Ellipse


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. As such, 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 to e = 1.
Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is:
Assuming a ≥ b, the foci are for. The standard parametric equation is:
Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane. 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:
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. 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, ἔλλειψις, 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:
The midpoint 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 the vertex points, which have distance to the center. The distance of the foci to the center is called the focal distance or linear eccentricity. The quotient is the eccentricity.
The case yields a circle and is included as a special type of ellipse.
The equation can be viewed in a different way :
is called the circular directrix 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 plane 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:
For an arbitrary point the distance to the focus is
and to the other focus. Hence the point is on the ellipse whenever:
Removing the radicals by suitable squarings and using produces the standard equation of the ellipse:
or, solved for y:
The width and height parameters are called the semi-major and semi-minor axes. The top and bottom points are the co-vertices. The distances from a point on the ellipse to the left and right foci are and.
It follows from the equation that the ellipse is symmetric with respect to the coordinate axes and hence with respect to the origin.

Parameters

Semi-major and semi-minor axes

Throughout this article is the semi-major axis, i.e. In general the canonical ellipse equation may have ; in this form the semi-major axis would be. This form can be converted to the standard form by transposing the variable names and and the parameter names and

Linear eccentricity

This is the distance from the center to a focus: .

Eccentricity

The eccentricity can be expressed as:
assuming An ellipse with equal axes 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. A calculation shows:
The semi-latus rectum is equal to the radius of curvature at the vertices.

Tangent

An arbitrary line 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 of the ellipse has the coordinate equation:
A vector parametric equation of the tangent is:
Proof:
Let be a point on an ellipse and be the equation of any line containing. Inserting the line's equation into the ellipse equation and respecting yields:
Using one finds that is a tangent vector at point, which proves the vector equation.
If and are two points of the ellipse such that, then the points lie on two conjugate diameters.

Shifted ellipse

If the standard ellipse is shifted to have center, its equation is
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 of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation
provided
To distinguish the degenerate cases from the non-degenerate case, let ∆ be the determinant
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, semi-minor axis, center coordinates, and rotation angle using the formulae:
These expressions can be derived from the canonical equation by an affine transformation of the coordinates :
Conversely, the canonical form parameters can be obtained from the general form coefficients by the equations:

Parametric representation

Standard [|parametric representation]

Using trigonometric functions, a parametric representation of the standard ellipse is:
The parameter t is not the angle of with the x-axis, but has a geometric meaning due to Philippe de La Hire.

Rational representation

With the substitution and trigonometric formulae one obtains
and the rational parametric equation of an ellipse
which covers any point of the ellipse except the left vertex.
For this formula represents the right upper quarter of the ellipse moving counter-clockwise with increasing The left vertex is the limit
Rational representations of conic sections are commonly used in Computer Aided Design.

Tangent slope as parameter

A parametric representation, which uses the slope of the tangent at a point of the ellipse
can be obtained from the derivative of the standard representation :
With help of trigonometric formulae
one obtains:
Replacing and of the standard representation yields:
Here is the slope of the tangent at the corresponding ellipse point, is the upper and the lower half of the ellipse. The vertices, having vertical tangents, are not covered by the representation.
The equation of the tangent at point has the form. The still unknown can be determined by inserting the coordinates of the corresponding ellipse point :
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:
;parametric representation
An affine transformation of the Euclidean plane has the form, where is a regular matrix and is an arbitrary vector. If are the column vectors of the matrix, the unit circle,, is mapped onto the ellipse:
Here is the center and are the directions of two conjugate diameters, in general not perpendicular.
;vertices
The four vertices of the ellipse are, for a parameter defined by:
This is derived as follows. The tangent vector at point is:
At a vertex parameter, the tangent is perpendicular to the major/minor axes, so:
Expanding and applying the identities gives the equation for.
;implicit representation
Solving the parametric representation for by Cramer's rule and using, one gets the implicit representation
;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 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 measured from the major axis, the ellipse's equation is

Polar form relative to focus

If instead we use polar coordinates with the origin at one focus, with the angular coordinate still measured from the major axis, the ellipse's equation is
where the sign in the denominator is negative if the reference direction points towards the center, 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, the polar form is
The angle in these formulas is called the true anomaly of the point. The numerator of these formulas is the semi-latus rectum.

Eccentricity and the directrix property

Each of the two lines parallel to the minor axis, and at a distance of from it, is called a directrix of the ellipse.
The proof for the pair follows from the fact that and satisfy the equation
The second case is proven analogously.
The converse is also true and can be used to define an ellipse :
The choice, which is the eccentricity of a circle, is not allowed in this context. One may consider the directrix of a circle to be the line at infinity.
;Proof
Let, and assume is a point on the curve.
The directrix has equation. With, the relation produces the equations
The substitution yields
This is the equation of an ellipse, or a parabola, or a hyperbola. All of these non-degenerate conics have, in common, the origin as a vertex.
If, introduce new parameters so that, and then the equation above becomes
which is the equation of an ellipse with center, the x-axis as major axis, and
the major/minor semi axis.
;General ellipse
If the focus is and the directrix, one obtains the equation

Focus-to-focus reflection property

An ellipse possesses the following property:
; Proof
Because the tangent is perpendicular to the normal, the statement is true for the tangent and the supplementary angle of the angle between the lines to the foci, too.
Let be the point on the line with the distance to the focus, is the semi-major axis of the ellipse. Let line be the bisector of the supplementary angle to the angle between the lines. In order to prove that is the tangent line at point, one checks that any point on line which is different from cannot be on the ellipse. Hence has only point in common with the ellipse and is, therefore, the tangent at point.
From the diagram and the triangle inequality one recognizes that holds, which means:. But if is a point of the ellipse, the sum should be.
; 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.

Conjugate diameters

A circle has the following property:
An affine transformation preserves parallelism and midpoints of line segments, so this property is true for any ellipse.
; Definition:
Two diameters of an ellipse are conjugate if the midpoints of chords parallel to lie on
From the diagram one finds:
Conjugate diameters in an ellipse generalize orthogonal diameters in a circle.
In the parametric equation for a general ellipse given above,
any pair of points belong to a diameter, and the pair belong to its conjugate diameter.

Theorem of Apollonios on conjugate diameters

For an ellipse with semi-axes the following is true:
; Proof:
Let the ellipse be in the canonical form with parametric equation
The two points are on conjugate diameters. From trigonometric formulae one obtains and
The area of the triangle generated by is
and from the diagram it can be seen that the area of the parallelogram is 8 times that of. Hence

Orthogonal tangents

For the ellipse the intersection points of orthogonal tangents lie on the circle .
This circle is called orthoptic or director circle of the ellipse.

Drawing ellipses

Ellipses appear in descriptive geometry as images of circles. There exist various tools to draw an ellipse. Computers provide the fastest and most accurate method for drawing an ellipse. However, technical tools to draw an ellipse without a computer exist. The principle of ellipsographs were known to Greek mathematicians such as Archimedes and Proklos.
If there is no ellipsograph available, one can draw an ellipse using an approximation by the four osculating circles at the vertices.
For any method described below, knowledge of the axes and the semi-axes is necessary.
If this presumption is not fulfilled one has to know at least two conjugate diameters. With help of Rytz's construction the axes and semi-axes can be retrieved.

de La Hire's point construction

The following construction of single points of an ellipse is due to de La Hire. It is based on the [|standard parametric representation] of an ellipse:
  1. Draw the two circles centered at the center of the ellipse with radii and the axes of the ellipse.
  2. Draw a line through the center, which intersects the two circles at point and, respectively.
  3. Draw a line through that is parallel to the minor axis and a line through that is parallel to the major axis. These lines meet at an ellipse point.
  4. Repeat steps and with different lines through the center.

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. 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 pencil pulls the loop taut to form a triangle. The tip of the pencil then traces an ellipse if it is moved while keeping the string taut. Using two pegs and a rope, gardeners use this procedure to outline an elliptical flower bed—thus it is called the gardener's ellipse.
A similar method for drawing with a closed string is due to the Irish bishop Charles Graves.

Paper strip methods

The two following methods rely on the parametric representation :
This representation can be modeled technically by two simple methods. In both cases center, the axes and semi axes have to be known.
;Method 1
The first method starts with
The point, where the semi axes meet is marked by. If the strip slides with both ends on the axes of the desired ellipse, then point P traces the ellipse. For the proof one shows that point has the parametric representation, where parameter is the angle of the slope of the paper strip.
A technical realization of the motion of the paper strip can be achieved by a Tusi couple. The device is able to draw any ellipse with a fixed sum, which is the radius of the large circle. This restriction may be a disadvantage in real life. More flexible is the second paper strip method.
A variation of the paper strip method 1 uses the observation that the midpoint of the paper strip is moving on the circle with center and radius. Hence, the paperstrip can be cut at point into halves, connected again by a joint at and the sliding end fixed at the center . After this operation the movement of the unchanged half of the paperstrip is unchanged. This variation requires only one sliding shoe.
; Method 2:
The second method starts with
One marks the point, which divides the strip into two substrips of length and. The strip is positioned onto the axes as described in the diagram. Then the free end of the strip traces an ellipse, while the strip is moved. For the proof, one recognizes that the tracing point can be described parametrically by, where parameter is the angle of slope of the paper strip.
This method is the base for several ellipsographs.
Similar to the variation of the paper strip method 1 a variation of the paper strip method 2 can be established by cutting the part between the axes into halves.
Most ellipsograph drafting instruments are based on the second paperstrip method.

Approximation by osculating circles

From Metric properties below, one obtains:
The diagram shows an easy way to find the centers of curvature at vertex and co-vertex, respectively:
  1. mark the auxiliary point and draw the line segment
  2. draw the line through, which is perpendicular to the line
  3. the intersection points of this line with the axes are the centers of the osculating circles.
The centers for the remaining vertices are found by symmetry.
With help of a French curve one draws a curve, which has smooth contact to the osculating circles.

Steiner generation

The following method to construct single points of an ellipse relies on the Steiner generation of a conic section:
For the generation of points of the ellipse one uses the pencils at the vertices. Let be an upper co-vertex of the ellipse and.
is the center of the rectangle. The side of the rectangle is divided into n equal spaced line segments and this division is projected parallel with the diagonal as direction onto the line segment and assign the division as shown in the diagram. The parallel projection together with the reverse of the orientation is part of the projective mapping between the pencils at and needed. The intersection points of any two related lines and are points of the uniquely defined ellipse. With help of the points the points of the second quarter of the ellipse can be determined. Analogously one obtains the points of the lower half of the ellipse.
Steiner generation can also be defined for hyperbolas and parabolas. It is sometimes called a parallelogram method because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.

As hypotrochoid

The ellipse is a special case of the hypotrochoid when R = 2r, as shown in the adjacent image. The special case of a moving circle with radius inside a circle with radius is called a Tusi couple.

Inscribed angles and three-point form

Circles

A circle with equation is uniquely determined by three points not on a line. A simple way to determine the parameters uses the inscribed angle theorem for circles:
Usually one measures inscribed angles by a degree or radian θ, but here the following measurement is more convenient:

Inscribed angle theorem for circles

For four points no three of them on a line, we have the following :
At first the measure is available only for chords not parallel to the y-axis, but the final formula works for any chord.

Three-point form of circle equation

For example, for the three-point equation is:
Using vectors, dot products and determinants this formula can be arranged more clearly, letting :
The center of the circle satisfies:
The radius is the distance between any of the three points and the center.

Ellipses

This section, we consider the family of ellipses defined by equations with a fixed eccentricity e. It is convenient to use the parameter:
and to write the ellipse equation as:
where q is fixed and vary over the real numbers.
Like a circle, such an ellipse is determined by three points not on a line.
For this family of ellipses, one introduces the following q-analog angle measure, which is not a function of the usual angle measure θ:

Inscribed angle theorem for ellipses

At first the measure is available only for chords which are not parallel to the y-axis. But the final formula works for any chord. The proof follows from a straightforward calculation. For the direction of proof given that the points are on an ellipse, one can assume that the center of the ellipse is the origin.

Three-point form of ellipse equation

For example, for and one obtains the three-point form
Analogously to the circle case, the equation can be written more clearly using vectors:
where is the modified dot product

Pole-polar relation

Any ellipse can be described in a suitable coordinate system by an equation. The equation of the tangent at a point of the ellipse is If one allows point to be an arbitrary point different from the origin, then
This relation between points and lines is a bijection.
The inverse function maps
Such a relation between points and lines generated by a conic is called pole-polar relation or polarity. The pole is the point, the polar the line.
By calculation one can confirm the following properties of the pole-polar relation of the ellipse:
  1. The intersection point of two polars is the pole of the line through their poles.
  2. The foci and respectively and the directrices and respectively belong to pairs of pole and polar.
Pole-polar relations exist for hyperbolas and parabolas, too.

Metric properties

All metric properties given below refer to an ellipse with equation.

Area

The area enclosed by an ellipse is:
where and are the lengths of the semi-major and semi-minor axes, respectively. The area formula is intuitive: start with a circle of radius and stretch it by a factor to make an ellipse. This scales the area by the same factor: It is also easy to rigorously prove the area formula using integration as follows. Equation can be rewritten as For this curve is the top half of the ellipse. So twice the integral of over the interval will be the area of the ellipse:
The second integral is the area of a circle of radius that is, So
An ellipse defined implicitly by has area
The area can also be expressed in terms of eccentricity and the length of the semi-major axis as .

Circumference

The circumference of an ellipse is:
where again is the length of the semi-major axis, is the eccentricity, and the function is the complete elliptic integral of the second kind,
The circumference of the ellipse may be evaluated in terms of using Gauss's arithmetic-geometric mean; this is a quadratically converging iterative method.
The exact infinite series is:
where is the double factorial. This series converges, but by expanding in terms of James Ivory and Bessel derived an expression that converges much more rapidly:
Srinivasa Ramanujan gives two close approximations for the circumference in §16 of "Modular Equations and Approximations to "; they are
and
The errors in these approximations, which were obtained empirically, are of order and 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 upper half of an ellipse is parameterized by
Then the arc length from to is:
This is equivalent to
where is the incomplete elliptic integral of the second kind with parameter
The inverse function, the angle subtended as a function of the arc length, is given by a certain elliptic function.
Some lower and upper bounds on the circumference of the canonical ellipse with are
Here the upper bound is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and the minor axes.

Curvature

The curvature is given by
radius of curvature at point :
Radius of curvature at the two vertices and the centers of curvature:
Radius of curvature at the two co-vertices and the centers of curvature:

In triangle geometry

Ellipses appear in triangle geometry as
  1. Steiner ellipse: ellipse through the vertices of the triangle with center at the centroid,
  2. inellipses: ellipses which touch the sides of a triangle. Special cases are the Steiner inellipse and the Mandart inellipse.

    As plane sections of quadrics

Ellipses appear as plane sections of the following quadrics:

Physics

Elliptical reflectors and acoustics

If the water's surface is disturbed at one focus of an elliptical water tank, the circular waves of that disturbance, after reflecting off the walls, converge simultaneously to a single point: the second focus. This is a consequence of the total travel length being the same along any wall-bouncing path between the two foci.
Similarly, if a light source is placed at one focus of an elliptic mirror, all light rays on the plane of the ellipse are reflected to the second focus. Since no other smooth curve has such a property, it can be used as an alternative definition of an ellipse. If the ellipse is rotated along its major axis to produce an ellipsoidal mirror, this property holds for all rays out of the source. Alternatively, a cylindrical mirror with elliptical cross-section can be used to focus light from a linear fluorescent lamp along a line of the paper; such mirrors are used in some document scanners.
Sound waves are reflected in a similar way, so in a large elliptical room a person standing at one focus can hear a person standing at the other focus remarkably well. The effect is even more evident under a vaulted roof shaped as a section of a prolate spheroid. Such a room is called a whisper chamber. The same effect can be demonstrated with two reflectors shaped like the end caps of such a spheroid, placed facing each other at the proper distance. Examples are the National Statuary Hall at the United States Capitol ; the Mormon Tabernacle at Temple Square in Salt Lake City, Utah; at an exhibit on sound at the Museum of Science and Industry in Chicago; in front of the University of Illinois at Urbana–Champaign Foellinger Auditorium; and also at a side chamber of the Palace of Charles V, in the Alhambra.

Planetary orbits

In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun 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, 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. 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.
For elliptical orbits, useful relations involving the eccentricity are:
where
Also, in terms of and, the semi-major axis is their arithmetic mean, the semi-minor axis is their geometric mean, and the semi-latus rectum is their harmonic mean. In other words,

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 Lissajous figure 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.
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.

Optics

In statistics, a bivariate random vector 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.

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. Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.
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. 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.

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.