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

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 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 pointsEdit

 
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 midpoint   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 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 coordinatesEdit

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

Standard equationEdit

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   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.

ParametersEdit

Principal axesEdit

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 eccentricityEdit

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

EccentricityEdit

The eccentricity can be expressed as:

 ,

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

Semi-latus rectumEdit

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).

TangentEdit

An arbitrary line   intersects an ellipse at  ,  , or   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:

  with  

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 ellipseEdit

If the standard ellipse is shifted to have center  , its equation is

 

The axes are still parallel to the x- and y-axes.

General ellipseEdit

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]:p.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 an affine transformation of the coordinates  :

 

Conversely, the canonical form parameters can be obtained from the general form coefficients by the equations:

 

Parametric representationEdit

 
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 representationEdit

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 representationEdit

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 (see Bezier curve).

Tangent slope as parameterEdit

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 ellipseEdit

 
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 formsEdit

Polar form relative to centerEdit

 
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]:p. 75

 

Polar form relative to focusEdit

 
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 propertyEdit

 
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 (with   being the radius of the circle) 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 propertyEdit

 
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.

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:   . The equality   is true from the Angle bisector theorem because   and   . 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 (see whispering gallery).

Conjugate diametersEdit

Definition of conjugate diametersEdit

 
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.

Theorem of Apollonios on conjugate diametersEdit

 
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 tangentsEdit

 
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 ellipsesEdit

 
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 constructionEdit

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 methodEdit

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 methodsEdit

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 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.

 
Ellipse construction: paper strip method 2
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 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 (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 of an ellipse with osculating circles

Approximation by osculating circlesEdit

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 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.

(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 generationEdit

 
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 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 hypotrochoidEdit

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

The ellipse is a special case of the hypotrochoid when  , 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 formEdit

CirclesEdit

 
Circle: inscribed angle theorem

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:

For four points   (see diagram) the following statement is true:
The four points are on a circle if and only if the angles at   and   are 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 circlesEdit

For four points   no 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 equationEdit

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

For example, for   the three-point equation is:

 , which can be rearranged to  

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.

 

EllipsesEdit

This section, we consider the family of ellipses defined by equations   with a fixed eccentricity  . It is convenient to use the parameter:

 

and to write the ellipse equation as:

 

where q is fixed and   vary 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.)

 
Inscribed angle theorem for an ellipse

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:[13][14]

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

Inscribed angle theorem for ellipsesEdit

Given four points  , no three of them on a line (see diagram).
The four points are on an ellipse with equation   if and only if the angles at   and   are 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 equationEdit

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

For example, for   and   one obtains the three-point form

  and after conversion  

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

 

where   is the modified dot product  

Pole-polar relationEdit

 
Ellipse: 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

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
  • line   onto 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:  ).
  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. Because they are even polar pairs with respect to the circle  , the directrices can be constructed by compass and straightedge (see Inversive geometry).

Pole-polar relations exist for hyperbolas and parabolas as well.

Metric propertiesEdit

All metric properties given below refer to an ellipse with equation

 

 

 

 

 

(1)

except for the section on the area enclosed by a tilted ellipse, where the generalized form of Eq.(1) will be given.

AreaEdit

The area   enclosed by an ellipse is:

 

 

 

 

 

(2)

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   (so its area is  ) and stretch it by a factor   to make an ellipse. This scales the area by the same factor:  [15] It is also easy to rigorously prove the area formula using integration as follows. Equation (1) 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   (obtained by solving for flattening, then computing the semi-minor axis).

 
The area enclosed by a tilted ellipse is  .

So far we have dealt with erect ellipses, whose major and minor axes are parallel to the   and   axes. However, some applications require tilted ellipses. In charged-particle beam optics, for instance, the enclosed area of an erect or tilted ellipse is an important property of the beam, its emittance. In this case a simple formula still applies, namely

 

 

 

 

 

(3)

where  ,   are intercepts and  ,   are maximum values. It follows directly from Appolonio's theorem.

CircumferenceEdit

 
Ellipses with same 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,

 

which is in general not an elementary function.

The circumference of the ellipse may be evaluated in terms of   using Gauss's arithmetic-geometric mean;[16] this is a quadratically converging iterative method.[17]

The exact infinite series is:

 

where   is the double factorial (extended to negative odd integers by the recurrence relation  , for  ). This series converges, but by expanding in terms of   James Ivory[18] and Bessel[19] 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  ";[20] they are

 

and

 

The errors in these approximations, which were obtained empirically, are of order   and   respectively.

Arc lengthEdit

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 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.[citation needed]

Some lower and upper bounds on the circumference of the canonical ellipse   with   are[21]

 

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.

CurvatureEdit

The curvature is given by   radius of curvature at point  :

 

Radius of curvature at the two vertices   and the centers of curvature: