Ellipse

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

"Elliptical" redirects here. For the exercise machine, see Elliptical trainer.

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

In mathematics, an ellipse is a curve on a plane that surrounds two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a special type of an ellipse that has both focal points at the same location. The shape of an ellipse (how 'elongated' it is) is represented by its eccentricity, which for an ellipse can be any number from 0 (the limiting case of a circle) to arbitrarily close to but less than 1.

Ellipses are the closed type of conic section: a plane curve that results from the intersection of a cone by a plane. (See figure to the right.) Ellipses have many similarities with the other two forms of conic sections: the parabolas and the hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder.

Analytically, an ellipse can also be defined as the set of points such that the ratio of the distance of each point on the curve from a given point (called a focus or focal point) to the distance from that same point on the curve to a given line (called the directrix) is a constant, called the eccentricity of the ellipse.

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 barycenter of the planet-Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids. Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection. It 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, ἔλλειψις (élleipsis, "omission"), was given by Apollonius of Perga in his Conics, emphasizing the connection of the curve with "application of areas".

Elements of an ellipse

Drawing ellipses

Drawing an ellipse with two pins, a loop, and a pen

Trammel of Archimedes (ellipsograph) animation

Ellipse construction applying the parallelogram method

Pins-and-string method

The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two drawing pins, a length of string, and a pencil.^[11] In this method, pins are pushed into the paper at two points, which become the ellipse's foci. A string tied at each end to the two pins and the tip of a pen pulls the loop taut to form a triangle. The tip of the pen then traces an ellipse if it is moved while keeping the string taut. Using two pegs and a rope, gardeners use this procedure to outline an elliptical flower bed—thus it is called the gardener's ellipse.^[12]

The Gardener's Ellipse- Worked Example

Create an ellipse 3.0m by 1.5m

Tools required:

5 pegs and a mallet to drive them

a ball of cotton (non-stretchy) string or bricklayers line

a calculator

Method

1. Place center peg at C (the center of the ellipse)

2. Determine the orientation of the long (major) axis and, if desired, place a long line (longer than the major axis) along that axis passing through C

3. Write down the major (a) and minor (b) axes dimensions:

\begin{align}a &= 1500 \\b &= 750\end{align}

Note: [a is only C to a; b is only C to b, that is, HALF of the full major/minor axes]

4. Calculate the positions of the foci (F1) and (F2):

\begin{align}f &= \sqrt{1500^2-750^2} \\ &=1299 \end{align}

Note:[C to F][F1 to F2 = 2598]

5. Place pegs accordingly at F1 and F2 on the orientation line you affixed at step 2.

6. Cut your string to length 2a (allowing for the knot and tail ends!)

2a = 3000

Note: [The string loop will always be equal to the length of the major axis]

7. Place the string on the ground around the pegs F1 and F2 and pull taught and then moving around the periphery, mark as desired on the ground (It doesn't matter where you start/stop)

Trammel method

An ellipse can also be drawn using a ruler, a set square, and a pencil:

Draw two perpendicular lines M,N on the paper; these are the major (M) and minor (N) axes of the ellipse. Mark three points A, B, C on the ruler. A->C being the length of the semi-major axis and B->C the length of the semi-minor axis. With one hand, move the ruler on the paper, turning and sliding it so as to keep point A always on line N, and B on line M. With the other hand, keep the pencil's tip on the paper, following point C of the ruler. The tip traces out an ellipse.

The trammel of Archimedes, or ellipsograph, is a mechanical device that implements this principle. The ruler is replaced by a rod with a pencil holder (point C) at one end, and two adjustable side pins (points A and B) that slide into two perpendicular slots cut into a metal plate.^[13] The mechanism can be used with a router to cut ellipses from board material. The mechanism is also used in a toy called the "nothing grinder".

Parallelogram method

In the parallelogram method, an ellipse is constructed point by point using equally spaced points on two horizontal lines and equally spaced points on two vertical lines. It is based on Steiner's theorem on the generation of conic sections. Similar methods exist for the parabola and hyperbola.

Mathematical definitions and properties

In Euclidean geometry

Definition

In Euclidean geometry, the ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points (the foci) is constant. The ellipse can also be defined as the set of points such that the distance from any point in that set to a given point in the plane (a focus) is a constant positive fraction less than 1 (the eccentricity) of the perpendicular distance of the point in the set to a given line (called the directrix). Yet another equivalent definition of the ellipse is that it is the set of points that are equidistant from one point in the plane (a focus) and a particular circle, the directrix circle (whose center is the other focus).

The equivalence of these definitions can be proved using the Dandelin spheres.

Equations

The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is $\displaystyle{\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2} = 1$ . This can be explained as follows. If we let an independent parameter $\theta$ increase from 0 to 2π, and let

x = a\cos\theta,

and

y = b\sin\theta,

then plotting x and y values for all angles of θ results in an ellipse. Note that $\theta$ is the eccentric anomaly and is not the angle traced out by a point on the ellipse (see below). For example, at θ = 0, x = a, y = 0 and at θ = π/2, y = b, x = 0. This can be seen as follows.

Squaring both equations gives:

x^2 = a^2\cos^2\theta.

And

y^2 = b^2\sin^2\theta.

Dividing these two equations by a² and b² respectively gives:

\frac{x^2}{a^2} = \cos^2\theta.

And

\frac{y^2}{b^2} = \sin^2\theta.

Adding these two equations together gives:

\frac{x^2}{a^2} + \frac{y^2}{b^2} = \cos^2\theta + \sin^2\theta.

Applying the Pythagorean identity to the right hand side gives:

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

This means any noncircular ellipse is a compressed (or stretched) circle. If a circle is treated like an ellipse, then the area of the ellipse would be proportional to the length of either axis (i.e. doubling the length of an axis in a circular ellipse would create an ellipse with double the area of the original circle).

Focus

The distance from the center C to either focus is f = ae, which can be expressed in terms of the major and minor radii:

f = \sqrt{a^2-b^2}.

The sum of the distances from any point P = P(x,y) on the ellipse to those two foci is constant and equal to the major axis (proof):

PF_1 + PF_2 = \sqrt{(x+f)^2+y^2} + \sqrt{(x-f)^2+y^2} = 2a

Eccentricity

The eccentricity of the ellipse (commonly denoted as either e or $\varepsilon$ ) is

e=\varepsilon=\sqrt{\frac{a^2-b^2}{a^2}} =\sqrt{1-\left(\frac{b}{a}\right)^2} =f/a

(where again a and b are one-half of the ellipse's major and minor axes respectively, and f is the focal distance) or, as expressed in terms using the flattening factor $g=1-\frac{b}{a}=1-\sqrt{1-e^2},$

e=\sqrt{g(2-g)}.

Other formulas for the eccentricity of an ellipse are listed in the article on eccentricity of conic sections. Formulas for the eccentricity of an ellipse that is expressed in the more general quadratic form are described in the article dedicated to conic sections.

Directrix

Each focus F of the ellipse is associated with a line parallel to the minor axis called a directrix. Refer to the illustration on the right, in which the ellipse is centered at the origin. The distance from any point P on the ellipse to the focus F is a constant fraction of that point's perpendicular distance to the directrix, resulting in the equality e = PF/PD. The ratio of these two distances is the eccentricity of the ellipse. This property (which can be proved using the Dandelin spheres) can be taken as another definition of the ellipse.
Besides the well-known ratio e = f/a, where f is the distance from the center to the focus and a is the distance from the center to the farthest vertices (most sharply curved points of the ellipse), it is also true that e = a/d, where d is the distance from the center to the directrix.

Circular directrix

The ellipse can also be defined as the set of points that are equidistant from one focus and a circle, the directrix circle, that is centered on the other focus. The radius of the directrix circle equals the ellipse's major axis, so the focus and the entire ellipse are inside the directrix circle.

Ellipse as hypotrochoid

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

The ellipse is a special case of the hypotrochoid when R = 2r, as shown in the image to the right.

Area

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

A_\text{ellipse} = \pi ab

where a and b are the semi-major and semi-minor axes (half of the ellipse's major and minor axes), respectively.

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

The area formula πab is intuitive: start with a circle of radius b (so its area is πb²) and stretch it by a factor a/b to make an ellipse. This intuitively justifies the area by the same factor: πb²(a/b) = πab. However, a more rigorous proof requires integration as follows:

For the ellipse in standard form, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ , and hence $y=\pm \sqrt{\frac{a^2b^2-b^2x^2}{a^2}}$ , with horizontal intercepts at ± a, the area $A_\text{ellipse}$ can be computed as twice the integral of the positive square root:

\begin{align} A_\text{ellipse} &= \int_{-a}^a 2b\sqrt{1-x^2/a^2}\,dx\\ &= \frac ba \int_{-a}^a 2\sqrt{a^2-x^2}\,dx. \end{align}

The second integral is the area of a circle of radius $a$ , i.e., $\pi a^2$ ; thus we have:

A_\text{ellipse} = \frac{b}{a}A_\text{circle} = \pi ab.

The area formula can also be proven in terms of polar coordinates using the coordinate transformation $\bold{T}(r,\theta) = ( ra \cos\theta , rb \sin\theta ).$

Any point inside the ellipse with x-intercept a and y-intercept b can be defined in terms of r and $\theta$ , where $0 \le r \le 1$ and $0 \le \theta \le 2\pi$ .

To define the area differential in such coordinates we use the Jacobian matrix of the coordinate transformation times $dr\,d\theta$ :

\begin{align} dA_\text{ellipse} &= \det \begin{pmatrix} \frac{\partial \bold{T}}{\partial r} & \frac{\partial \bold{T}}{\partial \theta}\\ \end{pmatrix} \,dr\,d\theta \\ &= \det \begin{pmatrix} a\cos\theta & -ra\sin\theta \\ b\sin\theta & rb\cos\theta \end{pmatrix} \,dr\,d\theta \\ &= abr\,dr\,d\theta. \end{align}

We now integrate over the ellipse to find the area:

A_\text{ellipse} = \iint\limits_{\text{ellipse}} dA_\text{ellipse} = \iint\limits_{\text{ellipse}} abr\,dr\,d\theta = ab \int_{0}^{2\pi} \int_{0}^{1}r\,dr\,d\theta = ab\pi.

Circumference

The circumference $C$ of an ellipse is:

C = 4 a E(e)

where again a is the length of the semi-major axis, e is the eccentricity $\left(\sqrt{1-\left(\frac{b}{a}\right)^2}\right)$ , and the function $E$ is the complete elliptic integral of the second kind,

E(e) = \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta,

which calculates the circumference of the ellipse in the first quadrant alone, and the formula for the circumference of an ellipse can thus be written

$C = 4a\int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta.$

(1)

The arc length of an ellipse, in general, has no closed-form solution in terms of elementary functions. Elliptic integrals were motivated by this problem. Equation (1) may be evaluated directly using the Carlson symmetric form.^[14] This gives a succinct and rapidly converging method for evaluating the circumference.^[15]

The exact infinite series is:

C = 2\pi a \left[{1 - \left(\frac{1}{2}\right)^2e^2 - \left(\frac{1\cdot 3}{2\cdot 4}\right)^2\frac{e^4}{3} - \left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\right)^2\frac{e^6}{5} - \cdots}\right]

C = 2\pi a \left[1 - \sum_{n=1}^\infty \left(\frac{(2n-1)!!}{2^n n!}\right)^2 \frac{e^{2n}}{2n-1}\right],

where $n!!$ is the double factorial. Unfortunately, this series converges rather slowly; however, by expanding in terms of $h = \frac{(a-b)^2}{(a+b)^2}$ , Ivory^[16] and Bessel^[17] derived an expression that converges much more rapidly,

C = \pi (a+b) \left[1 + \sum_{n=1}^\infty \left(\frac{(2n-1)!!}{2^n n!}\right)^2 \frac{h^n}{(2n-1)^2}\right].

Ramanujan gives two good approximations for the circumference in §16 of "Modular Equations and Approximations to π";^[18] they are

C \approx \pi \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]= \pi \left[3(a+b)-\sqrt{10ab+3(a^2+b^2)}\right]

and

C\approx\pi\left(a+b\right)\left(1+\frac{3h}{10+\sqrt{4-3h}}\right).

The errors in these approximations, which were obtained empirically, are of order $h^3$ and $h^5$ , respectively.

More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral.

The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions.

Some lower and upper bounds on the circumference of the canonical ellipse $\tfrac{x^2}{a^2}+\tfrac{y^2}{b^2}=1$ with a ≥b are^[19]

C\le 2\pi a,

\pi (a+b)\le C\le 4(a+b),

4\sqrt{a^2+b^2}\le C\le \sqrt{2} \pi \sqrt{a^2+b^2} .

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

Chords

The midpoints of a set of parallel chords of an ellipse are collinear.^[20]^:p.147

Latus rectum

The chords of an ellipse that are perpendicular to the major axis and pass through one of its foci are called the latera recta of the ellipse. The length of each latus rectum is $2 b 2 / a$ .

Curvature

The curvature is given by $\kappa = \frac{1}{a^2 b^2}\left(\frac{x^2}{a^4}+\frac{y^2}{b^4}\right)^{-\frac{3}{2}}.$

Angle bisection property

A local normal (perpendicular) to the ellipse at any point P bisects the angle $\angle F_1 P F_2$ to the foci. This is evident graphically in the parallelogram method of construction, and can be proven analytically, for example by using the parametric form in canonical position, as given below.

Reflexive property

When a ray of light originating from one focus reflects off the inner surface of an ellipse, it always passes through the other focus.

In projective geometry

In a projective geometry defined over a field, a conic section can be defined as the set of all points of intersection between corresponding lines of two pencils of lines in a plane that are related by a projective, but not perspective, map (see Steiner's theorem). By projective duality, a conic section can also be defined as the envelope of all lines that connect corresponding points of two lines related by a projective, but not perspective, map.

In a pappian projective plane (one defined over a field), all conic sections are equivalent to each other, and the different types of conic sections are determined by how they intersect the line at infinity, denoted by Ω. An ellipse is a conic section that does not intersect this line. A parabola is a conic section that is tangent to Ω, and a hyperbola is one that crosses Ω twice.^[21] Since an ellipse does not intersect the line at infinity, it properly belongs to the affine plane determined by removing the line at infinity and all of its points from the projective plane.

Affine space

An ellipse is also the result of projecting a circle, sphere, or ellipse in a three dimensional affine space onto a plane (flat), by parallel lines. This is a special case of conical (perspective) projection of any of those geometric objects in the affine space from a point O onto a plane P, when the point O lies in the plane at infinity of the affine space. In the setting of pappian projective planes, the image of an ellipse by any affine map (a projective map that leaves the line at infinity invariant) is an ellipse, and, more generally, the image of an ellipse by any projective map M such that the line M⁻¹(Ω) does not touch or cross the ellipse is an ellipse.

In analytic geometry

General ellipse

In analytic geometry, the ellipse is defined as the set of points $(X,Y)$ of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation^[22]^[23]

~A X^2 + B X Y + C Y^2 + D X + E Y + F = 0

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

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

\begin{vmatrix} A & B/2 & D/2\\B/2 & C & E/2\\D/2 & E/2 & F \end{vmatrix};

that is,

\Delta = \left( AC - \frac{B^2}{4} \right) F + \frac{BED}{4} - \frac{CD^2}{4} - \frac{AE^2}{4}.

Then the ellipse is a non-degenerate real ellipse if and only if C∆ < 0. If C∆ > 0, we have an imaginary ellipse, and if ∆ = 0, we have a point ellipse.^[24]^:p.63

The general equation's coefficients can be obtained from known semi-major axis $a$ , semi-minor axis $b$ , center coordinates $(x_c, y_c)$ and rotation angle $\Theta$ using the following formulae:

\begin{align}A &= a^2 (\sin\Theta)^2 + b^2 (\cos\Theta)^2\\ B &= 2 (b^2-a^2) \sin\Theta \cos\Theta\\ C &= a^2 (\cos\Theta)^2 + b^2 (\sin\Theta)^2\\ D &= -2 A x_c - B y_c\\ E &= -B x_c - 2 C y_c\\ F &= A x_c^2 + B x_c y_c + C y_c^2 - a^2 b^2\end{align}

These expressions can be derived from the canonical equation (see next section) by substituting the coordinates with expressions for rotation and translation of the coordinate system:

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

x_{can} = (x-x_c) \cos\Theta + (y-y_c) \sin\Theta

y_{can} = -(x-x_c) \sin\Theta + (y-y_c) \cos\Theta

Canonical form

Let $a>b$ . Through change of coordinates (a rotation of axes and a translation of axes) the general ellipse can be described by the canonical implicit equation

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

Here $(x,y)$ are the point coordinates in the canonical system, whose origin is the center $(X_c,Y_c)$ of the ellipse, whose $x$ -axis is the unit vector $(X_a,Y_a)$ coinciding with the major axis, and whose $y$ -axis is the perpendicular vector $(-Y_a,X_a)$ coinciding with the minor axis. That is, $x = X_a(X - X_c) + Y_a(Y - Y_c)$ and $y = -Y_a(X - X_c) + X_a(Y - Y_c)$ .

In this system, the center is the origin $(0,0)$ and the foci are $(-e a, 0)$ and $(+e a, 0)$ .

Any ellipse can be obtained by rotation and translation of a canonical ellipse with the proper semi-diameters. The expression of an ellipse centered at $(X_c,Y_c)$ is

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

Moreover, any canonical ellipse can be obtained by scaling the unit circle of $\reals^2$ , defined by the equation

X^2+Y^2=1\,

by factors a and b along the two axes.

For an ellipse in canonical form, we have

Y = \pm b\sqrt{1 - (X/a)^2} = \pm \sqrt{(a^2-X^2)(1 - e^2)}

The distances from a point $(X,Y)$ on the ellipse to the left and right foci are $a + e X$ and $a - e X$ , respectively.

Line segment as a type of degenerate ellipse

A line segment is a degenerate ellipse with semi-minor axis = 0 and eccentricity = 1, and with the focal points at the ends.^[25] Although the eccentricity is 1 this is not a parabola. A radial elliptic trajectory is a non-trivial special case of an elliptic orbit, where the ellipse is a line segment.

In trigonometry

General parametric form

An ellipse in general position can be expressed parametrically as the path of a point $(X(t),Y(t))$ , where

X(t)=X_c + a\,\cos t\,\cos \varphi - b\,\sin t\,\sin\varphi

Y(t)=Y_c + a\,\cos t\,\sin \varphi + b\,\sin t\,\cos\varphi

as the parameter t varies from 0 to 2π. Here $(X_c,Y_c)$ is the center of the ellipse, and $\varphi$ is the angle between the $X$ -axis and the major axis of the ellipse.

Parametric form in canonical position

Parametric equation for the ellipse (red) in canonical position. The eccentric anomaly t is the angle of the blue line with the X-axis.

For an ellipse in canonical position (center at origin, major axis along the X-axis), the equation simplifies to

X(t)=a\,\cos t

Y(t)=b\,\sin t

The parameter t (called the eccentric anomaly in astronomy) is not the angle of $(X(t),Y(t))$ with the X-axis (see diagram at right).

For a given point on an ellipse, formulae connecting the tangential angle $\phi$ , the polar angle from the ellipse center $\theta$ , and the parametric angle t^[26] are:^[27]^[28]^[29]^[30]

-\cot\phi = \frac{a}{b} \tan t = \frac{\tan \theta}{(1-g)^2} = \frac{\tan \theta}{1-e^2},

-\tan t = \frac{b}{a} \cot\phi = \sqrt{1-e^2} \cot\phi = (1-g) \cot\phi = \frac{-\tan\theta}{\sqrt{1-e^2}} = -\frac{a}{b} \tan\theta.

Polar form relative to center

Polar coordinates centered at the 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^[24]^{:p. 75}

r(\theta)=\frac{ab}{\sqrt{(b \cos \theta)^2 + (a\sin \theta)^2}}

Polar form relative to focus

Polar coordinates centered at 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{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.

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=\frac{a (1-e^2)}{1 - e\cos(\theta - \phi)}.

The angle $\theta$ in these formulas is called the true anomaly of the point. The numerator $a (1-e^2)$ of these formulas is the semi-latus rectum of the ellipse, usually denoted $l$ . It is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis.

Semi-latus rectum.

General polar form

The following equation on the polar coordinates (r, θ) describes a general ellipse with semidiameters a and b, centered at a point (r₀, θ₀), with the a axis rotated by φ relative to the polar axis:

r(\theta)=\frac{P(\theta)+Q(\theta)}{R(\theta)}

where r is the radius or central distance, and

P(\theta)=r_0 \left[\left(b^2-a^2\right) \cos\left(\theta+\theta_0 -2\varphi \right)+\left(a^2+b^2\right) \cos \left(\theta -\theta_0\right)\right]

Q(\theta)=\sqrt2 ab \sqrt{R(\theta)-2 r_0^2 \sin^2\left(\theta-\theta_0\right)}

R(\theta)=\left(b^2-a^2\right) \cos(2\theta-2\varphi)+a^2+b^2

Angular eccentricity

The angular eccentricity $\alpha$ is the angle whose sine is the eccentricity e; that is,

\alpha=\arcsin(e)=\arccos\left(\frac{b}{a}\right)=2\arctan\left(\sqrt{\frac{a-b}{a+b}}\right).

As a parametric rational polynomial

An ellipse can be parameterized as a rational quadratic polynomial, in other words described by the equations $x(t)=p(t)/r(t)$ and $y(t)=q(t)/r(t)$ where $p(t),q(t),$ and $r(t)$ are quadratic polynomials in $t.$ The tangent half-angle identities

u = \tan(t/2) = \sin t / (1 + \cos t) = (1 - \cos t) / \sin t

imply $u^2 = (1-\cos t) / (1 + \cos t)$ and this implies

\cos t = (1-u^2)/(u^2 + 1).

Substituting this equation for $\cos t$ into the first tangent half-angle identity yields

\sin t = u(1 + \cos t) = 2u/(u^2 + 1).

Substituting these values for $\sin t$ and $\cos t$ into the trigonometric parameterization above yields

\begin{array}{lcl} x(u) & = & a(1-u^2)/(u^2+1) \\ y(u) & = & 2bu/(u^2+1) \end{array}.

For $u\in [0,1],$ this formula represents the quarter ellipse centered at the origin with radii $a$ and $b$ moving counter-clockwise with increasing $u.$ It is easy to test this by computing $(x(0), y(0)) = (a,0)$ and $(x(1), y(1)) = (0,b).$

Degrees of freedom

An ellipse in the plane has five degrees of freedom (the same as a general conic section), defining its vertical and horizontal position, orientation, shape, and scale. In comparison, circles have only three degrees of freedom (horizontal position, vertical position and scale), while parabolae have four. Said another way, the set of all ellipses in the plane, with any natural metric (such as the Hausdorff distance) is a five-dimensional manifold.

The five degrees of freedom can be identified with, for example, the coefficients A,B,C,D,E of the implicit equation, or with the coefficients X_c, Y_c, φ, a, b of the general parametric form. Thus an ellipse is uniquely determined by any five points lying on it.

Applications

Ellipses in physics

Elliptical reflectors and acoustics

Planetary orbits

Main article: Elliptic orbit

In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun [approximately] at one focus, in his first law of planetary motion. Later, Isaac Newton explained this as a corollary of his law of universal gravitation.

More generally, in the gravitational two-body problem, if the two bodies are bound to each other (i.e., the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at the same focus.

Keplerian elliptical orbits are the result of any radially directed attraction force whose strength is inversely proportional to the square of the distance. Thus, in principle, the motion of two oppositely charged particles in empty space would also be an ellipse. (However, this conclusion ignores losses due to electromagnetic radiation and quantum effects, which become significant when the particles are moving at high speed.)

For elliptical orbits, useful relations involving the eccentricity $e$ are:

\begin{align}e &= \frac{r_{a}-r_{p}}{r_{a}+r_{p}}=\frac{r_{a}-r_{p}}{2a}\\ r_{a} &= (1+e)a\\ r_{p} &= (1-e)a\end{align}

where

$r_{a}$ is the radius at apoapsis (the farthest distance)
$r_{p}$ is the radius at periapsis (the closest distance)
$a$ is the length of the semi-major axis

Also, in terms of $r_{a}$ and $r_{p}$ , the semi-major axis $a$ is their arithmetic mean, the semi-minor axis $b$ is their geometric mean, and the semi-latus rectum $l$ is their harmonic mean. In other words,

\begin{align}a &= \frac{r_{a}+r_{p}}{2}\\ b &= \sqrt{r_{a}\cdot r_{p}}\\ l &= \frac{2}{\frac{1}{r_{a}}+\frac{1}{r_{p}}}=\frac{2r_{a}r_{p}}{r_{a}+r_{p}}\end{align}

Harmonic oscillators

The general solution for a harmonic oscillator in two or more dimensions is also an ellipse. Such is the case, for instance, of a long pendulum that is free to move in two dimensions; of a mass attached to a fixed point by a perfectly elastic spring; or of any object that moves under influence of an attractive force that is directly proportional to its distance from a fixed attractor. Unlike Keplerian orbits, however, these "harmonic orbits" have the center of attraction at the geometric center of the ellipse, and have fairly simple equations of motion.

Phase visualization

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

Elliptical gears

Two non-circular gears with the same elliptical outline, each pivoting around one focus and positioned at the proper angle, turn smoothly while maintaining contact at all times. Alternatively, they can be connected by a link chain or timing belt, or in the case of a bicycle the main chainring may be elliptical, or an ovoid similar to an ellipse in form. Such elliptical gears may be used in mechanical equipment to produce variable angular speed or torque from a constant rotation of the driving axle, or in the case of a bicycle to allow a varying crank rotation speed with inversely varying mechanical advantage.

Elliptical bicycle gears make it easier for the chain to slide off the cog when changing gears.^[31]

An example gear application would be a device that winds thread onto a conical bobbin on a spinning machine. The bobbin would need to wind faster when the thread is near the apex than when it is near the base.^[32]

Optics

In a material that is optically anisotropic (birefringent), the refractive index depends on the direction of the light. The dependency can be described by an index ellipsoid. (If the material is optically isotropic, this ellipsoid is a sphere.)
In lamp-pumped solid-state lasers, elliptical cylinder-shaped reflectors have been used to direct light from the pump lamp (coaxial with one ellipse focal axis) to the active medium rod (coaxial with the second focal axis).^[33]
In laser-plasma produced EUV light sources used in microchip lithography, EUV light is generated by plasma positioned in the primary focus of an ellipsoid mirror and is collected in the secondary focus at the input of the lithography machine.^[34]

Ellipses in statistics and finance

In statistics, a bivariate random vector (X, Y) is jointly elliptically distributed if its iso-density contours — loci of equal values of the density function — are ellipses. The concept extends to an arbitrary number of elements of the random vector, in which case in general the iso-density contours are ellipsoids. A special case is the multivariate normal distribution. The elliptical distributions are important in finance because if rates of return on assets are jointly elliptically distributed then all portfolios can be characterized completely by their mean and variance — that is, any two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return.^[35]^[36]

Ellipses in computer graphics

Drawing an ellipse as a graphics primitive is common in standard display libraries, such as the MacIntosh QuickDraw API, and Direct2D on Windows. Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. M. L. V. Pitteway extended Bresenham's algorithm for lines to conics in 1967.^[37] Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.^[38]

In 1970 Danny Cohen presented at the "Computer Graphics 1970" conference in England a linear algorithm for drawing ellipses and circles. In 1971, L. B. Smith published similar algorithms for all conic sections and proved them to have good properties.^[39] These algorithms need only a few multiplications and additions to calculate each vector.

It is beneficial to use a parametric formulation in computer graphics because the density of points is greatest where there is the most curvature. Thus, the change in slope between each successive point is small, reducing the apparent "jaggedness" of the approximation.

Drawing with Bézier paths

Composite Bézier curves may also be used to draw an ellipse to sufficient accuracy, since any ellipse may be construed as an affine transformation of a circle. The spline methods used to draw a circle may be used to draw an ellipse, since the constituent Bézier curves behave appropriately under such transformations.

Drawing with three points of a parallelogram

Rytz’s construction can be used to find the minor and major axes and their angle of an ellipse from conjugate diameters (which can be seen as three points of a parallelogram). The method uses the conjugate diameters of an ellipse to map the ellipse to an unit circle under affine transformation and calculate the ellipse parameters from that.

Ellipses in optimization theory

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

Notes

↑ The "major axis" and "minor axis" are sometimes called the "transverse diameter" and "conjugate diameter"; see Haswell, Charles Haynes (1920). Mechanics' and Engineers' Pocket-book of Tables, Rules, and Formulas. Harper & Brothers. This usage is now rare
↑ Herschel, Sir John Frederick William (1842). A treatise on astronomy. Lea & Blanchard. p. 256.
↑ Lankford, John (1997). History of Astronomy: An Encyclopedia. Taylor & Francis. p. 194. ISBN 978-0-8153-0322-0.
↑ Prasolov, Viktor Vasilʹevich; Tikhomirov, Vladimir Mikhaĭlovich (2001). Geometry. American Mathematical Society. p. 80. ISBN 978-0-8218-2038-4.
↑ Fenna, Donald (2007). Cartographic Science: A Compendium of Map Projections, With Derivations. CRC Press. p. 24. ISBN 978-0-8493-8169-0.
↑ AutoCAD release 13 command reference. Autodesk, Inc. 1994. p. 216.
↑ Salomon, David (2006). Curves And Surfaces for Computer Graphics. Birkhäuser. p. 365. ISBN 978-0-387-24196-8.
↑ Kreith, Frank; Goswami, D. Yogi (2005). The CRC Handbook Of Mechanical Engineering. CRC Press. pp. 11–8. ISBN 978-0-8493-0866-6. Circles and Ellipses (11.3.2)
↑ The Mathematical Association of America (1976), The American Mathematical Monthly, vol. 83, page 207
↑ Gibson, C. G. (2001), Elementary Geometry of Differentiable Curves: An Undergraduate Introduction, Cambridge University Press, p. 127, ISBN 9780521011075 .
↑ Besant 1907, p. 57
↑ Armengaud, Aîné (1853). "Ovals, Ellipses, Parabolas, Volutes, etc. §53". The Practical Draughtsman's Book of Industrial Design. Longman, Brown, Green, and Longmans. p. 16.
↑ Brown, Henry T. (1881). Five Hundred and Seven Mechanical Movements: Embracing All Those which are Most Important in Dynamics, Hydraulics, Hydrostatics, Pneumatics, Steam Engines, Mill and Other Gearing, Presses, Horology, and Miscellaneous Machinery; and Including Many Movements Never Before Published, and Several which Have Only Recently Come Into Use. Brown & Brown. pp. 40–41 section 152.
↑ Carlson, B. C. (1995). "Numerical computation of real or complex elliptic integrals". Numerical Algorithms 10 (1): 13–98. arXiv:math/9409227. Bibcode:1995NuAlg..10...13C. doi:10.1007/BF02198293.
↑ Python code for the circumference of an ellipse in terms of the complete elliptic integral of the second kind, retrieved 2013-12-28
↑ Ivory, J. (1798). "A new series for the rectification of the ellipsis". Transactions of the Royal Society of Edinburgh 4: 177–190. doi:10.1017/s0080456800030817.
↑ Bessel, F. W. (2010). "The calculation of longitude and latitude from geodesic measurements (1825)". Astron. Nachr. 331 (8): 852–861. arXiv:0908.1824. doi:10.1002/asna.201011352. English translation of Astron. Nachr. 4, 241–254 (1825).
↑ Ramanujan, Srinivasa, (1914). "Modular Equations and Approximations to π". Quart. J. Pure App. Math. 45: 350–372.
↑ Jameson, G.J.O. "inequalities for the perimeter of an ellipse", Mathematical Gazette 98, July 2014, 227-234. doi: 10.2307/3621497
↑ Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.
↑ Meserve 1983, p. 159
↑ Larson, Ron; Hostetler, Robert P.; Falvo, David C. (2006). "Chapter 10". Precalculus with Limits. Cengage Learning. p. 767. ISBN 0-618-66089-5.
↑ Young, Cynthia Y. (2010). "Chapter 9". Precalculus. John Wiley and Sons. p. 831. ISBN 0-471-75684-9.
1 2 Lawrence, J. Dennis, A Catalog of Special Plane Curves, Dover Publ., 1972.
↑ Seligman, Courtney (1993–2010). "Orbital Motions Ellipses and Other Conic Sections". Online Astronomy eText.
↑ If the ellipse is illustrated as a meridional one for the earth, the tangential angle is equal to geodetic latitude, the angle $\theta$ is the geocentric latitude, and parametric angle t is a parametric (or reduced) latitude of auxiliary circle
↑ Ellipse at MathWorld, derived from formula (58) and (60)
↑ clarifies problems with MathWorld formula (60)
↑ Auxiliary circle and various ellipse formulas
↑ Meeus, J. (1991). "Ch. 10: The Earth's Globe". Astronomical Algorithms. Willmann-Bell. p. 78. ISBN 0-943396-35-2.
↑ David Drew. "Elliptical Gears".
↑ Grant, George B. (1906). A treatise on gear wheels. Philadelphia Gear Works. p. 72.
↑ http://www.rp-photonics.com/lamp_pumped_lasers.html
↑ http://www.cymer.com/plasma_chamber_detail/
↑ Chamberlain, G. (February 1983). "A characterization of the distributions that imply mean—Variance utility functions". Journal of Economic Theory 29 (1): 185–201. doi:10.1016/0022-0531(83)90129-1.
↑ Owen, J.; Rabinovitch, R. (June 1983). "On the class of elliptical distributions and their applications to the theory of portfolio choice". Journal of Finance 38: 745–752. doi:10.1111/j.1540-6261.1983.tb02499.x. JSTOR 2328079.
↑ Pitteway, M.L.V. (1967). "Algorithm for drawing ellipses or hyperbolae with a digital plotter". The Computer Journal 10 (3): 282–9. doi:10.1093/comjnl/10.3.282.
↑ Van Aken, J.R. (September 1984). "An Efficient Ellipse-Drawing Algorithm". IEEE Computer Graphics and Applications 4 (9): 24–35. doi:10.1109/MCG.1984.275994.
↑ Smith, L.B. (1971). "Drawing ellipses, hyperbolae or parabolae with a fixed number of points". The Computer Journal 14 (1): 81–86. doi:10.1093/comjnl/14.1.81.

References

Besant, W.H. (1907). "Chapter III. The Ellipse". Conic Sections. London: George Bell and Sons. p. 50.
Coxeter, H.S.M. (1969). Introduction to Geometry (2nd ed.). New York: Wiley. pp. 115–9.
Meserve, Bruce E. (1983) [1959], Fundamental Concepts of Geometry, Dover, ISBN 0-486-63415-9
Miller, Charles D.; Lial, Margaret L.; Schneider, David I. (1990). Fundamentals of College Algebra (3rd ed.). Scott Foresman/Little. p. 381. ISBN 0-673-38638-4.

External links

Wikiquote has quotations related to: Ellipse

Wikimedia Commons has media related to Ellipses.

Apollonius' Derivation of the Ellipse at Convergence
The Shape and History of The Ellipse in Washington, D.C. by Clark Kimberling
Ellipse circumference calculator
Collection of animated ellipse demonstrations
Weisstein, Eric W., "Ellipse", MathWorld.
Weisstein, Eric W., "Ellipse as special case of hypotrochoid", MathWorld.
Ivanov, A.B. (2001), "Ellipse", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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