Ellipse

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 elongation of an ellipse 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 the above-mentioned eccentricity of the ellipse.

An ellipse may also be defined analytically as the set of points for each of which the sum of its distances to two foci is a fixed number.

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 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".

Ellipse-conic
An ellipse (red) obtained as the intersection of a cone with an inclined plane
Ellipse-def0
Ellipse: notations
Ellipse-var
Ellipses: examples

Definition of an ellipse as locus of points

Ellipse-def-e
Ellipse: Definition
Ellipse-def-dc
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 can be defined using two fixed points, , , called the foci and a distance, usually denoted . The ellipse defined with , and is the set of points such that the sum of the distances is constant and equal to . In order to omit the special case of a line segment, one assumes More formally, for a given , an ellipse is the set

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 called the minor axis. The major axis 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 circular directrix (related to focus ) of the ellipse.[1][2] This property should not be confused with the definition of an ellipse with help of a directrix (line) below.

Using Dandelin spheres one can prove that 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 coordinates

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

Equation

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 and semi-minor axes. 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.

Eccentricity

The eccentricity of an ellipse can be expressed in terms of the ratio of semi-minor and semi-major axes as

This definition relies on the major axis 2a of an ellipse not being shorter than its minor axis 2b. Ellipses with equal axes are simply circles, that is, ellipses with zero eccentricity. The degree of flatness of ellipses increases as their eccentricity does.

Semi-latus rectum

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 on the major axis.

Tangent

An arbitrary line intersects an ellipse at 0, 1 or 2 points. In the first case the line is called exterior line, in the second case tangent and secant in the third case. Through any point of an ellipse there is exactly one tangent.

The tangent at a point of the ellipse has the coordinate equation
A vector equation of the tangent is
  • with

Proof: Let be an ellipse point and the vector equation of a line (containing ). Inserting the line's equation into the ellipse equation and respecting yields:

In case of line and the ellipse have only point in common and is a tangent. The tangent direction is orthogonal to vector which is then a normal vector of the tangent and the tangent has the equation with a still unknown . Because is on the tangent and on the ellipse, one gets .
In case of line has a second point with the ellipse in common.

With help of (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 holds, then the points lie on two conjugate diameters of the Ellipse (see below). In case of the ellipse is a circle and "conjugate" means "orthogonal".

Equation of a shifted ellipse

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

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

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

Parametric representation

Ellipse-ratpar
Ellipse points calculated by the rational representation with equal spaced parameters ()

Standard parametric representation

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

Parameter t can be taken as shown in the diagram and is due to de la Hire.[3]

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.

Rational representation

With the substitution and trigonometric formulae one gets

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 popular with Computer Aided Design (see Bezier curve).

Tangent slope as parameter

A parametric representation, which uses the slope of the tangent at a point of the ellipse can be obtained from the derivative of the standard representation :

With help of trigonometric formulae one gets:

Replacing und of the standard representation one yields

Where is the slope of the tangent at the corresponding ellipse point, is the upper and the lower half of the ellipse. The points with vertical tangents (vertices) 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, which omits differential calculus and trigonometric formulae.

Shifted Ellipse

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.

Parameters a and b

The parameters and represent the lengths of line segments and are therefore non-negative real numbers. Throughout this article is the semi-major axis, i.e., In general the canonical ellipse equation may 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 form assumed in the remainder of this article simply by transposing the variable names and and the parameter names and

Definition of an ellipse by the directrix property

Ellipse-ll-e
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 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 locus of points for which the quotient of the distances to the point and to the line is that 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.)

Kegelschnitt-schar-ev
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 obtains 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 foci

Ellipse-reflex
Ellipse: the tangent bisects the supplementary angle of the angle between the lines to the foci.
Elli-norm-tang-n
Rays from one focus reflect off the ellipse to 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 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: . 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=1

Ellipse-aff
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 four 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 chords

Parallelproj-kreis-ellipse
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, the property (M) holds:

(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 does preserve 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 parametric equation

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

Orthogonal tangents

Orthoptic-ellipse-s
Ellipse with its orthoptic

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 diameters

Elli-apoll-cd
Ellipse: theorem of Apollonios on conjugate diameters

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

  • Let and be 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 the ellipse be in the canonical form with parametric 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 it can be seen that the area of the parallelogram is 8 times that of . Hence

Drawing ellipses

Zp-turm-tor
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 (Archimedes, Proklos).

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 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.

Elliko-g
Ellipse: gardener's method

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

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 technical 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.

Elliko-pap1

Ellipse construction: paper strip method 1

Tusi couple vs Paper strip plus Ellipses horizontal

Ellipses with Tusi couple. Two examples: red and cyan.

A variation of the paper strip method 1[4] 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. The advantage of this variation is: Only one expensive sliding shoe is necessary.
One should be aware that the end, which is sliding on the minor axis, has to be changed.

Ellipse-papsm-1a

Variation of the paper strip method 1

Ellipses with SliderCrank inner Ellipses

Animation of the variation of the paper strip method 1

Elliko-pap2
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.

Archimedes Trammel

Trammel of Archimedes (principle)

L-Ellipsenzirkel

Ellipsograph due to Benjamin Bramer

Ellipses with SliderCrank Ellipses at Slider Side

Variation of the paper strip method 2

Elliko-skm
Approximation of an ellipse with osculating circles

Approximation by osculating circles

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 , 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 generation of an ellipse

Ellipse-steiner-e
Ellipse: Steiner generation
Ellipse construction - parallelogram method
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 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 gets the points of the lower half of the ellipse.

The Steiner generation also exists 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.

Ellipsographs

Most technical instruments for drawing ellipses are based on the second paperstrip method.

Inscribed angles for ellipses and the 3-point-form

Circles

Inscribe-a-c
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 measure 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

One gets the equation of the circle determined by 3 points not on a line by a conversion of the equation

Using vectors, dot products and determinants this formula can be arranged more clearly:

For example, for the 3-pointform is

, which can be rearranged to

Ellipses

This section 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 .

Inscribe-a-e
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,:[5][6]

  • 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 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.

A consequence of the inscribed angle theorem for ellipses is the

3-point-form of an ellipse's equation

One gets the equation of the ellipse determined by 3 points not on a line by a conversion of the equation

Analogously to the circle case this formula can be written more clearly using vectors:

where is the modified dot product

For example, for and one gets the 3-point-form

and after conversion

Pole-polar relation for an ellipse

Ellipse-pol
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 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.

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

Metric properties

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

Area

The area enclosed by an ellipse is:

where and are the lengths of the semi-major and semi-minor axes, respectively. The area formula is intuitive: start with a circle of radius (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

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

The circumference of an ellipse is:

where again is the length of the semi-major axis, is the eccentricity and the function is the complete elliptic integral of the second kind,

The circumference of the ellipse may be evaluated in terms of using Gauss's arithmetic-geometric mean;[7] this is a quadratically converging iterative method.[8]

The exact infinite series is:

where is the double factorial. Unfortunately, this series converges rather slowly; however, by expanding in terms of Ivory[9] and Bessel[10] derived an expression that converges much more rapidly,

Ramanujan gives two good approximations for the circumference in §16 of "Modular Equations and Approximations to ";[11] 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.

Some lower and upper bounds on the circumference of the canonical ellipse with are[12]

Here the upper bound is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and the minor axes.

Curvature

The curvature is given by radius of curvature at point :

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

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

Ellipse as quadric

General ellipse

In analytic geometry, the ellipse is defined as a quadric: the set of points of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation[13][14]

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.[15]: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 form

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 for eccentricity e.

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 forms

Polar form relative to center

Ellipse Polar center
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[15]:p. 75

Polar form relative to focus

Ellipse Polar
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 hypotrochoid

Ellipse as hypotrochoid
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 in triangle geometry

Ellipses appear in triangle geometry as

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

Ellipses as plane sections of quadrics

Ellipses appear as plane sections of the following quadrics:

Ellipsoid Quadric

Ellipsoid

Quadric Cone

Elliptic cone

Elliptic Cylinder Quadric

Elliptic cylinder

Hyperboloid1

Hyperboloid of one sheet

Hyperboloid2

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.)

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 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 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.[16]

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.[17]

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).[18]
  • 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.[19]

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.[20][21]

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.[22] Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.[23]

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.[24] 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

Notes

  1. ^ Apostol, Tom M.; Mnatsakanian, Mamikon A. (2012), New Horizons in Geometry, The Dolciani Mathematical Expositions #47, The Mathematical Association of America, p. 251, ISBN 978-0-88385-354-2
  2. ^ The German term for this circle is Leitkreis which can be translated as "Director circle", but that term has a different meaning in the English literature (see Director circle).
  3. ^ K. Strubecker: Vorlesungen über Darstellende Geometrie, GÖTTINGEN, VANDENHOECK & RUPRECHT, 1967, p. 26
  4. ^ J. van Mannen: Seventeenth century instruments for drawing conic sections. In: The Mathematical Gazette. Vol. 76, 1992, p. 222–230.
  5. ^ E. Hartmann: Lecture Note 'Planar Circle Geometries', an Introduction to Möbius-, Laguerre- and Minkowski Planes, p. 55
  6. ^ W. Benz, Vorlesungen über Geomerie der Algebren, Springer (1973)
  7. ^ Carlson, B. C. (2010), "Elliptic Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
  8. ^ Python code for the circumference of an ellipse in terms of the complete elliptic integral of the second kind, retrieved 2013-12-28
  9. ^ 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.
  10. ^ Bessel, F. W. (2010). "The calculation of longitude and latitude from geodesic measurements (1825)". Astron. Nachr. 331 (8): 852–861. arXiv:0908.1824. Bibcode:2010AN....331..852K. 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. arXiv:0908.1823. Bibcode:1825AN......4..241B. doi:10.1002/asna.18260041601.
  11. ^ Ramanujan, Srinivasa, (1914). "Modular Equations and Approximations to π". Quart. J. Pure App. Math. 45: 350–372.
  12. ^ Jameson, G.J.O. (2014). "Inequalities for the perimeter of an ellipse". Mathematical Gazette. 98: 227–234. doi:10.1017/S002555720000125X.
  13. ^ Larson, Ron; Hostetler, Robert P.; Falvo, David C. (2006). "Chapter 10". Precalculus with Limits. Cengage Learning. p. 767. ISBN 0-618-66089-5.
  14. ^ Young, Cynthia Y. (2010). "Chapter 9". Precalculus. John Wiley and Sons. p. 831. ISBN 0-471-75684-9.
  15. ^ a b Lawrence, J. Dennis, A Catalog of Special Plane Curves, Dover Publ., 1972.
  16. ^ David Drew. "Elliptical Gears". [1]
  17. ^ Grant, George B. (1906). A treatise on gear wheels. Philadelphia Gear Works. p. 72.
  18. ^ Encyclopedia of Laser Physics and Technology - lamp-pumped lasers, arc lamps, flash lamps, high-power, Nd:YAG laser
  19. ^ "Archived copy". Archived from the original on 2013-05-17. Retrieved 2013-06-20.CS1 maint: Archived copy as title (link)
  20. ^ 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.
  21. ^ 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.
  22. ^ 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.
  23. ^ 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.
  24. ^ 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.

References

  • 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 links

Area

Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.

Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.

Circumference

In geometry, the circumference (from Latin circumferentia, meaning "carrying around") of a circle is the (linear) distance around it. That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter. The perimeter is the length around any closed figure and is the term used for most figures excepting the circle and some circular-like figures such as ellipses.

Informally, "circumference" may also refer to the edge itself rather than to the length of the edge.

Composite pattern

In software engineering, the composite pattern is a partitioning design pattern. The composite pattern describes a group of objects that is treated the same way as a single instance of the same type of object. The intent of a composite is to "compose" objects into tree structures to represent part-whole hierarchies. Implementing the composite pattern lets clients treat individual objects and compositions uniformly.

Conic section

In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, when Apollonius of Perga undertook a systematic study of their properties.

The conic sections of the Euclidean plane have various distinguishing properties. Many of these have been used as the basis for a definition of the conic sections. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, and some particular line, called a directrix, are in a fixed ratio, called the eccentricity. The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables. This equation may be written in matrix form, and some geometric properties can be studied as algebraic conditions.

In the Euclidean plane, the conic sections appear to be quite different from one another, but share many properties. By extending the geometry to a projective plane (adding a line at infinity) this apparent difference vanishes, and the commonality becomes evident. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically.

Curve fitting

Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables. Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data.

Diameter

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere.

In more modern usage, the length of a diameter is also called the diameter. In this sense one speaks of the diameter rather than a diameter (which refers to the line itself), because all diameters of a circle or sphere have the same length, this being twice the radius r.

For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the width is often defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers. For a curve of constant width such as the Reuleaux triangle, the width and diameter are the same because all such pairs of parallel tangent lines have the same distance.

For an ellipse, the standard terminology is different. A diameter of an ellipse is any chord passing through the center of the ellipse. For example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one. The longest diameter is called the major axis.

The word "diameter" is derived from Greek διάμετρος (diametros), "diameter of a circle", from διά (dia), "across, through" and μέτρον (metron), "measure". It is often abbreviated DIA, dia, d, or .

Eccentricity (mathematics)

In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape.

More formally two conic sections are similar if and only if they have the same eccentricity.

One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular:

The eccentricity of a circle is zero.

The eccentricity of an ellipse which is not a circle is greater than zero but less than 1.

The eccentricity of a parabola is 1.

The eccentricity of a hyperbola is greater than 1.

Ellipsanime

Ellipsanime (formely known as Le Studio Ellipse and Ellipse Programme) is a French animation studio that produces television programs. It was founded in 1987.

In February 2000 it merged with Expand SA; Expand sold the company to Dargaud in 2003 and it became Ellipsanime in 2004. Ellipse has worked with many other animation companies, with one good example being the Canadian animation studio Nelvana Limited.

Ellipsoid

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

An ellipsoid is a quadric surface;  that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse like"). It is bounded, which means that it may be enclosed in a sufficiently large sphere.

An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, or simply axes of the ellipsoid. If the three axes have different lengths, the ellipsoid is said to be tri-axial or (rarely) scalene, and the axes are uniquely defined.

If two of the axes have the same length, then the ellipsoid is an "ellipsoid of revolution", also called a spheroid. In this case, the ellipsoid is invariant under a rotation around the third axis, and there are thus infinitely many ways of choosing the two perpendicular axes of the same length. If the third axis is shorter, the ellipsoid is an oblate spheroid; if it is longer, it is a prolate spheroid. If the three axes have the same length, the ellipsoid is a sphere.

Flattening

Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is f and its definition in terms of the semi-axes of the resulting ellipse or ellipsoid is

The compression factor is b/a in each case. For the ellipse, this factor is also the aspect ratio of the ellipse.

There are two other variants of flattening (see below) and when it is necessary to avoid confusion the above flattening is called the first flattening. The following definitions may be found in standard texts and online web texts

Kepler's laws of planetary motion

In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun.

The orbit of a planet is an ellipse with the Sun at one of the two foci.

A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.Most planetary orbits are nearly circular, and careful observation and calculation are required in order to establish that they are not perfectly circular. Calculations of the orbit of Mars, whose published values are somewhat suspect, indicated an elliptical orbit. From this, Johannes Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits.

Kepler's work (published between 1609 and 1619) improved the heliocentric theory of Nicolaus Copernicus, explaining how the planets' speeds varied, and using elliptical orbits rather than circular orbits with epicycles.Isaac Newton showed in 1687 that relationships like Kepler's would apply in the Solar System to a good approximation, as a consequence of his own laws of motion and law of universal gravitation.

Line segment

In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints.

Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or otherwise a diagonal. When the end points both lie on a curve such as a circle, a line segment is called a chord (of that curve).

Parallelogram

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English.

The three-dimensional counterpart of a parallelogram is a parallelepiped.

The etymology (in Greek παραλληλ-όγραμμον, a shape "of parallel lines") reflects the definition.

President's Park

President's Park, located in Washington, D.C., encompasses the White House including the Eisenhower Executive Office Building, the Treasury Building (Washington, D.C.), and grounds; the White House Visitor Center; Lafayette Square; and The Ellipse. President's Park was the original name of Lafayette Square. The current President's Park is administered by the National Park Service. The park is officially referred to as President's Park or The White House and President's Park.

Semi-major and semi-minor axes

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. For the special case of a circle, the semi-major axis is the radius.

The length of the semi-major axis of an ellipse is related to the semi-minor axis's length through the eccentricity and the semi-latus rectum , as follows:

The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches. Thus it is the distance from the center to either vertex of the hyperbola.

A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping fixed. Thus and tend to infinity, faster than .

The semi-minor axis of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section.

The major and minor axes are the axes of symmetry for the curve: in an ellipse, the minor axis is the shorter one; in a hyperbola, it is the one that does not intersect the hyperbola.

Spheroid

A spheroid, or ellipsoid of revolution, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circular symmetry.

If the ellipse is rotated about its major axis, the result is a prolate (elongated) spheroid, shaped like an American football or rugby ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, shaped like a lentil. If the generating ellipse is a circle, the result is a sphere.

Due to the combined effects of gravity and rotation, the figure of the Earth (and of all planets) is not quite a sphere, but instead is slightly flattened in the direction of its axis of rotation. For that reason, in cartography the Earth is often approximated by an oblate spheroid instead of a sphere. The current World Geodetic System model uses a spheroid whose radius is 6,378.137 km (3,963.191 mi) at the Equator and 6,356.752 km (3,949.903 mi) at the poles.

The word spheroid originally meant "an approximately spherical body", admitting irregularities even beyond the bi- or tri-axial ellipsoidal shape, and that is how the term is used in some older papers on geodesy (for example, referring to truncated spherical harmonic expansions of the Earth).

Strewn field

The term strewnfield indicates the area where meteorites from a single fall are dispersed.

It is also often used with tektites produced by large meteorite impact.

Superellipse

A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape.

In the Cartesian coordinate system, the set of all points (xy) on the curve satisfy the equation

where n, a and b are positive numbers, and the vertical bars | | around a number indicate the absolute value of the number.

The Ellipse

The Ellipse (sometimes referred to as President's Park South) is a 52-acre (210,000 m²) park located south of the White House fence and north of Constitution Avenue and the National Mall. Properly, the Ellipse is the name of the five-furlong (1 km) circumference street within the park. The entire park, which features various monuments, is open to the public and is part of President's Park. The Ellipse is also the location for a number of annual events. D.C. locals can often be heard to say they are "on the Ellipse." which is understood to mean that the individual is on the field that is bounded by Ellipse Road.

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