic equation in has, except for a positive factor, the form

$T_{1}=\left({\frac {a^{2}}{c^{2}}}-1=\right)\quad \ {\frac {x_{0}^{2}}{a^{2}}}-1$ .

B. If a polar is not parallel to the axis, it has the principal form $y=mx+n$ . Because of $n \neq 0$ this can be transformed into the normal form $-mx/n+y/n=1$ Comparison with the normal form results in coordinates of the pole with the parameters of the main form:

$x_{0}=-{\frac {ma^{2}}{n}},\quad \ y_{0}={\frac {b^{2}}{n}}$ .

Inserting the main form $y=mx+n$ into the midpoint form of an ellipse yields for the abscissa of any intersection the condition ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {(mx+n)^{2}}{b^{2}}}=1$ ; the discriminant of this quadratic equation in has, except for a positive factor, the form

$T_{2}=\left({\frac {m^{2}a^{2}}{n^{2}}}+{\frac {b^{2}}{n^{2}}}-1=\right)\quad \ {\frac {x_{0}^{2}}{a^{2}}}+{\frac {y_{0}^{2}}{b^{2}}}-1.$

C. Overall, the term $T=T_{1}$ or $T=T_{2}$ allows the following distinction of pairwise disjoint cases for any polar:

For $T<0$ the polar no point in common with the ellipse, and the pole lies inside the ellipse. From this follows (2).
For polar has exactly one point in common with the ellipse, and the pole lies on the ellipse. So the polar is a tangent to the ellipse, the pole is its point of contact (see figure: $P_1,\ p_1$ ). From this follows (3).
For the polar two points in common with the ellipse, and the pole lies outside the ellipse. From this follows (4).

D. If a tangent is not perpendicular, solving the equation $T_{2}=0$ to and substituting gives the principal form of the tangent:

$\quad \ y=mx\pm {\sqrt {m^{2}a^{2}+b^{2}}}$ ;

Substituting into the coordinates $x_{0}=-{\tfrac {ma^{2}}{n}},y_{0}={\tfrac {b^{2}}{n}}$ the touch point gives the coordinates of the parameter representation of an ellipse with slope as parameter: $\quad \ {\vec {c}}_{\pm }(m)=\left(-{\frac {ma^{2}}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}\;,\;{\frac {b^{2}}{\pm {\sqrt {m^{2}a^{2}+b^{2}}}}}\right)^{\mathrm {T} },\,m\in \mathbb {R}$ ;

this parameter representation does not capture the main vertices.

E. Starting from the bilinear form of the ellipse given in the section "Equation", B., the polar to the point has the normal forms

${\vec {p}}^{\mathrm {T} }M{\vec {x}}=1$ with normal vector ${\vec {n}}^{\mathrm {T} }={\vec {p}}^{\mathrm {T} }M$ and

${\vec {x}}^{\mathrm {T} }M{\vec {p}}=1$ with normal vector $M{\vec {p}}={\vec {n}}$ .

If is a point of the ellipse, these equations also describe a tangent.

This coordinate-free computational representation of the polar is suitable for proofs. With the coordinate representations $P(x_{0},y_{0})$ and X(x,y) and the matrix coordinates for given in the section "Equation", the equation ${\tfrac {x_{0}}{a^{2}}}x+{\tfrac {y_{0}}{b^{2}}}y=1$ obtained again by evaluating the matrix products.

Proof of (5) ("Every common point of a polar and an ellipse is a tangent point from the associated pole to the ellipse."):
Since the ellipse points $S_{1},S_{2}$ lie on the polar to ${\vec {s}}_{1}^{\mathrm {T} }M{\vec {p}}=1$ and ${\vec {s}}_{2}^{\mathrm {T} }M{\vec {p}}=1$ . If one does not include in these equations $M{\vec {p}}$ , but ${\vec {s}}_{1}^{\mathrm {T} }M$ or ${\vec {s}}_{2}^{\mathrm {T} }M$ as a normal vector, they state that the tangents at the ellipse points $S_{1},S_{2}$ the point in common.

Proof of (6) ("The intersection of two polars is the pole of the straight line through the poles."):
For an intersection two polars to $P_{1}$ and $P_{2}$ holds ${\vec {s}}^{\mathrm {T} }M{\vec {p}}_{1}=1$ and ${\vec {s}}^{\mathrm {T} }M{\vec {p}}_{2}=1$ . If in these equations we do not take $M{\vec {p}}_{1}$ or $M{\vec {p}}_{2}$ , but ${\vec {s}}^{\mathrm {T} }M={\vec {n}}^{\mathrm {T} }$ as a normal vector, they state that on the polar to the points $P_{1}$ , $P_{2}$ lie. Further, consideration of the parametric form ${\vec {x}}={\vec {p}}_{1}+\lambda ({\vec {p}}_{2}-{\vec {p}}_{1})$ with

${\vec {n}}^{\mathrm {T} }{\vec {x}}={\vec {n}}^{\mathrm {T} }{\vec {p}}_{1}+\lambda ({\vec {n}}^{\mathrm {T} }{\vec {p}}_{2}-{\vec {n}}^{\mathrm {T} }{\vec {p}}_{1})=1+\lambda (1-1)=1={\vec {s}}^{\mathrm {T} }M{\vec {x}}$

the pointwise equality of the straight line $(P_{1}P_{2})$ with the polar to .

Conical section coulter with a common apex and a common half parameter

Ellipse with orthoptic curve (purple)

Ellipse: Pole-Polar Relationship

Focal Point Property

The tangent bisects the exterior angle of the focal rays

Ellipse with guidelines

Ellipse with two conjugate diameters

Guideline construction

Ellipse as affine image of the unit circle

Another definition of the ellipse uses a special geometric mapping, namely affinity. Here the ellipse is defined as an affine image of the unit circle.

Parameter representation

An affine mapping in the real plane is of the form ${\vec {x}}\to {\vec {f}}_{0}+A{\vec {x}}$ , where is a regular matrix (determinant not 0) and ${\vec {f}}_{0}$ is any vector. If ${\vec {f}}_{1},\;{\vec {f}}_{2}$ the column vectors of the matrix , then the unit circle $(\cos t,\sin t),0\leq \ t\leq 2\pi ,$ applied to the ellipse

${\vec {x}}={\vec {p}}(t)={\vec {f}}_{0}+{\vec {f}}_{1}\cos t+{\vec {f}}_{2}\sin t$

${\vec {f}}_{0}$ is the center and ${\vec {f}}_{1},\;{\vec {f}}_{2}$ are two conjugate radii (see below). below) of the ellipse. ${\vec {f}}_{1},\;{\vec {f}}_{2}$ generally not perpendicular to each other. That is, ${\vec {f}}_{0}\pm {\vec {f}}_{1}$ and ${\vec {f}}_{0}\pm {\vec {f}}_{2}$ are usually not the vertices of the ellipse. This definition of an ellipse provides a simple parameter representation (see below) of any ellipse.

Vertex, vertex shape

Since in a vertex the tangent is perpendicular to the associated ellipse diameter and the tangent direction in an ellipse point is ${\vec {p}}'(t)=-{\vec {f}}_{1}\sin t+{\vec {f}}_{2}\cos t$ the parameter $t_{0}$ of a vertex results from the equation

${\vec {p}}'(t)\cdot ({\vec {p}}(t)-{\vec {f}}_{0})=(-{\vec {f}}_{1}\sin t+{\vec {f}}_{2}\cos t)\cdot ({\vec {f}}_{1}\cos t+{\vec {f}}_{2}\sin t)=0$

and thus from $\cot(2t_{0})={\tfrac {{\vec {f}}_{1}^{\,2}-{\vec {f}}_{2}^{\,2}}{2{\vec {f}}_{1}\cdot {\vec {f}}_{2}}}$ .
(The formulas
were $\cos ^{2}t-\sin ^{2}t=\cos 2t,\ 2\sin t\cos t=\sin 2t$ used).

If ${\vec {f}}_{1}\cdot {\vec {f}}_{2}=0$ , $t_{0}=0$ and the parameter representation is already in vertex form.

The 4 vertices of the ellipse are ${\vec {p}}(t_{0}),{\vec {p}}(t_{0}\pm {\frac {\pi }{2}}),{\vec {p}}(t_{0}+\pi ).$

The vertex shape of the parameter representation of the ellipse is

${\vec {x}}={\vec {p}}(t)={\vec {f}}_{0}+({\vec {p}}(t_{0})-{\vec {f}}_{0})\cos(t-t_{0})+({\vec {p}}(t_{0}+{\tfrac {\pi }{2}})-{\vec {f}}_{0})\sin(t-t_{0}).$

Semi-axes

With the abbreviations $\;M={\vec {f}}_{1}^{2}+{\vec {f}}_{2}^{2},\ N=|\det({\vec {f}}_{1},{\vec {f}}_{2})|\;$ follows from Apollonios' two theorems:

$a^{2}+b^{2}=M,\quad ab=N$

If we solve for a,b , we get (see Steiner ellipse)

$a={\frac {1}{2}}({\sqrt {M+2N}}+{\sqrt {M-2N}})$

$b={\frac {1}{2}}({\sqrt {M+2N}}-{\sqrt {M-2N}})\ .$

Area

From the second theorem of Apollonios follows:
The area of an ellipse $\;{\vec {x}}={\vec {f}}_{0}+{\vec {f}}_{1}\cos t+{\vec {f}}_{2}\sin t\;$ is

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

For example 3, $\ A=\pi \;2{\sqrt {3}}\ .$

Examples

$\vec f_0=\begin{pmatrix} 0 \\ 0 \end{pmatrix},\ \vec f_1=\begin{pmatrix} a \\ 0 \end{pmatrix},\ \vec f_2=\begin{pmatrix} 0 \\ b \end{pmatrix}$ provides the usual parameter representation of the ellipse with the equation ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1:\quad {\vec {x}}={\vec {p}}(t)={\begin{pmatrix}a\cos t\\b\sin t\end{pmatrix}}$ .
${\vec {f}}_{0}={\begin{pmatrix}x_{0}\\y_{0}\end{pmatrix}},\ {\vec {f}}_{1}={\begin{pmatrix}a\cos \varphi \\a\sin \varphi \end{pmatrix}},\ {\vec {f}}_{2}={\begin{pmatrix}-b\sin \varphi \\b\cos \varphi \end{pmatrix}}$ gives the parameter representation of the ellipse, which ${\vec {f}}_{0}$ arises from ${\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1$ by rotating it by angle φ $\varphi$ and then shifting it by The parameter representation is already in vertex form. That is, ${\vec {f}}_{0}\pm {\vec {f}}_{1}$ and ${\vec {f}}_{0}\pm {\vec {f}}_{2}$ are the vertices of the ellipse.
The parameter representation

${\vec {x}}={\vec {p}}(t)={\begin{pmatrix}{\sqrt {3}}\\0\end{pmatrix}}\cos t+{\begin{pmatrix}1\\2\end{pmatrix}}\sin t$

of an ellipse is not in vertex form.

The vertex parameter is given by $\cot(2t_{0})=-{\tfrac {1}{\sqrt {3}}}$ to $t_{0}=-{\tfrac {\pi }{6}}$ .

The vertex form of the parameter representation is:

${\vec {x}}={\vec {p}}(t)={\begin{pmatrix}\ 1\\-1\end{pmatrix}}\cos(t+{\tfrac {\pi }{6}})+{\sqrt {3}}{\begin{pmatrix}1\\1\end{pmatrix}}\sin(t+{\tfrac {\pi }{6}})$

The vertices are: $(1,-1),(-1,1),({\sqrt {3}},{\sqrt {3}}),(-{\sqrt {3}},-{\sqrt {3}})$ and

the semi-axes: $a={\sqrt {2}},\ b={\sqrt {6}}.$

Implicit representation

Solving the parameter representation using Cramer's rule for $\;\cos t,\sin t\;$ and using $\;\cos ^{2}t+\sin ^{2}t-1=0\;$ , one obtains the implicit representation

$\det({\vec {x}}\!-\!{\vec {f}}\!_{0},{\vec {f}}\!_{2})^{2}+\det({\vec {f}}\!_{1},{\vec {x}}\!-\!{\vec {f}}\!_{0})^{2}-\det({\vec {f}}\!_{1},{\vec {f}}\!_{2})^{2}=0$ .

For example 3 we get: $\ x^{2}-xy+y^{2}-3=0\ .$

If we rotate the ellipse with the equation $\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1$ around the zero point (center) its equation has the form

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

Conversely, if the equation of a rotated ellipse is present and one wants to take advantage of the parameter representation described here, one determines the position vectors of two conjugate points. If one chooses as first point (e,0) , the result is:

${\vec {f}}_{1}={e \choose 0},\quad {\vec {f}}_{2}={\frac {e}{\sqrt {d^{2}-c^{2}}}}{-c \choose 1}\ .$

Example: For the ellipse with the equation $\;x^{2}+2xy+3y^{2}-1=0\;$ are

${\vec {f}}_{1}={1 \choose 0},\quad {\vec {f}}_{2}={\frac {1}{\sqrt {2}}}{-1 \choose 1}$

the position vectors of two conjugate points.

Ellipse in space

If the vectors are ${\vec {f}}_{0},\;{\vec {f}}_{1},\;{\vec {f}}_{2}$ from the $\mathbb {R} ^{3}$ , we obtain a parameter representation of an ellipse in space.

Ellipse as affine image of the unit circle

Transformation to vertex shape (example 3)

Sequence of ellipses: rotated and scaled so that two consecutive ellipses touch each other.

Peripheral angle theorem and 3-point form for ellipses

Circles

A circle with equation $(x-c)^{2}+(y-d)^{2}=r^{2},\ r>0$ is uniquely determined by three points $(x_{1},y_{1}),\;(x_{2},y_{2}),\;(x_{3},y_{3})$ not on a straight line. A simple method to determine the parameters $c,d,r$ uses the Peripheral Angle Theorem for circles:

Four points $P_{i}=(x_{i},y_{i}),\ i=1,2,3,4$ (see figure) lie on a circle exactly when the angles at $P_{3}$ and $P_{4}$ equal.

Usually one measures an inscribed angle in degrees or radians. To determine the equation of a circle through 3 points, the following angle measure is more suitable:

To measure the angle between two straight lines with equations $y=m_{1}x+d_{1},\ y=m_{2}x+d_{2},\ m_{1}\neq m_{2}$ , the following quotient is used here:

${\frac {1+m_{1}\cdot m_{2}}{m_{2}-m_{1}}}$

This quotient is the cotangent of the intersection angle of the two straight lines.

Peripheral angle theorem for circles:For four points $P_{i}=(x_{i},y_{i}),\ i=1,2,3,4$ no three on a straight line (see figure) holds:

The four points lie on a circle exactly when the angles at $P_{3}$ and $P_{4}$ are equal in the above angular measure, that is, when:

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

The angle measure is initially available only for secants that are not parallel to the axis. However, the simplified formula given is eventually valid for these exceptions as well.

A consequence of the Peripheral Angle Theorem in this form is:

3-point form of a circular equation:

The equation of the circle through the 3 points $P_{i}=(x_{i},y_{i})$ not on a straight line is obtained by transforming the equation (eliminating the denominators and quadratic completion):

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

This formula can be written more clearly by using the position vectors, the scalar product and the determinant:

${\frac {({\color {red}{\vec {x}}}-{\vec {x}}_{1})\cdot ({\color {red}{\vec {x}}}-{\vec {x}}_{2})}{\det({\color {red}{\vec {x}}}-{\vec {x}}_{1},{\color {red}{\vec {x}}}-{\vec {x}}_{2})}}={\frac {({\vec {x}}_{3}-{\vec {x}}_{1})\cdot ({\vec {x}}_{3}-{\vec {x}}_{2})}{\det({\vec {x}}_{3}-{\vec {x}}_{1},{\vec {x}}_{3}-{\vec {x}}_{2})}}\;.$

Example:

For $P_{1}=(2,0),\;P_{2}=(0,1),\;P_{3}=(0,0)$ first obtain the 3-point form

${\frac {(x-2)x+y(y-1)}{yx-(y-1)(x-2)}}=0$ and finally $(x-1)^{2}+(y-1/2)^{2}=5/4\ .$

Ellipses

In this section only ellipses are considered with equations

${\frac {(x-c)^{2}}{a^{2}}}+{\frac {(y-d)^{2}}{b^{2}}}=1\quad \leftrightarrow \quad (x-c)^{2}+{\frac {a^{2}}{b^{2}}}(y-d)^{2}=a^{2},\quad c,d,\in \mathbb {R} ,\ a>0\ ,$

for which the quotient ${\tfrac {a^{2}}{b^{2}}}$ is fixed (invariant). With the abbreviation ${\color {blue}q}={\tfrac {a^{2}}{b^{2}}}$ we obtain the more suitable form

$(x-c)^{2}+{\color {blue}q}\;(y-d)^{2}=a^{2},\quad c,d\in \mathbb {R} ,\quad a>0$ q >

The axes of such ellipses are parallel to the coordinate axes and their eccentricity (see above) is fixed. The major axis is parallel to the axis if $q>1$ and parallel to the y {\displaystyle if q < 1

As with the circle, such an ellipse is not uniquely determined by three points on a straight line.

For this more general case, one introduces the following angular measure:

To measure the angle between two straight lines with equations $y=m_{1}x+d_{1},\ y=m_{2}x+d_{2},\ m_{1}\neq m_{2}$ , the following quotient is used here:

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

Peripheral Angle Theorem for Ellipses:
For four points $P_{i}=(x_{i},y_{i}),\ i=1,2,3,4$ no three on a straight line (see figure) holds:

The four points lie exactly on an ellipse with the equation $(x-c)^{2}+q\;(y-d)^{2}=a^{2}$ , if the angles at $P_{3}$ and $P_{4}$ are equal in the above angular measure, that is, if:

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

The angle measure is initially available only for secants that are not parallel to the axis. However, the simplified formula given is eventually valid for these exceptions as well.

The proof results from simple recalculation. In the case "points on an ellipse ..." one can assume that the center of the ellipse is the origin.

A consequence of the Peripheral Angle Theorem in this form is:

3-Point Form of an Ellipse Equation:
The equation of the ellipse through the 3 points $P_{i}=(x_{i},y_{i})$ not on a straight line is obtained by transforming the equation (eliminating the denominators and quadratic completion):

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

This formula can be represented more clearly (as with the circle) by

${\frac {({\color {red}{\vec {x}}}-{\vec {x}}_{1})*({\color {red}{\vec {x}}}-{\vec {x}}_{2})}{\det({\color {red}{\vec {x}}}-{\vec {x}}_{1},{\color {red}{\vec {x}}}-{\vec {x}}_{2})}}={\frac {({\vec {x}}_{3}-{\vec {x}}_{1})*({\vec {x}}_{3}-{\vec {x}}_{2})}{\det({\vec {x}}_{3}-{\vec {x}}_{1},{\vec {x}}_{3}-{\vec {x}}_{2})}}\;,$

where ∗ ${\vec {u}}*{\vec {v}}=u_{x}v_{x}+{\color {blue}q}u_{y}v_{y}$ describes the scalar product suitable here.

Example:

For $P_{1}=(2,0),\;P_{2}=(0,1),\;P_{3}=(0,0)$ and $q=4$ first obtain the 3-point form

${\frac {(x-2)x+4y(y-1)}{yx-(y-1)(x-2)}}=0$ and finally ${\frac {(x-1)^{2}}{2}}+{\frac {(y-1/2)^{2}}{1/2}}=1$ .

Ellipse: Peripheral Angle Theorem

Circle: Peripheral Angle Theorem

Examples

If you look obliquely at a circle (for example, at the top surface of a circular cylinder), this circle appears as an ellipse; more precisely, a parallel projection generally maps circles onto ellipses.

In astronomy, ellipses often occur as orbits of celestial bodies. According to Kepler's first law, every planet moves on an ellipse around the sun, with the sun resting in one of the two foci. The same is true for the orbits of recurrent (periodic) comets, planetary moons or double stars. In general, every two-body problem of gravitational force results in elliptical, parabolic or hyperbolic orbits, depending on the energy.

Steiner ellipse (blue) with Steiner ellipse (red)

Example of an Inellipse

For every two- or three-dimensional harmonic oscillator, the motion occurs on an elliptical path. For example, the pendulum bob of a thread pendulum oscillates approximately on an elliptical path, if the motion of the pendulum thread is not only in one plane.

In triangle geometry there are Steiner ellipses, inellipses (Steiner ellipse, Mandart ellipse).

Collection of formulas (elliptic equations)

Ellipse equation (Cartesian coordinates)

center (0|0) ,

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

Resolved to $y^{2}$ :

$y^{2}=b^{2}\left(1-{\frac {x^{2}}{a^{2}}}\right)={\frac {(a^{2}-x^{2})(a^{2}-e^{2})}{a^{2}}}=(a^{2}-x^{2})(1-\varepsilon ^{2})$

The last form is convenient for representing an ellipse using the two orbital elements, numerical eccentricity and major semi-axis.

Center $(x_{0}|y_{0})$ , major axis parallel to axis:

${\frac {(x-x_{0})^{2}}{a^{2}}}+{\frac {(y-y_{0})^{2}}{b^{2}}}=1.$

Ellipse equation (parameter form)

Center (0|0) , major axis as axis:

${\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}a\cos t\\b\sin t\end{pmatrix}}\quad {\text{mit}}\quad 0\leq t<2\pi .$

Center $(x_{0}|y_{0})$ , major axis parallel to axis:

${\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}x_{0}+a\cos t\\y_{0}+b\sin t\end{pmatrix}}\quad {\text{mit}}\quad 0\leq t<2\pi .$

Center $(x_{0}|y_{0})$ , major axis rotates around α with respect to $\alpha$ axis:

${\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}x_{0}+a\cos t\,\cos \alpha -b\sin t\,\sin \alpha \\y_{0}+a\cos t\,\sin \alpha +b\sin t\,\cos \alpha \end{pmatrix}}\quad {\text{mit}}\quad 0\leq t<2\pi .$

Here denotes the parameter of this representation. This does not correspond to the polar angle φ $\varphi$ between the and the straight line leading through the origin and the respective ellipse point, but e. g. the polar angle between the -axis and the straight line leading through the origin and the point with the same -coordinate as the ellipse point but on the circle with radius (cf. construction according to de la Hire). In astronomy this parameter is called the eccentric anomaly for Kepler ellipses, for meridian ellipses in geodesy it is called parametric or reduced latitude, cf. reference ellipsoid.

For non-rotated ellipses, i.e. α $\alpha =0$ , depends on the polar angle φ $\varphi$ $\tan \varphi =y/x$ is defined by related to the parameter

$\tan \varphi ={\frac {b}{a}}\tan t={\sqrt {1-\varepsilon ^{2}}}\,\tan t$

This relation allows an illustrative interpretation of the parameter : If one stretches the coordinate of an ellipse point P=(x,y) by the factor a/b , then this new point lies P'=(x,y') on a circle with radius and the same center as the ellipse. The parameter is now the angle between the axis and the connecting line ${\overline {MP'}}$ :

$y'={\frac {a}{b}}y=x{\frac {a}{b}}\tan \varphi =x\tan t=a\sin t\quad \Longrightarrow \quad {\begin{pmatrix}x\\y'\end{pmatrix}}=a{\begin{pmatrix}\cos t\\\sin t\end{pmatrix}}$

Ellipse equation (polar coordinates with respect to the center)

Main axis horizontal, center as pole, polar axis along main axis to the right:

$r(\varphi )={\frac {ab}{\sqrt {a^{2}\sin ^{2}\varphi +b^{2}\cos ^{2}\varphi }}}={\frac {b}{\sqrt {1-\varepsilon ^{2}\cos ^{2}\varphi }}}\in [b,a]\quad {\text{mit}}\quad 0\leq \varphi <2\pi$

Expressed in Cartesian coordinates, parameterized by the angle of the polar coordinates, with the center of the ellipse at (0|0) and its major axis along the axis:

${\begin{pmatrix}x\\y\end{pmatrix}}={\frac {b}{\sqrt {1-\varepsilon ^{2}\cos ^{2}\varphi }}}{\begin{pmatrix}\cos \varphi \\\sin \varphi \end{pmatrix}}\quad {\text{mit}}\quad 0\leq \varphi <2\pi$

Derivation

From the ellipse equation in Cartesian coordinates $(x/a)^{2}+(y/b)^{2}=1$ and the parametrization of the Cartesian in polar coordinates $x=r\cos \varphi$ and $y=r\sin \varphi$ follows:

${\frac {r^{2}\cos ^{2}\varphi }{a^{2}}}+{\frac {r^{2}\sin ^{2}\varphi }{b^{2}}}=1\quad \Longrightarrow \quad r^{2}\left(b^{2}\cos ^{2}\varphi +a^{2}\sin ^{2}\varphi \right)=a^{2}b^{2}$

Rearrange and square root gives the radius depending on the polar angle.

Ellipse equation (polar coordinates with respect to a focal point)

Main axis horizontal, right focal point as pole, polar axis along main axis to the right (half parameter $p=b^{2}/a$ ):

$r_{\mathrm {R} }(\varphi _{\mathrm {R} })={\frac {a^{2}-e^{2}}{a+e\cos \varphi _{\mathrm {R} }}}={\frac {p}{1+\varepsilon \cos \varphi _{\mathrm {R} }}}\in [r_{\mathrm {peri} },r_{\mathrm {apo} }]\quad {\text{mit}}\quad 0\leq \varphi _{\mathrm {R} }<2\pi$

Main axis horizontal, left focal point as pole, polar axis along main axis to the right:

$r_{\mathrm {L} }(\varphi _{\mathrm {L} })={\frac {a^{2}-e^{2}}{a-e\cos \varphi _{\mathrm {L} }}}={\frac {p}{1-\varepsilon \cos \varphi _{\mathrm {L} }}}\in [r_{\mathrm {peri} },r_{\mathrm {apo} }]\quad {\text{mit}}\quad 0\leq \varphi _{\mathrm {L} }<2\pi$

The range of values of the radii extends from the periapsis distance $r_{\mathrm {peri} }$ to the apoapsis distance $r_{\mathrm {apo} }$ , which have the following values:

$r_{\mathrm {peri} }={\frac {p}{1+\varepsilon }}=a(1-\varepsilon )\ ,\qquad r_{\mathrm {apo} }={\frac {p}{1-\varepsilon }}=a(1+\varepsilon )$

Expressed in Cartesian coordinates, parameterized by the angle φ $\varphi _{\mathrm {R} }$ or φ $\varphi _{\mathrm {L} }$ of polar coordinates, where the right focal point of the ellipse is at (e|0) , the left focal point at (-e|0) :

${\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}e\\0\end{pmatrix}}+{\frac {p}{1+\varepsilon \cos \varphi _{\mathrm {R} }}}{\begin{pmatrix}\cos \varphi _{\mathrm {R} }\\\sin \varphi _{\mathrm {R} }\end{pmatrix}}\quad {\text{mit}}\quad 0\leq \varphi _{\mathrm {R} }<2\pi$

${\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}-e\\0\end{pmatrix}}+{\frac {p}{1-\varepsilon \cos \varphi _{\mathrm {L} }}}{\begin{pmatrix}\cos \varphi _{\mathrm {L} }\\\sin \varphi _{\mathrm {L} }\end{pmatrix}}\quad {\text{mit}}\quad 0\leq \varphi _{\mathrm {L} }<2\pi$

The angle φ $\varphi _{\mathrm {R} }$ or φ $\varphi _{\mathrm {L} }$ , depending on which pole is the reference point, is called the true anomaly in astronomy.

Derivation

Consider a triangle formed by the two fixed points $F_{\mathrm {L} }$ , $F_{\mathrm {R} }$ and any point on the ellipse is spanned.

The distances between these points are: ${\overline {F_{\mathrm {L} }F_{\mathrm {R} }}}=2e$ well as ${\overline {F_{\mathrm {L} }P}}=r_{\mathrm {L} }$ and by the definition of the ellipse ${\overline {F_{\mathrm {R} }P}}=2a-r_{\mathrm {L} }$ . Let the angle at $F_{\mathrm {L} }$ let φ $\varphi _{\mathrm {L} }=\angle F_{\mathrm {R} }F_{\mathrm {L} }P$ . With the cosine theorem now holds:

$(2a-r_{\mathrm {L} })^{2}=(2e)^{2}+r_{\mathrm {L} }^{2}-2(2e)r_{\mathrm {L} }\cos \varphi _{\mathrm {L} }\quad \Longrightarrow \quad r_{\mathrm {L} }(a-\underbrace {e} _{a\varepsilon }\cos \varphi _{\mathrm {L} })=\underbrace {a^{2}-e^{2}} _{pa}$

The derivation for the right pole is analogous. The distances are ${\overline {F_{\mathrm {L} }F_{\mathrm {R} }}}=2e$ and ${\overline {F_{\mathrm {R} }P}}=r_{\mathrm {R} }$ and ${\overline {F_{\mathrm {L} }P}}=2a-r_{\mathrm {R} }$ . Let the angle at $F_{\mathrm {R} }$ be π $\pi -\varphi _{\mathrm {R} }=\angle PF_{\mathrm {R} }F_{\mathrm {L} }$ , since φ $\varphi _{\mathrm {R} }=\angle (S_{\mathrm {R} },F_{\mathrm {R} },P)$ is defined, where $S_{\mathrm {R} }$ marks the right main vertex.

$(2a-r_{\mathrm {R} })^{2}=(2e)^{2}+r_{\mathrm {R} }^{2}-2(2e)r_{\mathrm {R} }\underbrace {\cos(\pi -\varphi _{\mathrm {R} })} _{-\cos \varphi _{\mathrm {R} }}\quad \Longrightarrow \quad r_{\mathrm {R} }(a+\underbrace {e} _{a\varepsilon }\cos \varphi _{\mathrm {R} })=\underbrace {a^{2}-e^{2}} _{pa}$

Alternative derivation

By equating the two representations of $r_{\mathrm {L} }^{2}-r_{\mathrm {R} }^{2}$ obtained:

$\left.{\begin{array}{l}r_{\mathrm {L} }^{2}-r_{\mathrm {R} }^{2}=\left[y^{2}+(x+e)^{2}\right]-\left[y^{2}+(x-e)^{2}\right]=4ex=4a\varepsilon x\\r_{\mathrm {L} }^{2}-r_{\mathrm {R} }^{2}=(r_{\mathrm {L} }+r_{\mathrm {R} })(r_{\mathrm {L} }-r_{\mathrm {R} })=2a(r_{\mathrm {L} }-r_{\mathrm {R} })\end{array}}\right\}\implies r_{\mathrm {L} }-r_{\mathrm {R} }=2\varepsilon x$

This corresponds on the one hand with $r_{\mathrm {R} }=2a-r_{\mathrm {L} }$ and $x=r_{\mathrm {L} }\cos \varphi _{\mathrm {L} }-e$

$2r_{\mathrm {L} }-2a=2\varepsilon (r_{\mathrm {L} }\cos \varphi _{\mathrm {L} }-e)\quad \Longrightarrow \quad r_{\mathrm {L} }(1-\varepsilon \cos \varphi _{\mathrm {L} })=a-\varepsilon e\equiv p$

and on the other hand with $r_{\mathrm {L} }=2a-r_{\mathrm {R} }$ and $x=r_{\mathrm {R} }\cos \varphi _{\mathrm {R} }+e$ :

$2a-2r_{\mathrm {R} }=2\varepsilon (r_{\mathrm {R} }\cos \varphi _{\mathrm {R} }+e)\quad \Longrightarrow \quad r_{\mathrm {R} }(1+\varepsilon \cos \varphi _{\mathrm {R} })=a-\varepsilon e\equiv p$

Eccentric anomaly and true anomaly φ $\varphi {\mathrm {R} }$ with respect to the right focal point and true anomaly φ $\varphi {\mathrm {L} }$ with respect to the left focal point as a function of the polar angle φ $\varphi$ for different numerical eccentricities ε $\varepsilon$

Formula collection (curve properties)

Tangent equation (Cartesian coordinates)

Center (0|0) , major axis as axis, touch point $(x_{B}|y_{B})$ :

${\frac {x_{B}x}{a^{2}}}+{\frac {y_{B}y}{b^{2}}}=1\quad \iff \quad {\frac {x_{B}(x-x_{B})}{a^{2}}}+{\frac {y_{B}(y-y_{B})}{b^{2}}}=0$

Center $(x_{0}|y_{0})$ Major axis parallel to axis, touch point $(x_{B}|y_{B})$ :

${\frac {(x_{B}-x_{0})(x-x_{0})}{a^{2}}}+{\frac {(y_{B}-y_{0})(y-y_{0})}{b^{2}}}=1$

Tangent equation (parameter form)

An (unnormalized) tangent vector to the ellipse has the shape:

${\vec {T}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\begin{pmatrix}a\cos t\\b\sin t\end{pmatrix}}={\begin{pmatrix}-a\sin t\\b\cos t\end{pmatrix}}={\begin{pmatrix}-ay/b\\bx/a\end{pmatrix}}$

The tangent equation in vectorial representation with center at (0|0) , major axis as axis and tangent point at $(x_{B}|y_{B})$ :

${\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}x_{B}\\y_{B}\end{pmatrix}}+\mu {\begin{pmatrix}-ay_{B}/b\\bx_{B}/a\end{pmatrix}}\quad {\text{mit}}\quad \mu \in \mathbb {R}$

Relationship between polar angle and normal angle

There is the following relationship between polar angle φ $\varphi$ normal angle β $\beta$ and ellipse parameter (see adjacent graph).

$\tan \varphi ={\frac {b}{a}}\tan t={\frac {b^{2}}{a^{2}}}\tan \beta =(1-\varepsilon ^{2})\tan \beta$

Derivation

The relation of the polar angle φ $\varphi$ and the slope angle of the normal β $\beta$ (see graph on the right) can be found, for example, like this:

Solving the tangent equation to

$y={\frac {b^{2}}{y_{B}}}-{\frac {b^{2}x_{B}}{a^{2}y_{B}}}x$

gives the tangent slope $\tan(\alpha )=-\tan(\pi -\alpha )=\Delta y/\Delta x$ as coefficient of to

$\tan \alpha =-{\frac {b^{2}}{a^{2}}}{\frac {x_{B}}{y_{B}}}=-{\frac {b^{2}}{a^{2}}}{\frac {1}{\tan \varphi }}.$

With $\tan \alpha =\tan(\beta +\pi /2)=-1/\tan \beta$ we obtain the sought relationship between β $\beta$ and φ $\varphi$ .

Normal equation (Cartesian coordinates)

Center (0|0) , major axis as axis, touch point $(x_{B}|y_{B})$ :

$\left(1-{\frac {y}{y_{B}}}\right){\frac {b^{2}}{a^{2}}}+{\frac {x}{x_{B}}}=1$

or also

$b^{2}\left({\frac {y}{y_{B}}}-1\right)=a^{2}\left({\frac {x}{x_{B}}}-1\right)$

Normal equation (parameter form)

An (unnormalized) normal vector to the ellipse has the shape:

${\vec {N}}={\begin{pmatrix}b\cos t\\a\sin t\end{pmatrix}}={\begin{pmatrix}bx/a\\ay/b\end{pmatrix}}$

The normal equation in vectorial representation with center at (0|0) , major axis as axis and touch point at $(x_{B}|y_{B})$ :

${\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}x_{B}\\y_{B}\end{pmatrix}}+\mu {\begin{pmatrix}bx_{B}/a\\ay_{B}/b\end{pmatrix}}\quad {\text{mit}}\quad \mu \in \mathbb {R}$

Radii and centers of curvature

Radius of curvature at point $(x_{p}|y_{p})$ :

$r=a^{2}b^{2}\left({\frac {x_{p}^{2}}{a^{4}}}+{\frac {y_{p}^{2}}{b^{4}}}\right)^{3/2}={\frac {1}{a^{4}b^{4}}}{\sqrt {\left(a^{4}y_{p}^{2}+b^{4}x_{p}^{2}\right)^{3}}}$

Center of the circle of curvature, center of curvature $M(\xi |\eta )$ :

$\xi ={\frac {e^{2}x_{p}^{3}}{a^{4}}}\qquad \eta =-{\frac {e^{2}y_{p}^{3}}{b^{4}}}\qquad \vert \,e^{2}=a^{2}\varepsilon ^{2}=a^{2}-b^{2}$

Radius and center of curvature in one of the two principal vertices $(\pm a|0)$ :

$r_{\mathrm {H} }=p={\frac {b^{2}}{a}}\qquad M_{\mathrm {H} }\left(\xi =\pm {\frac {e^{2}}{a}}\,{\bigg |}\,\eta =0\right)$

Radius and center of curvature in one of the two minor vertices $(0|\pm b)$ :

$r_{\mathrm {N} }={\frac {a^{2}}{b}}\qquad M_{\mathrm {N} }\left(\xi =0\,{\bigg |}\,\eta =\pm {\frac {e^{2}}{b}}\right)$

The angles of the elliptic tangent

Formula collection (area and perimeter)

Area

With semiaxes and :

$A=\pi ab=\pi a^{2}{\sqrt {1-\varepsilon ^{2}}}$

If the ellipse is given by an implicit equation

$\alpha x^{2}+\beta xy+\gamma y^{2}+1=0$

then its area is

$A={\frac {2\pi }{\sqrt {4\alpha \gamma -\beta ^{2}}}}.$

Ellipse sector

For an ellipse with semiaxes and and a sector $\varphi \in \left]0,{\frac {\pi }{2}}\right[$ enclosing with the major semiaxis the angle φ

$A_{\mathrm {Sektor} }={\frac {ab}{2}}\arctan \left({\frac {a}{b}}\tan(\varphi )\right)$

If one describes the ellipse sector instead of the polar angle by the parameter from the parameter representation $(x,y)=(a\cos t,b\sin t)$ , we obtain the formula

$A_{\mathrm {Sektor} }={\frac {ab}{2}}t.$

Scope

Formula

The circumference an ellipse with large semi-axis and small semi-axis calculated to be

$U=4a\cdot E(\varepsilon )$ ,

where E(k) stands for the complete elliptic integral of second kind. The numerical eccentricity ε $\varepsilon$ calculated for ellipses as

$\varepsilon ={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}$ .

Derivation

The circumference of an ellipse cannot be expressed exactly by elementary functions. However, it can be represented with the help of an integral, which is therefore called an elliptic integral.

The formula for the arc length of a curve ${\mathcal {C}}$ is

$L=\int \limits _{\mathcal {C}}|\gamma '(t)|\;\mathrm {d} t$ .

For the ellipse with parameter representation taking into account the symmetry for the perimeter U, we get {\displaystyle $\;(a\cos t,b\sin t),\;0\leq t\;<2\pi ,\;$

$U=4\int _{0}^{\pi /2}{\sqrt {a^{2}\cdot \sin ^{2}t+b^{2}\cdot \cos ^{2}t}}\;\mathrm {d} t$ .

Factoring out $a^{2}$ , using $\;\sin ^{2}t=1-\cos ^{2}t\;$ and ε $\;\varepsilon ^{2}=1-{\frac {b^{2}}{a^{2}}}\;$ leads to

$U=4a\cdot \int _{0}^{\frac {\pi }{2}}{\sqrt {1-\varepsilon ^{2}\cdot \cos ^{2}t}}\;\mathrm {d} t\ .$

By substituting $\vartheta ={\frac {\pi }{2}}-t,\;\mathrm {d} t=-\mathrm {d} \vartheta$ we obtain the following form:

$E(\varepsilon )=\int _{0}^{\frac {\pi }{2}}{\sqrt {1-\varepsilon ^{2}\cdot \sin ^{2}\vartheta }}\;\mathrm {d} \vartheta$ .

The integral $E(\varepsilon )$ is called a complete elliptic integral of the second kind.

The circumference the ellipse is thus

$U=4a\cdot E(\varepsilon ),\quad \varepsilon ={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}$ .

The circumference thus depends on the numerical eccentricity ε $\varepsilon$ and the major semi-axis With the help of the adjacent diagram, for a given eccentricity ε $\varepsilon$ the value of the factor $k=4E(\varepsilon )$ $U=k\cdot a$ read for the circumference . for each ellipse lies between the extreme cases k=4 ( $\varepsilon =1$ , degenerate ellipse as line) and $k=2\pi$ ( $\varepsilon =0$ ellipse becomes a circle).

Series development

${\begin{aligned}U&=2a\pi \left(1-\sum _{i=1}^{\infty }\left(\prod _{j=1}^{i}{\frac {2j-1}{2j}}\right)^{2}{\frac {\varepsilon ^{2i}}{2i-1}}\right)\\&=2a\pi \left[1-\left({\frac {1}{2}}\right)^{2}\varepsilon ^{2}-\left({\frac {1\cdot 3}{2\cdot 4}}\right)^{2}{\frac {\varepsilon ^{4}}{3}}-\ldots -\left({\frac {1\cdot 3\cdot 5\dotsm (2n-1)}{2\cdot 4\cdot 6\dotsm 2n}}\right)^{2}{\frac {\varepsilon ^{2n}}{2n-1}}-\ldots \right]\end{aligned}}$

For ε $\varepsilon$ close to 1, this series expansion converges extremely slowly. Therefore, numerical integration is recommended, e.g., according to the Romberg method.

A series that converges faster is based on the Gauss-Kummer series. For an ellipse with semi-axes and (with a>b λ = a $\lambda ={\tfrac {a-b}{a+b}}$ defined. Then it follows:

${\begin{aligned}U&=\pi (a+b)\left(1+\sum _{n=0}^{\infty }\left({\frac {\binom {2n}{n}}{(n+1)2^{2n+1}}}\right)^{2}\ \lambda ^{2n+2}\right)\\&=\pi (a+b)\left(1+{\frac {\lambda ^{2}}{4}}+{\frac {\lambda ^{4}}{64}}+{\frac {\lambda ^{6}}{256}}\ +{\frac {25\lambda ^{8}}{16\,384}}+{\frac {49\lambda ^{10}}{65\,536}}+{\frac {441\lambda ^{12}}{1\,048\,576}}+{\frac {1\,089\lambda ^{14}}{4\,194\,304}}+\dotsb +\ \left({\frac {1\cdot 3\cdot 5\dotsm (2n-1)}{2\cdot 4\cdot 6\cdot 8\dotsb (2n+2)}}\right)^{2}\lambda ^{2n+2}+\dotsb \right)\end{aligned}}$

Approximations

Approximation using the arithmetic mean of the semi-axes

$U\approx \pi (a+b)$

Accuracy of this formula

Exc. ε	q = b / a	Error
= 0,000	1,000	0 (circle: exact)
< 0,051	> 0,9987	< 10−7
< 0,090	> 0,996	< 10−6
< 0,1582	> 0,9874	< 10−5
< 0,277	> 0,961	< 0,01 %
< 0,46	> 0,885	< 0,1 %
< 0,75	> 0,66	< 1 %
< 0,83	> 0,55	< 2 %
< 0,927	> 0,37	< 5 %
< 0,978	> 0,21	< 10 %
< 0,999	> 0,044	< 18,3 %
< 1,000	> 0,000	< 21,46 %

Approximation using the root mean square of the semi-axes

$U\approx 2\pi {\sqrt {{\frac {1}{2}}(a^{2}+b^{2})}}=\pi {\sqrt {2(a^{2}+b^{2})}}$

$U\approx 2\pi {\sqrt {{\frac {1}{2}}(a^{2}+a^{2}(1-\epsilon ^{2}))}}=2\pi a{\sqrt {1-{\frac {\epsilon ^{2}}{2}}}}$

Accuracy of this formula

Exc. ε	q = b / a	Error
= 0,000	= 1,0000	0 (circle: exact)
< 0,016	> 0,9999	< 10−9
< 0,026	> 0,9997	< 10−8
< 0,047	> 0,9989	< 10−7
< 0,084	> 0,9965	< 10−6
< 0,149	> 0,9888	< 10−5
< 0,262	> 0,9651	< 0,01 %
< 0,450	> 0,8930	< 0,1 %
< 0,720	> 0,6937	< 1 %
< 0,808	> 0,5891	< 2 %
< 0,914	> 0,4037	< 5 %
< 0,977	> 0,2104	< 10 %
< 1,000	> 0,0000	< 14,91 %

Approximation formula according to Ramanujan

$U\approx \pi \left((a+b)\,+\,{\frac {3(a-b)^{2}}{10(a+b)+{\sqrt {a^{2}+14ab+b^{2}}}}}\right)$

respectively

$U\approx \pi (a+b)\left(1+{\frac {3\lambda ^{2}}{10+{\sqrt {4-3\lambda ^{2}}}}}\right)$ , where λ $\quad \lambda ={\frac {a-b}{a+b}}$ .

This approximation is very accurate in a wide ε $\varepsilon$ range of $0\leq \varepsilon \leq 0{,}9$ and always yields a slightly too small value throughout the range, which $\varepsilon$ increases monotonically with ε

The relative error is:

Area	rel. Error
0,0000 ≤ ε ≤ 0,8820	< 10−9
0,8820 < ε ≤ 0,9242	< 10−8
0,9242 < ε ≤ 0,9577	< 10−7
0,9577 < ε ≤ 0,9812	< 10−6
0,9812 < ε ≤ 0,9944	< 10−5
0,9944 < ε ≤ 0,9995	< 10−4
0,9995 < ε ≤ 1,0000	< 0,000403

For ε $\varepsilon =\lambda =1$ instead of 4 we get the minimum too small value ${\tfrac {14}{11}}\pi$ .

Character

Unicode contains four ellipsis symbols in the Miscellaneous Symbols and Arrows block, which can be used as graphic characters or decorative characters in any text (including continuous text):

Unicode	Characters	Name	LaTeX
U+2B2C	⬬	black horizontal ellipse (Full horizontal ellipse)	`\EllipseSolid`
U+2B2D	⬭	white horizontal ellipse (Hollow horizontal ellipse)	`\Ellipse`
U+2B2E	⬮	black vertical ellipse (Full vertical ellipse)	^Note
U+2B2F	⬯	white vertical ellipse (Hollow vertical ellipse)	^Note

^Note By rotating the horizontal variant using the package rotating, which is preinstalled with the usual LaTeX distributions.

LaTeX also knows a hollow horizontal ellipse with shadow on the right: \EllipseShadow.

Questions and Answers

Q: What is an ellipse?

A: An ellipse is a shape that looks like an oval or a flattened circle. In geometry, it is a plane curve which results from the intersection of a cone by a plane in a way that produces a closed curve.

Q: How does one create an ellipse?

A: An ellipse can be made by putting two pins into cardboard and then looping string around those two pins and putting a pencil in the loop and pulling as far as possible without breaking the string in all directions.

Q: What are circles special cases of?

A: Circles are special cases of ellipses, created when the cutting plane is perpendicular to the cone's axis.

Q: How many foci does an ellipse have?

A: An ellipse has two foci.

Q: What equation describes an ellipse?

A: The equation for an ellipse is (x - h)²/a² + (y - k)²/b² = 1 where h and k represent the center of the ellipse and 2a represents the length from each end of the longer skinnier side, while 2b represents the length between each end of its shorter side. C represents the length between its foci and center, such that A²-B²=C².

Q: Where do we see examples of elliptical orbits?

A: Elliptical orbits can be seen in planets, with their sun at one focus point.

Ellipse

Definition of an ellipse as a geometric place

Ellipse in Cartesian coordinates

Equation

Half parameter

Tangent

Equation of a displaced ellipse

Parameter representations

Properties

Focal Point Property

Directrix

Conjugate diameters

Orthogonal tangents

Pole-Polar Relationship

Derivation of the positional relationships of pole and polar; alternative derivation of a tangent and an ellipse equation.

Ellipse as affine image of the unit circle

Parameter representation

Implicit representation

Ellipse in space

Peripheral angle theorem and 3-point form for ellipses

Circles

Ellipses

Examples

Collection of formulas (elliptic equations)

Ellipse equation (Cartesian coordinates)

Ellipse equation (parameter form)

Ellipse equation (polar coordinates with respect to the center)

Ellipse equation (polar coordinates with respect to a focal point)

Formula collection (curve properties)

Tangent equation (Cartesian coordinates)

Tangent equation (parameter form)

Relationship between polar angle and normal angle

Normal equation (Cartesian coordinates)

Normal equation (parameter form)

Radii and centers of curvature

Formula collection (area and perimeter)

Area

Scope

Formula

Derivation

Series development

Approximations

Approximation using the arithmetic mean of the semi-axes

Approximation using the root mean square of the semi-axes

Approximation formula according to Ramanujan

Character

See also

Questions and Answers

Q: What is an ellipse?

Q: How does one create an ellipse?

Q: What are circles special cases of?

Q: How many foci does an ellipse have?

Q: What equation describes an ellipse?

Q: Where do we see examples of elliptical orbits?

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