This article is about the geometric figure. For other uses, see Ellipse (disambiguation).
"Elliptical" redirects here. For the exercise machine, see Elliptical trainer.
Not to be confused with Ellipsis.
An ellipse (red) obtained as the intersection of a cone with an inclined plane
Ellipse: notations
Ellipses: examples

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

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

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

Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in our solar system is approximately an ellipse with the barycenter of the planet–Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids. Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection. It is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency. A similar effect leads to elliptical polarization of light in optics.

The name, ἔλλειψις (élleipsis, "omission"), was given by Apollonius of Perga in his Conics, emphasizing the connection of the curve with "application of areas".

Contents

Definition of an ellipse as locus of pointsEdit

 
Ellipse: Definition
 
Ellipse: definition with director circle

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

  • An ellipse is a set of points, such that for any point   of the set, the sum of the distances   to two fixed points  ,the foci, is constant, usually denoted by   In order to omit the special case of a line segment, one presumes  
 

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. It contains the 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.

The equation   can be viewed in a different way (see picture):
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 director circle (related to focus  ) of the ellipse. This property should not be confused with the definition of an ellipse with help of a directrix (line) below.

Using Dandelin spheres one proves easily the important statement:

  • Any plane section of a cone with a plane, which does not contain the apex and whose slope is less than the slope of the lines on the cone, is an ellipse.

Ellipse in Cartesian coordinatesEdit

 
shape parameters:
a: semi-major axis,
b: semi-minor axis
c: linear eccentricity,
p: semi latus rectum.

EquationEdit

If Cartesian coordinates are introduced such that the origin is the center of the ellipse and 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 second focus  . Hence the point   is on the ellipse if the following condition is fulfilled

 

Remove the square roots by suitable squarings and use the relation   to obtain the equation of the ellipse:

  •   or solved for y
 

The shape parameters   are called the semi major axis and semi minor axis. The points   are the co-vertices.

It follows from the equation that the ellipse is symmetric with respect to both of the coordinate axes and hence symmetric with respect to the origin.

Semi-latus rectumEdit

The length of the chord through one of the foci, which is perpendicular to the major axis of the ellipse is called the latus rectum. One half of it is the semi-latus rectum  . A calculation shows

  •  

The semi-latus rectum   may also be viewed as the radius of curvature of the osculating circles at the vertices  .

TangentEdit

The simplest way to determine the equation of the tangent at a point   is to implicitly differentiate the equation   of the ellipse. This produces

 

With respect to  , the equation of the tangent at point   is,

  •  

Or as vector equation:

  •  

Equation of a shifted ellipseEdit

If the ellipse is shifted such that its center is   the equation is

  •  

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

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: animation of the de la Hire method

Using the sine and cosine functions  , a parametric representation of the ellipse   can be obtained, :

  •  

(It is  . )

Parameter t can be taken as shown in the diagram and is due to de la Hire.[1]

The parameter t (called the eccentric anomaly in astronomy) is not the angle of   with the x-axis (see diagram at right). For other interpretations of parameter t see section Drawing ellipses.

With the substitution   and trigonometric formulae one gets

 

and the rational parametric equation of an ellipse

  •  

For   this formula represents the quarter ellipse centered at the origin with radii   and   moving counter-clockwise with increasing   It is easy to test this by computing   and  

A shifted ellipse with center   can be described by

  •  

A parametric representation of an arbitrary ellipse is contained in the section Ellipse as an affine image of the unit circle x²+y²=1 below.

Remarks on the parameters a and bEdit

The parameters   and   represent the length of line segments and are therefore non-negative real numbers. Throughout this article   is the semi-major axis, i.e   In real life one may obtain an equation   with no further information on the parameters  . For any choice of   one gets an ellipse with   the semi-major axis, if   and   the semi-major axis, if  .

Definition of an ellipse by the directrix propertyEdit

 
Ellipse: directrix property

The two lines at distance   and parallel to the minor axis are called directrices 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 inverse statement is also true and can be used to define a 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 set of points (locus of points), for which the quotient of the distances to the point and to the line is  
 
is an ellipse.

The choice  , which is the eccentricity of a circle, is in this context not allowed. One may consider the directrix of a circle to be the line at infinity.

(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  .

General case

If the focus is   and the directrix   one gets the equation

 

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

The Normal bisects the angle between the lines to the fociEdit

 
Ellipse: the tangent bisects the angle between the lines to the foci
 
rays from one focus pass through the other focus

For an ellipse the following statement is true:

  • 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 complementary angle of 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. Line   is the bisector of the angle between the lines  . In order to prove that   is the tangent line at point  , one checks that any point   on line   which is different from   cannot be on the ellipse. Hence   has only point   in common with the ellipse and is, therefore, the tangent at point  .
From the diagram and the triangle inequality one recognizes that   holds, which means:  . But if   is a point of the ellipse, the sum should be  .

application

The rays from one focus are reflected by the ellipse to the second focus. This property has optical and acoustic applications similar to the reflective property of a parabola (see whispering gallery).

Ellipse as an affine image of the unit circle x²+y²=1Edit

 
Ellipse as an affine image of the unit circle

Another definition of an Ellipse uses affine transformations:

  • Any Ellipse is the affine image of the unit circle with equation  .

An affine transformation of the Euclidean plane has the form  , where   is a regular matrix (its determinant is not 0) and   is an arbitrary vector. If   are the column vectors of the matrix  , the unit circle   is mapped onto the Ellipse

  •  

  is the center,   are the directions of two conjugate diameters of the ellipse. In general the vectors   are not perpendicular. That means, in general  and  are not the vertices of the ellipse.

The tangent vector at point   is

 

Because at a vertex the tangent is perpendicular to the major/minor axis (diameters) of the ellipse one gets the parameter   of a vertex from the equation

 

and hence

  •  .

(The formulae   were used.)

If  , then  .

The 4 vertices of the ellipse are  

The advantage of this definition is that one gets a simple parametric representation of an arbitrary ellipse, even in the space, if the vectors   are vectors of the Euclidean space.

Conjugate diameters and the midpoints of parallel chordsEdit

 
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

For a circle, obviously

(M) the midpoints of parallel chords lie on a diameter.

The diameter and the parallel chords are orthogonal. An affine transformation in general does not preserve orthogonality but parallelism and midpoints of line segments. Hence: property (M) (which omits the term orthogonal) is true for any ellipse.

Definition

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

From the diagram one finds:

(T) Two diameters  , of an ellipse are conjugate, if the tangents at   and   are parallel to   and visa versa.

The term conjugate diameters is a kind of generalization of orthogonal.

Considering the parameter equation

 

of an ellipse, any pair   of points belong to a diameter and the pair   belongs to its conjugate diameter.

Orthogonal tangentsEdit

 
Ellipse with its orthoptic
Main article: Orthoptic (geometry)

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

This circle is called orthoptic of the given ellipse.

Theorem of Apollonios on conjugate diametersEdit

 
Ellipse: theorem of Apollonios on conjugate diameters

For an ellipse with semi-axes   the following is true:

  • Let be   and   halves of two conjugate diameters (see diagram) then
(1)  ,
(2) the triangle formed by   has the constant area  
(3) the parallelogram of tangents adjacent to the given conjugate diameters has the  
proof

Let be the ellipse in the canonical form with parameter equation. Then the ellipse has the parameter equation

 .

The two points   are on conjugate diameters (see previous section). From trigonometric formulae one gets   and

 

The area of the triangle generated by   is:

 

and from the diagram one recognizes: the area of the parallelogram is 8 times of  . Hence

 

Drawing ellipsesEdit

 
Central projection of circles (gate)

Ellipses appear in descriptive geometry as images (parallel or central projection) of circles (for details: see Ellipses in DG (German)). So it is essential to have tools to draw an ellipse. Nowadays the best tool is the computer. During the times before this tool was not available and one was restricted to compass and ruler for the construction of single points of an ellipse. But there are technical tools (ellipsographs) to draw an ellipse in a continuous way like a compass for drawing a circle, too. The principle of ellipsographs were known to Greek mathematicians (Archimedes, Proklos) already.

If there is no ellipsograph available, the best and quickest way to draw an ellipse is to draw 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 necessery (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.

 
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 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.

 
Ellipse construction: paper strip method 1
 
Ellipses with Tusi couple. Two examples: red and cyan.

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 P traces the ellipse. For the proof one shows that point   has the parametric representation  , where parameter   is the angle of the slope of the paper strip.

A technichal realization of the motion of the paper strip can be achieved by a Tusi couple (s. 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 nice application: If one stands somewhere in the middle of a ladder, which stands on a slippery ground and leans on a slippery wall, the ladder slides down and the persons feet trace an ellipse.

 
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).

 
Approximation of an ellipse with osculating circles

Approximation by osculating circlesEdit

From section metric properties one gets:

  • 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  , resp.:

(1) mark the auxiliary point   and draw the linesegment  
(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 oasculating 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 of an ellipseEdit

 
Ellipse: Steiner generation
 
Ellipse: Steiner generation

The following method to construct single points of an ellipse relies on the Steiner generation of a non degenerate 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 a 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 asign 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 gets the points of the lower half of the ellipse.

Remark:

  1. The Steiner generation exists for hyperbolas and parabolas, too.
  2. The Steiner generation 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.

EllipsographsEdit

Most technical instruments for drawing ellipses base on the 2. paperstrip method.

For more principles of ellipsographs:

Inscribed angles for ellipses and the 3-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 degree or radian . In order to get an equation of a circle determined by three points, the following measurement is more convenient:

  • In order to measure an angle between two lines with equations   one uses the quotient
 
This expression is the cotangent of the angle between the two lines.

Inscribed angle theorem for circles:

For four points  , no three of them on a line (see diagram), the following statement is true:
The four points are on a circle, if and only if the angles at   and   are equal. In the sense of the measurement above, that means, if
 

At first the masure is available for chords, which are not parallel to the y-axis, only. But the final formula works for any chord.

A consequence of the inscribed angle theorem for circles is the

3-point-form of a circle's equation:

The equation of the circle determined by 3 points   not on a line is the solution of the equation
 
for  .

EllipsesEdit

In this section one considers ellipses with an equation

 

where the ratio   is fixed. With the abbreviation   one gets the more convenient form

  •   and   fixed.

Such ellipses have their axes parallel to the coordinate axes and their eccentricity fixed. Their major axes are parallel to the x-axis if   and parallel to the y-axis if  .

 
Inscribed angle theorem for an ellipse

Like a circle, such an ellipse is determined by three points not on a line.

In this more general case one introduces the following measurement of an angle,:[2][3]

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

Inscribed angle theorem for ellipses:

For four points  , no three of them on a line (see diagram), the following statement is true:
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 means, if
 

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

A consequence of the inscribed angle theorem for ellipses is the

3-point-form of an ellipse's equation:

The equation of the ellipse determined by 3 points   not on a line is the solution of the equation
 
for  .

Pole-polar relation for an ellipseEdit

 
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 just polarity. The pole is the point, the polar the line. See Pole and polar.

By calculation one checks 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:  ).

Remarks:

  1. The intersection point of two polars is the pole of the line through their poles.
  2. The foci   and   respectively and the directrices   and   respectively belong to pairs of pole and polar.

Pole-polar relations exist for hyperbolas and parabolas, too.

Metric propertiesEdit

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

AreaEdit

The area   enclosed by an ellipse is:

  •  

where   and   are the lengths of the semi-major and semi-minor axes, respectively. The area formula   is intuitive: start with a circle of radius   (so its area is  ) and stretch it by a factor   to make an ellipse. This scales the area by the same factor:   It is also easy to rigorously prove the area formula using integration as follows. Equation (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  

CircumferenceEdit

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

 

 

 

 

 

(3)

The arc length of an ellipse, in general, has no closed-form solution in terms of elementary functions. Elliptic integrals were motivated by this problem. Equation (3) may be evaluated directly using the Carlson symmetric form.[4] This gives a succinct and quadratically converging iterative method for evaluating the circumference using the arithmetic-geometric mean.[5]

The exact infinite series is:

 
 

where   is the double factorial. Unfortunately, this series converges rather slowly; however, by expanding in terms of   Ivory[6] and Bessel[7] derived an expression that converges much more rapidly,

 

Ramanujan gives two good approximations for the circumference in §16 of "Modular Equations and Approximations to  ";[8] they are

 

and

 

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

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

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

Some lower and upper bounds on the circumference of the canonical ellipse   with   are[9]

 
 
 

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 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:

 

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

 

Ellipse as quadricEdit

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[10][11]

 

provided  

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

 

that is,

 

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.[12]:p.63

The general equation's coefficients can be obtained from known semi-major axis  , semi-minor axis  , center coordinates   and rotation angle   using the following formulae:

 

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

 
 
 

Canonical formEdit

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

 

Here   are the point coordinates in the canonical system, whose origin is the center   of the ellipse, whose  -axis is the unit vector   coinciding with the major axis, and whose  -axis is the perpendicular vector   coinciding with the minor axis. That is,   and  .

In this system, the center is the origin   and the foci are   and  .

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

 

Moreover, any canonical ellipse can be obtained by scaling the unit circle of  , defined by the equation

 

by factors a and b along the two axes.

For an ellipse in canonical form, we have

 

The distances from a point   on the ellipse to the left and right foci are   and  , respectively.

The canonical form coefficients can be obtained from the general form coefficients using the following equations:

 

where   is the angle from the positive horizontal axis to the ellipse's major axis.

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[12]: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 of the ellipse, usually denoted  . It is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis.

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

Ellipses as plane sections of quadricsEdit

Ellipses appear as plane sections of the following quadrics:

ApplicationsEdit

PhysicsEdit

Elliptical reflectors and acousticsEdit

See also: Fresnel zone

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 orbitsEdit

Main article: Elliptic orbit

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

More generally, in the gravitational two-body problem, if the two bodies are bound to each other (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. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at the same focus.

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

For elliptical orbits, useful relations involving the eccentricity   are:

 

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 of   and  , the semi-major axis   is their arithmetic mean, the semi-minor axis   is their geometric mean, and the semi-latus rectum   is their harmonic mean. In other words,

 .

Harmonic oscillatorsEdit

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 visualizationEdit

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

Elliptical gearsEdit

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.[13]

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.[14]

OpticsEdit

  • 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).[15]
  • 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.[16]

Statistics and financeEdit

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.[17][18]

Computer graphicsEdit

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.[19] Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.[20]

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.[21] 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 theoryEdit

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 alsoEdit

NotesEdit

  1. ^ K. Strubecker: Vorlesungen über Darstellende Geometrie, GÖTTINGEN, VANDENHOECK & RUPRECHT, 1967, p. 26
  2. ^ E. Hartmann: Lecture Note 'Planar Circle Geometries', an Introduction to Möbius-, Laguerre- and Minkowski Planes, p. 55
  3. ^ W. Benz, Vorlesungen über Geomerie der Algebren, Springer (1973)
  4. ^ Carlson, B. C. (1995). "Numerical computation of real or complex elliptic integrals". Numerical Algorithms. 10 (1): 13–98. arXiv:math/9409227 . Bibcode:1995NuAlg..10...13C. doi:10.1007/BF02198293. 
  5. ^ Python code for the circumference of an ellipse in terms of the complete elliptic integral of the second kind, retrieved 2013-12-28 
  6. ^ Ivory, J. (1798). "A new series for the rectification of the ellipsis". Transactions of the Royal Society of Edinburgh. 4: 177–190. doi:10.1017/s0080456800030817. 
  7. ^ Bessel, F. W. (2010). "The calculation of longitude and latitude from geodesic measurements (1825)". Astron. Nachr. 331 (8): 852–861. arXiv:0908.1824 . doi:10.1002/asna.201011352.  Englisch translation of Bessel, F. W. (1825). "Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermesssungen". Astron. Nachr. 4: 241–254. Bibcode:1825AN......4..241B. doi:10.1002/asna.18260041601. 
  8. ^ Ramanujan, Srinivasa, (1914). "Modular Equations and Approximations to π". Quart. J. Pure App. Math. 45: 350–372. 
  9. ^ Jameson, G.J.O. (2014). "Inequalities for the perimeter of an ellipse". Mathematical Gazette. 98: 227–234. doi:10.2307/3621497. 
  10. ^ Larson, Ron; Hostetler, Robert P.; Falvo, David C. (2006). "Chapter 10". Precalculus with Limits. Cengage Learning. p. 767. ISBN 0-618-66089-5. 
  11. ^ Young, Cynthia Y. (2010). "Chapter 9". Precalculus. John Wiley and Sons. p. 831. ISBN 0-471-75684-9. 
  12. ^ a b Lawrence, J. Dennis, A Catalog of Special Plane Curves, Dover Publ., 1972.
  13. ^ David Drew. "Elliptical Gears". [1]
  14. ^ Grant, George B. (1906). A treatise on gear wheels. Philadelphia Gear Works. p. 72. 
  15. ^ Encyclopedia of Laser Physics and Technology - lamp-pumped lasers, arc lamps, flash lamps, high-power, Nd:YAG laser
  16. ^ [2][dead link]
  17. ^ Chamberlain, G. (February 1983). "A characterization of the distributions that imply mean—Variance utility functions". Journal of Economic Theory. 29 (1): 185–201. doi:10.1016/0022-0531(83)90129-1. 
  18. ^ Owen, J.; Rabinovitch, R. (June 1983). "On the class of elliptical distributions and their applications to the theory of portfolio choice". Journal of Finance. 38: 745–752. doi:10.1111/j.1540-6261.1983.tb02499.x. JSTOR 2328079. 
  19. ^ Pitteway, M.L.V. (1967). "Algorithm for drawing ellipses or hyperbolae with a digital plotter". The Computer Journal. 10 (3): 282–9. doi:10.1093/comjnl/10.3.282. 
  20. ^ Van Aken, J.R. (September 1984). "An Efficient Ellipse-Drawing Algorithm". IEEE Computer Graphics and Applications. 4 (9): 24–35. doi:10.1109/MCG.1984.275994. 
  21. ^ Smith, L.B. (1971). "Drawing ellipses, hyperbolae or parabolae with a fixed number of points". The Computer Journal. 14 (1): 81–86. doi:10.1093/comjnl/14.1.81. 

ReferencesEdit

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

External linksEdit