Wikipedia / Ellipse
Wikipedia / Ellipse
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of both distances to the two focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity (the limiting case (the limiting case of infinite elongation, no longer an ellipse but a parabola).
An ellipse has a simple algebraic formula for its area, but for its perimeter (also known as circumference), integration is required to obtain the exact solution.
The largest and smallest diameters of an ellipse, also known as its width and height, are typically denoted and . An ellipse has four extreme points: two vertices at the endpoints of the major axis and two co-vertices at the endpoints of the minor axis.
Analytically (the linear eccentricity) is the distance from the center to a focus. The standard parametrization
Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane (see figure). Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. An angled cross section of a right circular cylinder is also an ellipse.
An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant, called the eccentricity
Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in the Solar System is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the barycenter of the Sunplanet pair). The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids. A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under parallel or perspective projection. The ellipse is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoidss with the same frequency: a similar effect leads to elliptical polarization of light in optics.
The name, (, "omission"), was given by Apollonius of Perga in his Conics.
Definition as locus of points
An ellipse can be defined geometrically as a set or locus of points in the Euclidean plane:
of the line segment joining the foci is called the center of the ellipse. The line through the foci is called the major axis, and the line perpendicular to it through the center is the minor axis. The major axis intersects the ellipse at two vertices of the foci to the center is called the focal distance is defined as the eccentricity.
yields a circle and is included as a special type of ellipse.
can be viewed in a different way (see figure):
is called the circular directrix (related to focus of the ellipse. This property should not be confused with the definition of an ellipse using a directrix line below.
Using Dandelin spheres, one can prove that any section of a cone with a plane is an ellipse, assuming the plane does not contain the apex and has slope less than that of the lines on the cone.
In Cartesian coordinates
Standard equation
The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, the x-axis is the major axis, and:
Removing the radicals (see diagram) produces the standard equation of the ellipse: or, solved for y
are called the semi-major and semi-minor axes are the co-vertices.
It follows from the equation that the ellipse is symmetric with respect to the coordinate axes and hence with respect to the origin.
Parameters
Principal axes
Throughout this article, the semi-major and semi-minor axes
Linear eccentricity
.
Eccentricity
) has zero eccentricity, and is a circle.
Semi-latus rectum
The length of the chord through one focus, perpendicular to the major axis, is called the latus rectum. One half of it is the semi-latus rectum. A calculation shows:
is equal to the radius of curvature at the vertices (see section curvature).
Tangent
intersects an ellipse at 0, 1, or 2 points, respectively called an exterior line, tangent and secant
A vector parametric equation
Proof: There are then cases:
is a tangent. The tangent direction has perpendicular vector.
has a second point in common with the ellipse, and is a secant.
, which proves the vector equation.
, then the points lie on two conjugate diameters (see below, the ellipse is a circle and "conjugate" means "orthogonal".)
Shifted ellipse
The axes are still parallel to the x- and y-axes.
General ellipse
In analytic geometry, the ellipse is defined as a quadric of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation
To distinguish the degenerate cases from the non-degenerate case, let ∆ be the determinant
Then the ellipse is a non-degenerate real ellipse if and only if C∆ < 0. If C∆ > 0, we have an imaginary ellipse, and if ∆ = 0, we have a point ellipse.
Conversely, the canonical form parameters can be obtained from the general-form coefficients by the equations:
where is the 2-argument arctangent function.
Parametric representation
Standard parametric representation
Using trigonometric functions
The parameter t (called the eccentric anomaly with the x-axis, but has a geometric meaning due to Philippe de La Hire (see below).
Rational representation
and the rational
.
is considered to be a point on the real projective line
Rational representations of conic sections are commonly used in computer-aided design (see Bézier curve).
Tangent slope as parameter
With help of trigonometric formulae
, having vertical tangents, are not covered by the representation.
This description of the tangents of an ellipse is an essential tool for the determination of the orthoptic of an ellipse. The orthoptic article contains another proof, without differential calculus and trigonometric formulae.
General ellipse
Another definition of an ellipse uses affine transformations:
Any ellipse is an affine image of the unit circle.
Parametric representation
is a regular matrix (with non-zero determinant
are the directions of two conjugate diameters, in general not perpendicular.
Vertices
Area
From Apollonios theorem (see below) one obtains:
Semiaxes
Solving this nonlinear system
Implicit representation
Conversely: If the equation
point to two conjugate points and the tools developed above are applicable.
Example
Rotated standard ellipse
one obtains a parametric representation of the standard ellipse rotated
Ellipse in space
to be vectors in space.
Polar forms
Polar form relative to center
In polar coordinates measured from the major axis, the ellipse's equation is is the eccentricity (not Euler's number).
Polar form relative to focus
points towards the center (as illustrated on the right), and positive if that direction points away from the center.
is called the true anomaly is the semi-latus rectum.
Eccentricity and the directrix property
from it, is called a directrix of the ellipse (see diagram).
The second case is proven analogously.
The converse is also true and can be used to define an ellipse (in a manner similar to the definition of a parabola):
, and any real number
, which is the eccentricity of a circle, is not allowed in this context in the Euclidean plane. However, one may consider the directrix of a circle to be the line at infinity in the projective plane.
, a hyperbola.)
Proof
produces the equations
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).
, the x.
Construction of a directrix
are inverse with respect to the circle inversion.
General ellipse
(The right side of the equation uses the Hesse normal form.)
Focus-to-focus reflection property
An ellipse possesses the following property:
.
Proof
By the triangle inequality and the angle bisector theorem so must be the tangent line.
Application
The rays from one focus are reflected by the ellipse to the second focus. This property has optical and acoustic applications similar to the reflective property of a parabola (see whispering gallery). It also serves to formulate Alhazen's problem of reflection on a circle tangent to the ellipse.
Additionally, because of the focus-to-focus reflection property of ellipses, if the rays are allowed to continue propagating, reflected rays will eventually align closely with the major axis.
Conjugate diameters
Definition of conjugate diameters
A circle has the following property:
The midpoints of parallel chords lie on a diameter.
An affine transformation preserves parallelism and midpoints of line segments, so this property is true for any ellipse. (Note that the parallel chords and the diameter are no longer orthogonal.)
Definition:
of an ellipse are conjugate
From the diagram one finds:
.
Conjugate diameters in an ellipse generalize orthogonal diameters in a circle.
belong to its conjugate diameter.
one gets: The points
(signs: (+,+) or (−,−) )
(signs: (−,+) or (+,−) )
are conjugate and
Theorem of Apollonios on conjugate diameters
the following is true:
be halves of two conjugate diameters (see diagram) then
.
The triangle the angle between the half diameters. Hence the area of the ellipse (see section metric properties.
The parallelogram
Proof:
Orthogonal tangents
the intersection points of orthogonal.
This circle is called orthoptic or director circle of the ellipse (not to be confused with the circular directrix defined above).
Drawing ellipses
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 was known to the 5th century mathematician Proclus, and the tool now known as an elliptical trammel was invented by Leonardo da Vinci.
If there is no ellipsograph available, one can draw an ellipse using an approximation by the four osculating circles at the vertices.
For any method described below, knowledge of the axes and the semi-axes is necessary (or equivalently: the foci and the semi-major axis). If this presumption is not fulfilled one has to know at least two conjugate diameters. With help of Rytz's construction the axes and semi-axes can be retrieved.
de La Hire's point construction
The following construction of single points of an ellipse is due to de La Hire. It is based on the standard parametric representation of an ellipse:
Draw the two circles and the axes of the ellipse.
Draw a line through the center, respectively.
Draw a line that is parallel to the minor axis and a line (see diagram).
Repeat steps (2) and (3) with different lines through the center.
Pins-and-string method
The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two drawing pins. 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. The Byzantine architect Anthemius of Tralles () described how this method could be used to construct an elliptical reflector, and it was elaborated in a now-lost 9th-century treatise by Al-Ḥasan ibn Mūsā.
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
have to be known.
Method 1
The first method starts with
.
is the angle of the slope of the paper strip.
A technical realization of the motion of the paper strip can be achieved by a Tusi couple (see animation). The device is able to draw any ellipse with a fixed, 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.
(see diagram). After this operation the movement of the unchanged half of the paperstrip is unchanged. This variation requires only one sliding shoe.
Method 2:
The second method starts with
.
is the angle of slope of the paper strip.
This method is the base for several ellipsographs (see section below).
Similar to the variation of the paper strip method 1 a variation of the paper strip method 2 can be established (see diagram) by cutting the part between the axes into halves. Most ellipsograph drafting instruments are based on the second paperstrip method.
Approximation by osculating circles
From Metric properties below, one obtains:
, respectively:
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
The following method to construct single points of an ellipse relies on the Steiner generation of a conic section:
Given two pencils, then the intersection points of corresponding lines form a non-degenerate projective conic section.
.
the points of the second quarter of the ellipse can be determined. Analogously one obtains the points of the lower half of the ellipse.
Steiner generation can also be defined for hyperbolas and parabolas. It is sometimes called a parallelogram method because one can use other points rather than the vertices, which starts with a parallelogram instead of a rectangle.
As hypotrochoid
The ellipse is a special case of the hypotrochoid is called a Tusi couple.
Inscribed angles and three-point form
Circles
uses the inscribed angle theorem for circles:
(see diagram) the following statement is true:
are equal.
Usually one measures inscribed angles by a degree or radian θ, but here the following measurement is more convenient:
Inscribed angle theorem for circles
no three of them on a line, we have the following (see diagram):
At first the measure is available only for chords not parallel to the y-axis, but the final formula works for any chord.
Three-point form of circle equation
the three-point equation is:
Using vectors, dot products and determinants
Ellipses
with a fixed
where q, the major axis is parallel to the x, it is parallel to the y-axis.)
Like a circle, such an ellipse is determined by three points not on a line.
For this family of ellipses, one introduces the following q-analog angle measure, which is not a function of the usual angle measure θ:
Inscribed angle theorem for ellipses
, no three of them on a line (see diagram).
At first the measure is available only for chords which are not parallel to the y-axis. But the final formula works for any chord. The proof follows from a straightforward calculation. For the direction of proof given that the points are on an ellipse, one can assume that the center of the ellipse is the origin.
Three-point form of ellipse equation
one obtains the three-point form
is the modified dot product
Pole-polar relation
to be an arbitrary point different from the origin, then
, not through the center of the ellipse.
This relation between points and lines is a bijection.
The inverse function maps
and
Such a relation between points and lines generated by a conic is called pole-polar relation or polarity. The pole is the point; the polar the line.
By calculation one can confirm the following properties of the pole-polar relation of the ellipse:
For a point (pole) on the ellipse, the polar is the tangent at this point (see diagram:
outside (see diagram:
For a point within the ellipse, the polar has no point with the ellipse in common (see diagram:
The intersection point of two polars is the pole of the line through their poles.
, the directrices can be constructed by compass and straightedge (see Inversive geometry).
Pole-polar relations exist for hyperbolas and parabolas as well.
Metric properties
All metric properties given below refer to an ellipse with equation except for the section on the area enclosed by a tilted ellipse, where the generalized form of Eq.() will be given.
Area
The area enclosed by an ellipse is:
However, using the same approach for the circumference would be fallacious – compare the integrals. It is also easy to rigorously prove the area formula using integration as follows. Equation (
(obtained by solving for flattening, then computing the semi-minor axis).
So far we have dealt with erect axes. However, some applications require tilted ellipses. In charged-particle beam optics, for instance, the enclosed area of an erect or tilted ellipse is an important property of the beam, its emittance. In this case a simple formula still applies, namely
are maximum values. It follows directly from Apollonios's theorem.
Circumference
is the complete elliptic integral of the second kind which is in general not an elementary function.
using Gauss's arithmetic-geometric mean; this is a quadratically converging iterative method (see here for details).
The exact infinite series is the double factorial).
James Ivory, Bessel and Kummer derived a series that converges much more rapidly. It is most concisely written in terms of the n = 1/2. For eccentricities less than 0.5 the error is at the limits of double-precision floating-point term.
Srinivasa Ramanujan gave two close approximations"; respectively. term.
Arc length
More generally, the arc length of a portion of the circumference, as a function of the angle subtended (or of any two points on the upper half of the ellipse), is given by an incomplete elliptic integral
are
is the circumference of a circumscribed concentric circle is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and the minor axes.
Given an ellipse whose axes are drawn, we can construct the endpoints of a particular elliptic arc whose length is one eighth of the ellipse's circumference using only straightedge and compass in a finite number of steps; for some specific shapes of ellipses, such as when the axes have a length ratio of , it is additionally possible to construct the endpoints of a particular arc whose length is one twelfth of the circumference. (The vertices and co-vertices are already endpoints of arcs whose length is one half or one quarter of the ellipse's circumference.) However, the general theory of straightedge-and-compass elliptic division appears to be unknown, unlike in the case of the circle and the lemniscate. The division in special cases has been investigated by Legendre in his classical treatise.
Curvature
The curvature is given by:
and the radius of curvatureThe radius of curvature of an ellipse, as a function of angle where e is the eccentricity.
Radius of curvature at the two vertices
Radius of curvature at the two co-verticesThe locus of all the centers of curvature is called an evolute. In the case of an ellipse, the evolute is an astroid.
In triangle geometry
Ellipses appear in triangle geometry as
Steiner ellipse: ellipse through the vertices of the triangle with center at the centroid,
inellipses: ellipses which touch the sides of a triangle. Special cases are the Steiner inellipse and the Mandart inellipse.
As plane sections of quadrics
Ellipses appear as plane sections of the following quadrics:
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 scannerss.
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.)
where
is the radius at apoapsis, i.e., the farthest distance of the orbit to the barycenter of the system, which is a focus of the ellipse
is the radius at periapsis, the closest distance
is the length of the semi-major axis
is their arithmetic mean is their geometric mean, and the semi-latus rectum is their harmonic mean
Harmonic oscillators
The general solution for a harmonic oscillator in two or more dimensions is also an ellipse. Such is the case, for instance, of a long pendulum that is free to move in two dimensions; of a mass attached to a fixed point by a perfectly elastic spring; or of any object that moves under influence of an attractive force that is directly proportional to its distance from a fixed attractor. Unlike Keplerian orbits, however, these "harmonic orbits" have the center of attraction at the geometric center of the ellipse, and have fairly simple equations of motion.
Phase visualization
In electronics, the relative phase of two sinusoidal signals can be compared by feeding them to the vertical and horizontal inputs of an oscilloscope. If the Lissajous figure display is an ellipse, rather than a straight line, the two signals are out of phase.
Elliptical gears
Two non-circular gears with the same elliptical outline, each pivoting around one focus and positioned at the proper angle, turn smoothly while maintaining contact at all times. Alternatively, they can be connected by a link chain or timing belt, or in the case of a bicycle the main chainring may be elliptical, or an ovoid similar to an ellipse in form. Such elliptical gears may be used in mechanical equipment to produce variable angular speed or torque from a constant rotation of the driving axle, or in the case of a bicycle to allow a varying crank rotation speed with inversely varying mechanical advantage.
Elliptical bicycle gears make it easier for the chain to slide off the cog when changing gears.
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.
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).
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.
Statistics and finance
In statistics, a bivariate random vector 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 the financial field 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.
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. Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.
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. 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 solving this problem.
See also
Cartesian oval, a generalization of the ellipse
Elliptic coordinates, an orthogonal coordinate system based on families of ellipses and hyperbolae
Elliptical distribution, in statistics
n-ellipse, a generalization of the ellipse for n foci
Spheroid, the ellipsoid obtained by rotating an ellipse about its major or minor axis
Stadium (geometry), a two-dimensional geometric shape constructed of a rectangle with semicircles at a pair of opposite sides
Steiner circumellipse, the unique ellipse circumscribing a triangle and sharing its centroid
Superellipse, a generalization of an ellipse that can look more rectangular or more "pointy"
True, eccentric, and mean anomaly
