# Ellipse

# Ellipse

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. 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 (the limiting case of a circle) to *e* = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola).

Analytically, the equation of a standard ellipse centered at the origin with width 2*a* and height 2*b* is:

`Assuming`

*a*≥*b*, the foci are (±*c*, 0) 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 cylinder is also an ellipse.

`An ellipse may also be defined in terms of one focus point and a line outside the ellipse called thedirectrix: for all points on the ellipse, the ratio between the distance to thefocusand 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

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 midpointof 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 distanceto the center. The distanceof the foci to the center is called the*focal distance*or linear eccentricity. The quotientis the*eccentricity*.`The caseyields a circle and is included as a special type of ellipse.`

`The equationcan be viewed in a different way (see figure):Ifis the circle with midpointand radius, then the distance of a pointto the circleequals 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 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:

- the foci are the points,the vertices are.

`For an arbitrary pointthe distance to the focusisand to the other focus. Hence the pointis on the ellipse whenever:`

`Removing theradicalsby suitable squarings and usingproduces the standard equation of the ellipse:`

or, solved for *y:*

`The width and height parametersare called thesemi-major and semi-minor axes. The top and bottom pointsare the`

*co-vertices*. The distances from a pointon the ellipse to the left and right foci areand.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 *a* ≥ *b*

*a*≥

*b*

`Throughout this articleis the semi-major axis, i.e.In general the canonical ellipse equationmay have(and hence the ellipse would be taller than it is wide); in this form the semi-major axis would be. This form can be converted to the standard form by transposing the variable namesandand the parameter namesand`

Linear eccentricity *c*

*c*

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

Eccentricity *e*

*e*

The eccentricity can be expressed as:

,

`assumingAn ellipse with equal axes () has zero eccentricity, and is a circle.`

Semi-latus rectum *l*

*l*

`The length of the chord through one of the foci, 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 rectumis equal to the *radius of curvature*of theosculating circlesat the vertices.`

Tangent

`An arbitrary lineintersects 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 pointof the ellipsehas the coordinate equation:A vectorparametric equationof the tangent is:

with

**Proof:**Letbe a point on an ellipse andbe the equation of any linecontaining. Inserting the line's equation into the ellipse equation and respectingyields:Case (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 .

Case (2): Then line has a second point in common with the ellipse, and is a secant.

`Using (1) one finds thatis a tangent vector at point, which proves the vector equation.`

`Ifandare two points of the ellipse such that, then the points lie on two`

*conjugate diameters*(seebelow). (If, the ellipse is a circle and "conjugate" means "orthogonal".)Shifted ellipse

The construction of points based on the parametric equation and the interpretation of parameter *t*, which is due to de la Hire

`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

`Inanalytic geometry, the ellipse is defined as a quadric: the set of pointsof theCartesian planethat, in non-degenerate cases, satisfy theimplicitequation`

^{[3]}^{[4]}`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.^{[5]} ^{[]}

`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 equationby 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

`Usingtrigonometric functions, a parametric representation of the standard ellipseis:`

`The parameter`

*t*(called the *eccentric anomaly- in astronomy) is not the angle of

*x*-axis, but has a geometric meaning due toPhilippe de La Hire(see *Drawing ellipses- below).
^{[6]}

Rational representation

`With the substitutionand trigonometric formulae one obtains`

and the *rational* parametric equation of an ellipse

`which covers any point of the ellipseexcept the left vertex.`

`Forthis formula represents the right upper quarter of the ellipse moving counter-clockwise with increasingThe left vertex is the limitRational representations of conic sections are commonly used inComputer Aided Design(seeBezier curve).`

Tangent slope as parameter

`A parametric representation, which uses the slopeof 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:

`Replacingandof the standard representation yields:`

`Hereis the slope of the tangent at the corresponding ellipse point,is the upper andthe lower half of the ellipse. The vertices, having vertical tangents, are not covered by the representation.The equation of the tangent at pointhas the form. The still unknowncan 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

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 .

`An affine transformation of the Euclidean plane has the form, whereis a regularmatrix(with non-zerodeterminant) andis an arbitrary vector. Ifare the column vectors of the matrix, the unit circle,, is mapped onto the ellipse:`

`Hereis the center andare the directions of twoconjugate diameters, in general not perpendicular. The four vertices of the ellipse are, for a parameterdefined by:`

`(If, then.) This is derived as follows. The tangent vector at pointis:`

`At a vertex parameter, the tangent is perpendicular to the major/minor axes, so:`

`Expanding and applying the identitiesgives the equation for.`

`This definition gives a parametric representation of an arbitrary ellipse, even in space, if we allowto be vectors in space.`

Polar forms

Polar form relative to center

Polar coordinates centered at the center

`Inpolar coordinates, with the origin at the center of the ellipse and with the angular coordinatemeasured from the major axis, the ellipse's equation is`

^{[5]}:p. 75Polar form relative to focus

Polar coordinates centered at focus

`If instead we use polar coordinates with the origin at one focus, with the angular coordinatestill measured from the major axis, the ellipse's equation is`

`where the sign in the denominator is negative if the reference directionpoints 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 anglein these formulas is called thetrue anomalyof the point. The numerator of these formulas is thesemi-latus rectum.`

Eccentricity and the directrix property

Ellipse: directrix property

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

`Each of the two lines parallel to the minor axis, and at a distance offrom 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 pairfollows from the fact thatandsatisfy 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 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.`

`(The choiceyields aparabola, and if, ahyperbola.)`

- Proof

`Let, and assumeis a point on the curve. The directrixhas equation. With, the relationproduces the equations`

- and

`The substitutionyields`

`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 parametersso 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 isand the directrix, one obtains the equation`

`(The right side of the equation uses theHesse normal formof a line to calculate the distance.)`

Focus-to-focus reflection property

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 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 (see diagram), too.

`Letbe the point on the linewith the distanceto the focus,is the semi-major axis of the ellipse. Let linebe the bisector of the supplementary angle to the angle between the lines. In order to prove thatis the tangent line at point, one checks that any pointon linewhich is different fromcannot be on the ellipse. Hencehas only pointin common with the ellipse and is, therefore, the tangent at point.From the diagram and thetriangle inequalityone recognizes thatholds, which means:. But ifis 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 (see whispering gallery).

Conjugate diameters

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 diametersof an ellipse are`

*conjugate*if the midpoints of chords parallel tolie onFrom the diagram one finds:

- Two diametersof an ellipse are conjugate whenever the tangents atandare 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 pointsbelong to a diameter, and the pairbelong to its conjugate diameter.`

Theorem of Apollonios on conjugate diameters

Ellipse: theorem of Apollonios on conjugate diameters

`For an ellipse with semi-axesthe following is true:`

Let and be halves of two conjugate diameters (see diagram) then

,

the

*triangle*formed by has the constant areathe 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 pointsare on conjugate diameters (see previous section). From trigonometric formulae one obtainsand`

`The area of the triangle generated byis`

`and from the diagram it can be seen that the area of the parallelogram is 8 times that of. Hence`

Orthogonal tangents

Ellipse with its orthoptic

`For the ellipsethe 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

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

the knowledge of the axes and the semi-axes is necessary (or equivalent: 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

Ellipse: gardener's method

`The following construction of single points of an ellipse is due tode La Hire.`

^{[7]}It is based on thestandard parametric representationof an ellipse:- (1) Draw the two

*circles*centered at the center of the ellipse with radiiand the axes of the ellipse.(2) Draw a

*line through the center*, which intersects the two circles at pointand, respectively.(3) Draw a

*line*throughthat is parallel to the minor axis and a

*line*throughthat 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.

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 confocal ellipses with a *closed* string is due to the Irish bishop Charles Graves.

Paper strip methods

Ellipse construction: paper strip method 2

Approximation of an ellipse with osculating circles

The two following methods rely on the parametric representation (see section *parametric representation*, above):

`This representation can be modeled technically by two simple methods. In both cases center, the axes and semi axeshave 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 P traces the ellipse. For the proof one shows that pointhas the parametric representation, where parameteris the angle of the slope of the paper strip.`

`A technical realization of the motion of the paper strip can be achieved by aTusi 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 midpointof the paper strip is moving on the circle with center(of the ellipse) and radius. Hence, the paperstrip can be cut at pointinto halves, connected again by a joint atand the sliding endfixed at the center(see diagram). After this operation the movement of the unchanged half of the paperstrip is unchanged.`

^{[8]}This variation requires only one sliding shoe.- Method 2

The second method starts with

a strip of paper of length .

`One marks the point, which divides the strip into two substrips of lengthand. 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 parameteris the angle of slope of the paper strip.`

This method is the base for several *ellipsographs* (see section below).

Similar to the variation of the paper strip method 1 a *variation of the paper strip method 2* can be established (see diagram) 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 radius of curvature at the vertices is:

The radius of curvature at the co-vertices is:

`The diagram shows an easy way to find the centers of curvatureat vertexand co-vertex, respectively:`

- (1) mark the auxiliary pointand 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.

(proof: simple calculation.)

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

Ellipse: Steiner generation

Ellipse: Steiner generation

The following method to construct single points of an ellipse relies on the Steiner generation of a conic section:

Given two pencils of lines at two points (all lines containing and , respectively) and a projective but not perspective mapping of onto , then the intersection points of corresponding lines form a non-degenerate projective conic section.

`For the generation of points of the ellipseone uses the pencils at the vertices. Letbe an upper co-vertex of the ellipse and.is the center of the rectangle. The sideof the rectangle is divided into n equal spaced line segments and this division is projected parallel with the diagonalas direction onto the line segmentand 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 atandneeded. The intersection points of any two related linesandare points of the uniquely defined ellipse. With help of the pointsthe 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

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

`The ellipse is a special case of thehypotrochoidwhen`

*R*= 2*r*, as shown in the adjacent image. The special case of a moving circle with radiusinside a circle with radiusis called aTusi couple.Inscribed angles and three-point form

Circles

Circle: inscribed angle theorem

`A circle with equationis uniquely determined by three pointsnot on a line. A simple way to determine the parametersuses the *inscribed angle theorem`

- for circles:

- For four points(see diagram) the following statement is true:The four points are on a circle if and only if the angles atandare equal.

Usually one measures inscribed angles by a degree or radian *θ,* but here the following measurement is more convenient:

In order to measure the angle between two lines with equations one uses the quotient:

Inscribed angle theorem for circles

`For four pointsno three of them on a line, we have the following (see diagram):`

The four points are on a circle, if and only if the angles at and are equal. In terms of the angle measurement above, this means:

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

- As a consequence, one obtains an equation for the circle determined by three non-colinear points:

`For example, forthe three-point equation is:`

- , which can be rearranged to

`Using vectors,dot productsanddeterminantsthis formula can be arranged more clearly, letting:`

`The center of the circlesatisfies:`

The radius is the distance between any of the three points and the center.

Ellipses

Inscribed angle theorem for an ellipse

`This section, we consider the family of ellipses defined by equationswith a`

*fixed*eccentricity*e*. It is convenient to use the parameter:and to write the ellipse equation as:

`where`

*q*is fixed andvary over the real numbers. (Such ellipses have their axes parallel to the coordinate axes: if, the major axis is parallel to the*x*-axis; if, it is parallel to the*y*-axis.)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*:^{[9]}^{[10]}

In order to measure an angle between two lines with equations one uses the quotient:

Inscribed angle theorem for ellipses

- Given four points, no three of them on a line (see diagram).The four points are on an ellipse with equationif and only if the angles atandare equal in the sense of the measurement above—that is, if

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

- A consequence, one obtains an equation for the ellipse determined by three non-colinear points:

`For example, forandone obtains the three-point form`

- and after conversion

Analogously to the circle case, the equation can be written more clearly using vectors:

`whereis the modifieddot product`

Pole-polar relation

Ellipse: pole-polar relation

`Any ellipse can be described in a suitable coordinate system by an equation. The equation of the tangent at a pointof the ellipse isIf one allows pointto be an arbitrary point different from the origin, then`

point is mapped onto the line , not through the center of the ellipse.

This relation between points and lines is a bijection.

The inverse function maps

line onto the point and

- lineonto the point

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:

For a point (pole)

*on*the ellipse the polar is the tangent at this point (see diagram: ).For a pole

*outside*the ellipse the intersection points of its polar with the ellipse are the tangency points of the two tangents passing (see diagram: ).For a point

*within*the ellipse the polar has no point with the ellipse in common. (see diagram: ).

The intersection point of two polars is the pole of the line through their poles.

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

`Theareaenclosed by an ellipse is:`

`whereandare the lengths of the semi-major and semi-minor axes, respectively. The area formulais intuitive: start with a circle of radius(so its area is) and stretch it by a factorto make an ellipse. This scales the area by the same factor:It is also easy to rigorously prove the area formula usingintegrationas follows. Equation (**1**) can be rewritten asForthis curve is the top half of the ellipse. So twice the integral ofover the intervalwill be the area of the ellipse:`

`The second integral is the area of a circle of radiusthat is,So`

`An ellipse defined implicitly byhas area`

`The area can also be expressed in terms of eccentricity and the length of the semi-major axis as(obtained by solving for flattening, then computing the semi-minor axis).`

Circumference

`Thecircumferenceof an ellipse is:`

`where againis the length of the semi-major axis,is the eccentricity, and the functionis thecomplete elliptic integral of the second kind,`

`The circumference of the ellipse may be evaluated in terms ofusingGauss's arithmetic-geometric mean;`

^{[11]}this is a quadratically converging iterative method.^{[12]}The exact infinite series is:

`whereis thedouble factorial. This series converges, but by expanding in terms ofJames Ivory`

^{[13]}and Bessel^{[14]}derived an expression that converges much more rapidly:`Srinivasa Ramanujangives two closeapproximationsfor the circumference in §16 of "Modular Equations and Approximations to";`

^{[15]}they areand

`The errors in these approximations, which were obtained empirically, are of orderandrespectively.`

More generally, the arc length of a portion of the circumference, as a function of the angle subtended (or *x*-coordinates of any two points on the upper half of the ellipse), is given by an incomplete elliptic integral. The upper half of an ellipse is parameterized by

`Then the arc lengthfromtois:`

This is equivalent to

`whereis 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 ellipsewithare`

^{[16]}`Here the upper boundis the circumference of acircumscribedconcentric circlepassing through the endpoints of the ellipse's major axis, and the lower boundis theperimeterof aninscribedrhombuswithverticesat the endpoints of the major and the minor axes.`

Curvature

`Thecurvatureis given byradius of curvatureat 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

Steiner ellipse: ellipse through the vertices of the triangle with center at the centroid,

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:

Ellipsoid

Elliptic cone

Elliptic cylinder

Hyperboloid of one sheet

Hyperboloid of two sheets

Applications

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. (In the special case of a circle with a source at its center all light would be reflected back to the center.) If the ellipse is rotated along its major axis to produce an ellipsoidal mirror (specifically, a prolate spheroid), 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 (where John Quincy Adams is said to have used this property for eavesdropping on political matters); 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 [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 (that is, 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. 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.)

`Forelliptical orbits, useful relations involving the eccentricityare:`

where

is the radius at apoapsis (the farthest distance)

is the radius at periapsis (the closest distance)

is the length of the semi-major axis

`Also, in terms ofand, the semi-major axisis theirarithmetic mean, the semi-minor axisis theirgeometric mean, and thesemi-latus rectumis theirharmonic 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.^{[17]}

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.^{[18]}

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).

^{[19]}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.

^{[20]}

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.^{[21]}^{[22]}

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.^{[23]} Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.^{[24]}

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.^{[25]} 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.

See also

Apollonius of Perga, the classical authority

Cartesian oval, a generalization of the ellipse

Circumconic and inconic

Ellipse fitting

Ellipsoid, a higher dimensional analog of an ellipse

Elliptic coordinates, an orthogonal coordinate system based on families of ellipses and hyperbolae

Elliptic partial differential equation

Elliptical distribution, in statistics

Geodesics on an ellipsoid

Great ellipse

Hyperbola

Kepler's laws of planetary motion

Matrix representation of conic sections

*n*-ellipse, a generalization of the ellipse for*n*fociOval

Rytz’s construction, a method for finding the ellipse axes from conjugate diameters or a parallelogram

Spheroid, the ellipsoid obtained by rotating an ellipse about its major or minor axis

Stadium (geometry), a two-dimensional geometric shape constructed of a rectangle with semicircles at a pair of opposite sides

Steiner circumellipse, the unique ellipse circumscribing a triangle and sharing its centroid

Steiner inellipse, the unique ellipse inscribed in a triangle with tangencies at the sides' midpoints

Superellipse, a generalization of an ellipse that can look more rectangular or more "pointy"

True, eccentric, and mean anomaly

## References

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*Leitkreis*which can be translated as "Director circle", but that term has a different meaning in the English literature (see Director circle).

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*Vorlesungen über Darstellende Geometrie.*Vandenhoeck & Ruprecht, Göttingen 1967, S. 26.

*Seventeenth century instruments for drawing conic sections.*In:

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*Vorlesungen über Geomerie der Algebren*, Springer (1973)

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