Ellipse

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of both distances to the two 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 formula for its area, but for its perimeter (also known as circumference), integration is required to obtain the exact solution.

The largest and smallest diameters of an ellipse, also known as its width and height, are typically denoted 2a and 2b. An ellipse has four extreme points: two vertices at the endpoints of the major axis and two co-vertices at the endpoints of the minor axis.

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

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 right circular 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, called the eccentricity: \[e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}.\]

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

Condensed: the full section is in Wikipedia.

An ellipse can be defined geometrically as a set or locus of points in the Euclidean plane:

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 \(e = \tfrac{c}{a}\) is defined as the eccentricity.

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 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.

The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, the x-axis is the major axis, and:

• the foci are the points \(F_1 = (c,\, 0),\ F_2=(-c,\, 0)\),
• the vertices are \(V_1 = (a,\, 0),\ V_2 = (-a,\, 0)\).

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

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

The width and height parameters \(a,\; b\) are called 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.

An arbitrary line \(g\) intersects an ellipse at 0, 1, or 2 points, respectively called an exterior line, tangent and secant. Through any point of an ellipse there is a unique tangent. The tangent at a point \((x_1,\, y_1)\) of the ellipse \(\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1\) has the coordinate equation: \[\frac{x_1}{a^2}x + \frac{y_1}{b^2}y = 1.\]

A vector parametric equation of the tangent is: \[\vec x = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + s \left(\begin{array}{r} -y_1 a^2 \\ x_1 b^2 \end{array}\right) , \quad s \in \R.\]

Proof: Let \((x_1,\, y_1)\) be a point on an ellipse and \(\vec{x} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + s \begin{pmatrix} u \\ v \end{pmatrix}\) be the equation of any line \(g\) containing \((x_1,\, y_1)\). Inserting the line's equation into the ellipse equation and respecting \(\frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} = 1\) yields: \[\frac{\left(x_1 + su\right)^2}{a^2} + \frac{\left(y_1 + sv\right)^2}{b^2} = 1\ \quad\Longrightarrow\quad 2s\left(\frac{x_1u}{a^2} + \frac{y_1v}{b^2}\right) + s^2\left(\frac{u^2}{a^2} + \frac{v^2}{b^2}\right) = 0\ .\] There are then cases:

1. \(\frac{x_1}{a^2}u + \frac{y_1}{b^2}v = 0.\) Then line \(g\) and the ellipse have only point \((x_1,\, y_1)\) in common, and \(g\) is a tangent. The tangent direction has perpendicular vector \(\begin{pmatrix} \frac{x_1}{a^2} & \frac{y_1}{b^2} \end{pmatrix}\), so the tangent line has equation \(\frac{x_1}{a^2}x + \tfrac{y_1}{b^2}y = k\) for some \(k\). Because \((x_1,\, y_1)\) is on the tangent and the ellipse, one obtains \(k = 1\).
2. \(\frac{x_ 1}{a^2}u + \frac{y_1}{b^2}v \ne 0.\) Then line \(g\) 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 \(\frac{x_1u}{a^2} + \tfrac{y_1v}{b^2} = 0\), then the points lie on two conjugate diameters (see below). (If \(a = b\), the ellipse is a circle and "conjugate" means "orthogonal".)

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 the x- and y-axes.

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 \[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\] provided \(B^2 - 4AC < 0.\)

To distinguish the degenerate cases from the non-degenerate case, let ∆ be the determinant \[\Delta = \begin{vmatrix} A & \frac{1}{2}B & \frac{1}{2}D \\ \frac{1}{2}B & C & \frac{1}{2}E \\ \frac{1}{2}D & \frac{1}{2}E & F \end{vmatrix} = ACF + \tfrac14 BDE - \tfrac14(AE^2 + CD^2 + FB^2).\]

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.

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: \[\begin{aligned} A &= a^2 \sin^2\theta + b^2 \cos^2\theta & B &= 2\left(b^2 - a^2\right) \sin\theta \cos\theta \\[1ex] C &= a^2 \cos^2\theta + b^2 \sin^2\theta & D &= -2A x_\circ - B y_\circ \\[1ex] E &= - B x_\circ - 2C y_\circ & F &= A x_\circ^2 + B x_\circ y_\circ + C y_\circ^2 - a^2 b^2. \end{aligned}\]

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:

\[\begin{aligned} a, b &= \frac{-\sqrt{2 \big(A E^2 + C D^2 - B D E + (B^2 - 4 A C) F\big)\big((A + C) \pm \sqrt{(A - C)^2 + B^2}\big)}}{B^2 - 4 A C}, \\ x_\circ &= \frac{2CD - BE}{B^2 - 4AC}, \\[5mu] y_\circ &= \frac{2AE - BD}{B^2 - 4AC}, \\[5mu] \theta &= \tfrac12 \operatorname{atan2}(-B,\, C-A), \end{aligned}\]

Condensed: the full section is in Wikipedia.

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

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).

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

and the rational 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 \(u.\) The left vertex is the limit \(\lim_{u \to \pm \infty} (x(u),\, y(u)) = (-a,\, 0)\;.\)

Alternately, if the parameter \([u:v]\) is considered to be a point on the real projective line \(\mathbf{P}(\mathbf{R})\), 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 \([1:0] \mapsto (-a,\, 0).\)

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

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 \(\vec x(t) = (a \cos t,\, b \sin t)\): \[\vec x'(t) = (-a\sin t,\, b\cos t) \quad \rightarrow \quad m = -\frac{b}{a}\cot t\quad \rightarrow \quad \cot t = -\frac{ma}{b}.\]

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

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

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 \(\vec c_\pm(m)\): \[y = mx \pm \sqrt{m^2 a^2 + b^2}\, .\]

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.

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 \(\vec p(t_0),\;\vec p\left(t_0 \pm \tfrac{\pi}{2}\right),\; \vec p\left(t_0 + \pi\right)\), 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

Semiaxes

Implicit representation

\(x^2+2cxy+d^2y^2-e^2=0\ ,\) with \(\; d^2-c^2 >0 \; ,\)

Rotated standard ellipse

Ellipse in space

Condensed: the full section is in Wikipedia.

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 \[r(\theta) = \frac{ab}{\sqrt{(b \cos \theta)^2 + (a\sin \theta)^2}}=\frac{b}{\sqrt{1 - (e\cos\theta)^2}}\] where \(e\) is the eccentricity (not Euler's number).

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 the true anomaly of the point. The numerator \(\ell=a (1-e^2)\) is the semi-latus rectum.

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

For an arbitrary point \(P\) 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 \(\left|PF_1\right|^2 = (x - c)^2 + y^2,\ \left|Pl_1\right|^2 = \left(x - \tfrac{a^2}{c}\right)^2\) 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 \(F\) (focus), any line \(l\) (directrix) not through \(F\), and any real number \(e\) 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 \(e,\) 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 the line at infinity in the projective plane.

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

Proof

Let \(F = (f,\, 0),\ e > 0\), and assume \((0,\, 0)\) is a point on the curve. The directrix \(l\) 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 \(x^2\left(e^2 - 1\right) + 2xf(1 + e) - y^2 = 0.\)

Construction of a directrix

General ellipse

Condensed: the full section is in Wikipedia.

An ellipse possesses the following property:

The normal at a point \(P\) 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 \(L\) be the point on the line \(\overline{PF_2}\) with distance \(2a\) to the focus \(F_2\), where \(a\) is the semi-major axis of the ellipse. Let line \(w\) be the external angle bisector of the lines \(\overline{PF_1}\) and \(\overline{PF_2}.\) Take any other point \(Q\) on \(w.\) By the triangle inequality and the angle bisector theorem, \(2a = \left|LF_2\right| < {}\)\(\left|QF_2\right| + \left|QL\right| = {}\)\(\left|QF_2\right| + \left|QF_1\right|,\) so \(Q\) must be outside the ellipse. As this is true for every choice of \(Q,\) \(w\) only intersects the ellipse at the single point \(P\) 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 (see whispering gallery). It also serves to formulate Alhazen's problem of reflection on a circle tangent to the ellipse.

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.