Ellipse

An ellipse (red) obtained as the intersection of a cone with an inclined plane.
Ellipse: notations
Ellipses: examples with increasing eccentricity

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 , a number ranging from (the limiting case of a circle) to (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 and height is:

Assuming , 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 (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. 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 (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.

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

Definition as locus of points[edit]

Ellipse: definition by sum of distances to foci
Ellipse: definition by focus and circular directrix

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

Given two fixed points called the foci and a distance which is greater than the distance between the foci, the ellipse is the set of points such that the sum of the distances is equal to :

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 two vertices , 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 (see figure):

If is the circle with center and radius , then the distance of a point to the circle equals the distance to the focus :

is called the circular directrix (related to focus ) of the ellipse.[1][2] 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[edit]

Shape parameters:
  • a: semi-major axis,
  • b: semi-minor axis,
  • c: linear eccentricity,
  • p: semi-latus rectum (usually ).

Standard equation[edit]

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 ,
  • the vertices are .

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 (see diagram) produces the standard equation of the ellipse:[3]

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[edit]

Principal axes[edit]

Throughout this article, the semi-major and semi-minor axes are denoted and , respectively, i.e.

In principle, the canonical ellipse equation may have (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 and and the parameter names and

Linear eccentricity[edit]

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

Eccentricity[edit]

The eccentricity can be expressed as:

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

Semi-latus rectum[edit]

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:[4]

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

Tangent[edit]

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:

There are then cases:

  1. Then line and the ellipse have only point in common, and is a tangent. The tangent direction has perpendicular vector , so the tangent line has equation for some . Because is on the tangent and the ellipse, one obtains .
  2. Then line has a second point in common with the ellipse, and is a secant.

Using (1) 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 (see below). (If , the ellipse is a circle and "conjugate" means "orthogonal".)

Shifted ellipse[edit]

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[edit]

A general ellipse in the plane can be uniquely described as a bivariate quadratic equation of Cartesian coordinates, or using center, semi-major and semi-minor axes, and angle

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[5][6]

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.[7]: 63 

The general equation's coefficients can be obtained from known semi-major axis , semi-minor axis , center coordinates , and rotation angle (the angle from the positive horizontal axis to the ellipse's major axis) using the formulae:

These expressions can be derived from the canonical equation

by a Euclidean transformation of the coordinates :

Conversely, the canonical form parameters can be obtained from the general-form coefficients by the equations:[3]

where atan2 is the 2-argument arctangent function.

Parametric representation[edit]

The construction of points based on the parametric equation and the interpretation of parameter t, which is due to de la Hire
Ellipse points calculated by the rational representation with equal spaced parameters ().

Standard parametric representation[edit]

Using trigonometric functions, a parametric representation of the standard ellipse is:

The parameter t (called the eccentric anomaly in astronomy) is not the angle of with the x-axis, but has a geometric meaning due to Philippe de La Hire (see § Drawing ellipses below).[8]

Rational representation[edit]

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

Alternately, if the parameter is considered to be a point on the real projective line , then the corresponding rational parametrization is

Then

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

Tangent slope as parameter[edit]

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[edit]

Ellipse as an affine image of the unit circle

Another definition of an ellipse uses affine transformations:

Any ellipse is an affine image of the unit circle with equation .
Parametric representation

An affine transformation of the Euclidean plane has the form , where is a regular matrix (with non-zero determinant) 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:

(If , then .) 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

Area

From Apollonios theorem (see below) one obtains:
The area of an ellipse is

Semiaxes

With the abbreviations the statements of Apollonios's theorem can be written as:

Solving this nonlinear system for yields the semiaxes:

Implicit representation

Solving the parametric representation for by Cramer's rule and using , one obtains the implicit representation

Conversely: If the equation

with

of an ellipse centered at the origin is given, then the two vectors

point to two conjugate points and the tools developed above are applicable.

Example: For the ellipse with equation the vectors are

Whirls: nested, scaled and rotated ellipses. The spiral is not drawn: we see it as the locus of points where the ellipses are especially close to each other.
Rotated Standard ellipse

For one obtains a parametric representation of the standard ellipse rotated by angle :

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[edit]

Polar form relative to center[edit]

Polar coordinates centered at the 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[7]: 75 

where is the eccentricity, not Euler's number.

Polar form relative to focus[edit]

Polar coordinates centered at 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 (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 , 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[edit]

Ellipse: 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 (see diagram).

For an arbitrary point of the ellipse, the quotient of the distance to one focus and to the corresponding directrix (see diagram) is equal to the eccentricity:

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 (in a manner similar to the definition of a parabola):

For any point (focus), any line (directrix) not through , and any real number with the ellipse is the locus of points for which the quotient of the distances to the point and to the line is that is:

The extension to , 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 yields a parabola, and if , a hyperbola.)

Pencil of conics with a common vertex and common semi-latus rectum
Proof

Let , and assume is a point on the curve. The directrix has equation . With , the relation produces the equations

and

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 (see diagram).

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 .

Construction of a directrix
Construction of a directrix

Because of point of directrix (see diagram) and focus are inverse with respect to the circle inversion at circle (in diagram green). Hence can be constructed as shown in the diagram. Directrix is the perpendicular to the main axis at point .

General ellipse

If the focus is and the directrix , one obtains the equation

(The right side of the equation uses the Hesse normal form of a line to calculate the distance .)

Focus-to-focus reflection property[edit]

Ellipse: the tangent bisects the supplementary angle of the angle between the lines to the foci.
Rays from one focus reflect off the ellipse to pass through the other focus.

An ellipse possesses the following property:

The normal at a point bisects the angle between the lines .
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 be the point on the line with distance to the focus , where is the semi-major axis of the ellipse. Let line be the external angle bisector of the lines and Take any other point on By the triangle inequality and the angle bisector theorem, therefore must be outside the ellipse. As this is true for every choice of only intersects the ellipse at the single point 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).

Conjugate diameters[edit]

Definition of conjugate diameters[edit]

Orthogonal diameters of a circle with a square of tangents, midpoints of parallel chords and an affine image, which is an ellipse with conjugate diameters, a parallelogram of tangents and midpoints of chords.

A circle has the following property:

The midpoints of parallel chords lie on a diameter.

An affine transformation preserves parallelism and midpoints of line segments, so this property is true for any ellipse. (Note that the parallel chords and the diameter are no longer orthogonal.)

Definition

Two diameters of an ellipse are conjugate if the midpoints of chords parallel to lie on

From the diagram one finds:

Two diameters of an ellipse are conjugate whenever the tangents at and are parallel to .

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.

For the common parametric representation of the ellipse with equation one gets: The points

(signs: (+,+) or (−,−) )
(signs: (−,+) or (+,−) )
are conjugate and

In case of a circle the last equation collapses to

Theorem of Apollonios on conjugate diameters[edit]

Theorem of Apollonios
For the alternative area formula

For an ellipse with semi-axes the following is true:[9][10]

Let and be halves of two conjugate diameters (see diagram) then
  1. .
  2. The triangle with sides (see diagram) has the constant area , which can be expressed by , too. is the altitude of point and the angle between the half diameters. Hence the area of the ellipse (see section metric properties) can be written as .
  3. The parallelogram of tangents adjacent to the given conjugate diameters has the
Proof

Let the ellipse be in the canonical form with parametric equation

The two points are on conjugate diameters (see previous section). 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[edit]

Ellipse with its orthoptic

For the ellipse the intersection points of orthogonal tangents lie on the circle .

This circle is called orthoptic or director circle of the ellipse (not to be confused with the circular directrix defined above).

Drawing ellipses[edit]

Central projection of circles (gate)

Ellipses appear in descriptive geometry as images (parallel or central projection) 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 (ellipsographs) 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 (or equivalently: the foci and the semi-major axis). 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[edit]

The following construction of single points of an ellipse is due to de La Hire.[11] 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 (see diagram).
  4. Repeat steps (2) and (3) with different lines through the center.
Ellipse: gardener's method

Pins-and-string method[edit]

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 is tied at each end to the two pins; its length after tying is . 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 confocal ellipses with a closed string is due to the Irish bishop Charles Graves.

Paper strip methods[edit]

The two following methods rely on the parametric representation (see § Standard parametric representation, above):

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

a strip of paper of length .

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 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 (see animation). 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 (of the ellipse) 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 (see diagram). After this operation the movement of the unchanged half of the paperstrip is unchanged.[12] This variation requires only one sliding shoe.