Ellipse Sign 411

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity $e$, a number ranging from $e = 0$ (the limiting case of a circle) to $e = 1$ (the limiting case of infinite elongation, no longer an ellipse but a parabola).

An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution.

Analytically, the equation of a standard ellipse centered at the origin with width $2a$ and height $2b$ is:
: $\frac+\frac = 1 .$

Assuming $a \ge b$, the foci are $(\pm c, 0)$ for $c = \sqrt$. The standard parametric equation is:
: $(x,y) = (a\cos(t),b\sin(t)) \quad \text \quad 0\leq t\leq 2\pi.$

Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane (see figure). Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. An angled cross section of a cylinder is also an ellipse.

An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant. This constant ratio is the above-mentioned eccentricity:
: $e = \frac = \sqrt.$

Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in the Solar System is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the barycenter of the Sunplanet pair). The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids. A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under parallel or perspective projection. The ellipse is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical polarization of light in optics.

The name, (, "omission"), was given by Apollonius of Perga in his ''Conics''.

Definition as locus of points

An ellipse can be defined geometrically as a set or locus of points in the Euclidean plane:
: Given two fixed points $F_1, F_2$ called the foci and a distance $2a$ which is greater than the distance between the foci, the ellipse is the set of points $P$ such that the sum of the distances $, PF_1, ,\ , PF_2,$ is equal to $2a$:$E = \left\\ .$

The midpoint $C$ of the line segment joining the foci is called the ''center'' of the ellipse. The line through the foci is called the ''major axis'', and the line perpendicular to it through the center is the ''minor axis''. The major axis intersects the ellipse at two '' vertices'' $V_1,V_2$, which have distance $a$ to the center. The distance $c$ of the foci to the center is called the ''focal distance'' or linear eccentricity. The quotient $e=\tfrac$ is the ''eccentricity''.

The case $F_1=F_2$ yields a circle and is included as a special type of ellipse.

The equation $, PF_2, + , PF_1 , = 2a$ can be viewed in a different way (see figure):
: If $c_2$ is the circle with center $F_2$ and radius $2a$, then the distance of a point $P$ to the circle $c_2$ equals the distance to the focus $F_1$:
:: $, PF_1, =, Pc_2, .$

$c_2$ is called the ''circular directrix'' (related to focus $F_2$) of the ellipse. This property should not be confused with the definition of an ellipse using a directrix line below.

Using Dandelin spheres, 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

Standard equation

The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, the ''x''-axis is the major axis, and:
: the foci are the points $F_1 = (c,\, 0),\ F_2=(-c,\, 0)$,
: the vertices are $V_1 = (a,\, 0),\ V_2 = (-a,\, 0)$.

For an arbitrary point $(x,y)$ the distance to the focus $(c,0)$ is
$\sqrt$ and to the other focus $\sqrt$. Hence the point $(x,\, y)$ is on the ellipse whenever:
:$\sqrt + \sqrt = 2a\ .$

Removing the radicals by suitable squarings and using $b^2 = a^2-c^2$ (see diagram) produces the standard equation of the ellipse:
:$\frac + \frac = 1,$

or, solved for ''y:''
:$y = \pm\frac\sqrt = \pm \sqrt.$

The width and height parameters $a,\; b$ are called the semi-major and semi-minor axes. The top and bottom points $V_3 = (0,\, b),\; V_4 = (0,\, -b)$ are the ''co-vertices''. The distances from a point $(x,\, y)$ on the ellipse to the left and right foci are $a + ex$ and $a - ex$.

It follows from the equation that the ellipse is ''symmetric'' with respect to the coordinate axes and hence with respect to the origin.

Parameters

Principal axes

Throughout this article, the semi-major and semi-minor axes are denoted $a$ and $b$, respectively, i.e. $a \ge b > 0 \ .$

In principle, the canonical ellipse equation $\tfrac + \tfrac = 1$ may have $a < b$ (and hence the ellipse would be taller than it is wide). This form can be converted to the standard form by transposing the variable names $x$ and $y$ and the parameter names $a$ and $b.$

Linear eccentricity

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

Eccentricity

The eccentricity can be expressed as:
: $e = \frac = \sqrt,$

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

Semi-latus rectum

The length of the chord through one focus, perpendicular to the major axis, is called the ''latus rectum''. One half of it is the ''semi-latus rectum'' $\ell$. A calculation shows:

: $\ell = \fraca = a \left(1 - e^2\right).$

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

Tangent

An arbitrary line $g$ intersects an ellipse at 0, 1, or 2 points, respectively called an ''exterior line'', ''tangent'' and ''secant''. Through any point of an ellipse there is a unique tangent. The tangent at a point $(x_1,\, y_1)$ of the ellipse $\tfrac + \tfrac = 1$ has the coordinate equation:

:$\fracx + \fracy = 1.$

A vector parametric equation of the tangent is:
: $\vec x = \beginx_1 \\ y_1\end + s\begin \;\! -y_1 a^2 \\ \;\ \ \ x_1 b^2 \end\$ with $\ s \in \mathbb\ .$

Proof:
Let $(x_1,\, y_1)$ be a point on an ellipse and $\vec = \beginx_1 \\ y_1\end + s\beginu \\ v\end$ be the equation of any line $g$ containing $(x_1,\, y_1)$. Inserting the line's equation into the ellipse equation and respecting $\frac + \frac = 1$ yields:
: $\frac + \frac = 1\ \quad\Longrightarrow\quad 2s\left(\frac + \frac\right) + s^2\left(\frac + \frac\right) = 0\ .$
There are then cases:
# $\fracu + \fracv = 0.$ Then line $g$ and the ellipse have only point $(x_1,\, y_1)$ in common, and $g$ is a tangent. The tangent direction has perpendicular vector $\begin\frac & \frac\end$, so the tangent line has equation $\fracx + \tfracy = k$ for some $k$. Because $(x_1,\, y_1)$ is on the tangent and the ellipse, one obtains $k = 1$.
# $\fracu + \fracv \ne 0.$ Then line $g$ has a second point in common with the ellipse, and is a secant.

Using (1) one finds that $\begin -y_1a^2 & x_1b^2 \end$ is a tangent vector at point $(x_1,\, y_1)$, which proves the vector equation.

If $(x_1, y_1)$ and $(u, v)$ are two points of the ellipse such that $\frac + \tfrac = 0$, then the points lie on two ''conjugate diameters'' (see below). (If $a = b$, the ellipse is a circle and "conjugate" means "orthogonal".)

Shifted ellipse

If the standard ellipse is shifted to have center $\left(x_\circ,\, y_\circ\right)$, its equation is
: $\frac + \frac = 1 \ .$

The axes are still parallel to the x- and y-axes.

General ellipse

In analytic geometry, the ellipse is defined as a quadric: the set of points $(X,\, Y)$ of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation

: $AX^2 + B X Y + C Y^2 + D X + E Y + F = 0$

provided $B^2 - 4AC < 0.$

To distinguish the degenerate cases from the non-degenerate case, let ''∆'' be the determinant

:$\Delta = \begin A & \fracB & \fracD \\ \fracB & C & \fracE \\ \fracD & \fracE & F \end = \left(AC - \frac\right) F + \frac - \frac - \frac.$

Then the ellipse is a non-degenerate real ellipse if and only if ''C∆'' < 0. If ''C∆'' > 0, we have an imaginary ellipse, and if ''∆'' = 0, we have a point ellipse.Lawrence, J. Dennis, ''A Catalog of Special Plane Curves'', Dover Publ., 1972.

The general equation's coefficients can be obtained from known semi-major axis $a$, semi-minor axis $b$, center coordinates $\left(x_\circ,\, y_\circ\right)$, and rotation angle $\theta$ (the angle from the positive horizontal axis to the ellipse's major axis) using the formulae:

:$\begin A &= a^2 \sin^2\theta + b^2 \cos^2\theta \\ B &= 2\left(b^2 - a^2\right) \sin\theta \cos\theta \\ C &= a^2 \cos^2\theta + b^2 \sin^2\theta \\ D &= -2A x_\circ - B y_\circ \\ E &= - B x_\circ - 2C y_\circ \\ F &= A x_\circ^2 + B x_\circ y_\circ + C y_\circ^2 - a^2 b^2. \end$

These expressions can be derived from the canonical equation $\tfrac + \tfrac = 1$ by an affine transformation of the coordinates $(x,\, y)$:

:$\begin x &= \left(X - x_\circ\right) \cos\theta + \left(Y - y_\circ\right) \sin\theta \\ y &= -\left(X - x_\circ\right) \sin\theta + \left(Y - y_\circ\right) \cos\theta. \end$

Conversely, the canonical form parameters can be obtained from the general form coefficients by the equations:

:\begin
a, b &= \frac \\
x_\circ &= \frac \\ pt y_\circ &= \frac \\ pt \theta &= \begin
\arctan\left(\frac\left(C - A - \sqrt\right)\right)
& \text B \ne 0 \\
0 & \text B = 0,\ A < C \\
90^\circ & \text B = 0,\ A > C \\
\end
\end

Parametric representation

Standard parametric representation

Using trigonometric functions, a parametric representation of the standard ellipse $\tfrac+\tfrac = 1$ is:
: $(x,\, y) = (a \cos t,\, b \sin t),\ 0 \le t < 2\pi\ .$

The parameter ''t'' (called the ''eccentric anomaly'' in astronomy) is not the angle of $(x(t),y(t))$ with the ''x''-axis, but has a geometric meaning due to Philippe de La Hire (see '' Drawing ellipses'' below).

Rational representation

With the substitution $u = \tan\left(\frac\right)$ and trigonometric formulae one obtains
:$\cos t = \frac\ ,\quad \sin t = \frac$

and the ''rational'' parametric equation of an ellipse
: \begin
x(u) &= a\frac \\ 0mu y(u) &= b\frac
\end\;,\quad -\infty < u < \infty\;,

which covers any point of the ellipse $\tfrac + \tfrac = 1$ except the left vertex $(-a,\, 0)$.

For u \in ,\, 1 this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing $u.$ The left vertex is the limit $\lim_ (x(u),\, y(u)) = (-a,\, 0)\;.$

Alternately, if the parameter :v/math> is considered to be a point on the real projective line $\mathbf(\mathbf)$, then the corresponding rational parametrization is
:
:v\mapsto \left(a\frac, b\frac \right).

Then :0\mapsto (-a,\, 0).

Rational representations of conic sections are commonly used in computer-aided design (see Bezier curve).

Tangent slope as parameter

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

With help of trigonometric formulae one obtains:
:$\cos t = \frac = \frac\ ,\quad\quad \sin t = \frac = \frac.$

Replacing $\cos t$ and $\sin t$ of the standard representation yields:
: $\vec c_\pm(m) = \left(-\frac,\;\frac\right),\, m \in \R.$

Here $m$ is the slope of the tangent at the corresponding ellipse point, $\vec c_+$ is the upper and $\vec c_-$ the lower half of the ellipse. The vertices$(\pm a,\, 0)$, having vertical tangents, are not covered by the representation.

The equation of the tangent at point $\vec c_\pm(m)$ has the form $y = mx + n$. The still unknown $n$ can be determined by inserting the coordinates of the corresponding ellipse point $\vec c_\pm(m)$:
: $y = mx \pm\sqrt\; .$

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

General ellipse

Another definition of an ellipse uses affine transformations:
: Any ''ellipse'' is an affine image of the unit circle with equation $x^2 + y^2 = 1$.

;Parametric representation
An affine transformation of the Euclidean plane has the form $\vec x \mapsto \vec f\!_0 + A\vec x$, where $A$ is a regular matrix (with non-zero determinant) and $\vec f\!_0$ is an arbitrary vector. If $\vec f\!_1, \vec f\!_2$ are the column vectors of the matrix $A$, the unit circle $(\cos(t), \sin(t))$, $0 \leq t \leq 2\pi$, is mapped onto the ellipse:
: $\vec x = \vec p(t) = \vec f\!_0 + \vec f\!_1 \cos t + \vec f\!_2 \sin t \ .$

Here $\vec f\!_0$ is the center and $\vec f\!_1,\; \vec f\!_2$ are the directions of two conjugate diameters, in general not perpendicular.

;Vertices
The four vertices of the ellipse are $\vec p(t_0),\;\vec p\left(t_0 \pm \tfrac\right),\; \vec p\left(t_0 + \pi\right)$, for a parameter $t = t_0$ defined by:
: $\cot (2t_0) = \frac.$

(If $\vec f\!_1 \cdot \vec f\!_2 = 0$, then $t_0 = 0$.) This is derived as follows. The tangent vector at point $\vec p(t)$ is:
: $\vec p\,'(t) = -\vec f\!_1\sin t + \vec f\!_2\cos t \ .$

At a vertex parameter $t = t_0$, the tangent is perpendicular to the major/minor axes, so:
: $0 = \vec p'(t) \cdot \left(\vec p(t) -\vec f\!_0\right) = \left(-\vec f\!_1\sin t + \vec f\!_2\cos t\right) \cdot \left(\vec f\!_1 \cos t + \vec f\!_2 \sin t\right).$

Expanding and applying the identities $\; \cos^2 t -\sin^2 t=\cos 2t,\ \ 2\sin t \cos t = \sin 2t\;$ gives the equation for $t = t_0\; .$

;Area
From Apollonios theorem (see below) one obtains:

The area of an ellipse $\;\vec x = \vec f_0 +\vec f_1 \cos t +\vec f_2 \sin t\;$ is
:$A=\pi, \det(\vec f_1, \vec f_2), \ .$

;Semiaxes
With the abbreviations
$\; M=\vec f_1^2+\vec f_2^2, \ N = \left, \det(\vec f_1,\vec f_2)\$ the statements of Apollonios's theorem can be written as:
:$a^2+b^2=M, \quad ab=N \ .$
Solving this nonlinear system for $a,b$ yields the semiaxes:
:$a=\frac(\sqrt+\sqrt)$
:$b=\frac(\sqrt-\sqrt)\ .$

;Implicit representation
Solving the parametric representation for $\; \cos t,\sin t\;$ by Cramer's rule and using $\;\cos^2t+\sin^2t -1=0\;$, one obtains the implicit representation
:$\det(\vec x\!-\!\vec f\!_0,\vec f\!_2)^2+\det(\vec f\!_1,\vec x\!-\!\vec f\!_0)^2-\det(\vec f\!_1,\vec f\!_2)^2=0$.

Conversely: If the equation
:$x^2+2cxy+d^2y^2-e^2=0\ ,$ with $\; d^2-c^2 >0 \; ,$
of an ellipse centered at the origin is given, then the two vectors
:$\vec f_1=,\quad \vec f_2=\frac\$
point to two conjugate points and the tools developed above are applicable.

''Example'': For the ellipse with equation $\;x^2+2xy+3y^2-1=0\;$ the vectors are
:$\vec f_1=,\quad \vec f_2=\frac$.

;Rotated Standard ellipse
For $\vec f_0= ,\;\vec f_1= a ,\;\vec f_2= b$ one obtains a parametric representation of the standard ellipse rotated by angle $\theta$:
:$x=x_\theta(t)=a\cos\theta\cos t-b\sin\theta\sin t\ ,$
:$y=y_\theta(t)=a\sin\theta\cos t+b\cos\theta\sin t\ .$

;Ellipse in space
The definition of an ellipse in this section gives a parametric representation of an arbitrary ellipse, even in space, if one allows $\vec f\!_0, \vec f\!_1, \vec f\!_2$ to be vectors in space.

Polar forms

Polar form relative to center

In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate $\theta$ measured from the major axis, the ellipse's equation is
: $r(\theta) = \frac=\frac$
where $e$ is the eccentricity, not Euler's number

Polar form relative to focus

If instead we use polar coordinates with the origin at one focus, with the angular coordinate $\theta = 0$ still measured from the major axis, the ellipse's equation is
: $r(\theta)=\frac$

where the sign in the denominator is negative if the reference direction $\theta = 0$ points towards the center (as illustrated on the right), and positive if that direction points away from the center.

In the slightly more general case of an ellipse with one focus at the origin and the other focus at angular coordinate $\phi$, the polar form is

:$r(\theta)=\frac.$

The angle $\theta$ in these formulas is called the true anomaly