Ellipses: examples with increasing eccentricity
In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity e, a number ranging from e = 0 (the limiting case of a circle) to e = 1 (the limiting case of infinite elongation, no longer an ellipse but a parabola).
Analytically, the equation of a standard ellipse centered at the origin with width 2a and height 2b is:
Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane (see figure). Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. An angled cross section of a cylinder is also an ellipse.
Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in the solar system is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the barycenter of the Sun–planet pair). The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids. A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under parallel or perspective projection. The ellipse is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical polarization of light in optics.
The name, ἔλλειψις (élleipsis, "omission"), was given by Apollonius of Perga in his Conics.
Definition as locus of points
Ellipse: Definition by sum of distances to foci
Ellipse: Definition by focus and circular directrix
An ellipse can be defined geometrically as a set or locus of points in the Euclidean plane:
Given two fixed points called the foci and a distance which is greater than the distance between the foci, the ellipse is the set of points such that the sum of the distances is equal to :
Using Dandelin spheres, one can prove that any plane section of a cone with a plane is an ellipse, assuming the plane does not contain the apex and has slope less than that of the lines on the cone.
In Cartesian coordinates
The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, the x-axis is the major axis, and:
- the foci are the points,the vertices are.
or, solved for y:
It follows from the equation that the ellipse is symmetric with respect to the coordinate axes and hence with respect to the origin.
Semi-major and semi-minor axes a ≥ b
Linear eccentricity c
The eccentricity can be expressed as:
Semi-latus rectum l
A vectorparametric equationof the tangent is:
Case (1): Then line and the ellipse have only point in common, and is a tangent. The tangent direction has perpendicular vector , so the tangent line has equation for some . Because is on the tangent and the ellipse, one obtains .
Case (2): Then line has a second point in common with the ellipse, and is a secant.
The construction of points based on the parametric equation and the interpretation of parameter t, which is due to de la Hire
The axes are still parallel to the x- and y-axes.
Conversely, the canonical form parameters can be obtained from the general form coefficients by the equations:
Standard parametric representation
- in astronomy) is not the angle of
and the rational parametric equation of an ellipse
Tangent slope as parameter
With help of trigonometric formulae one obtains:
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.
Ellipse as an affine image of the unit circle
Another definition of an ellipse uses affine transformations:
Any ellipse is an affine image of the unit circle with equation .
Polar form relative to center
Polar coordinates centered at the center
Polar form relative to focus
Polar coordinates centered at focus
Eccentricity and the directrix property
Ellipse: directrix property
Pencil of conics with a common vertex and common semi-latus rectum
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 second case is proven analogously.
The converse is also true and can be used to define an ellipse (in a manner similar to the definition of a parabola):
For any point (focus), any line (directrix) not through , and any real number with the ellipse is the locus of points for which the quotient of the distances to the point and to the line is that is:
- General ellipse
Focus-to-focus reflection property
Ellipse: the tangent bisects the supplementary angle of the angle between the lines to the foci.
Rays from one focus reflect off the ellipse to pass through the other focus.
An ellipse possesses the following property:
The normal at a point bisects the angle between the lines .
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.
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).
Orthogonal diameters of a circle with a square of tangents, midpoints of parallel chords and an affine image, which is an ellipse with conjugate diameters, a parallelogram of tangents and midpoints of chords
A circle has the following property:
- The midpoints of parallel chords lie on a diameter.
An affine transformation preserves parallelism and midpoints of line segments, so this property is true for any ellipse. (Note that the parallel chords and the diameter are no longer orthogonal.)
From the diagram one finds:
- Two diametersof an ellipse are conjugate whenever the tangents atandare parallel to.
Conjugate diameters in an ellipse generalize orthogonal diameters in a circle.
In the parametric equation for a general ellipse given above,
Theorem of Apollonios on conjugate diameters
Ellipse: theorem of Apollonios on conjugate diameters
Let and be halves of two conjugate diameters (see diagram) then
the triangle formed by has the constant area
the parallelogram of tangents adjacent to the given conjugate diameters has the
Let the ellipse be in the canonical form with parametric equation
Ellipse with its orthoptic
This circle is called orthoptic or director circle of the ellipse (not to be confused with the circular directrix defined above).
Central projection of circles (gate)
Ellipses appear in descriptive geometry as images (parallel or central projection) of circles. There exist various tools to draw an ellipse. Computers provide the fastest and most accurate method for drawing an ellipse. However, technical tools (ellipsographs) to draw an ellipse without a computer exist. The principle of ellipsographs were known to Greek mathematicians such as Archimedes and Proklos.
If there is no ellipsograph available, one can draw an ellipse using an approximation by the four osculating circles at the vertices.
For any method described below
the knowledge of the axes and the semi-axes is necessary (or equivalent: the foci and the semi-major axis).
If this presumption is not fulfilled one has to know at least two conjugate diameters. With help of Rytz's construction the axes and semi-axes can be retrieved.
de La Hire's point construction
Ellipse: gardener's method
- (1) Draw the two circles centered at the center of the ellipse with radiiand the axes of the ellipse.(2) Draw a line through the center, which intersects the two circles at pointand, respectively.(3) Draw a line throughthat is parallel to the minor axis and a line throughthat is parallel to the major axis. These lines meet at an ellipse point (see diagram).(4) Repeat steps (2) and (3) with different lines through the center.
The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two drawing pins, a length of string, and a pencil. In this method, pins are pushed into the paper at two points, which become the ellipse's foci. A string tied at each end to the two pins and the tip of a pencil pulls the loop taut to form a triangle. The tip of the pencil then traces an ellipse if it is moved while keeping the string taut. Using two pegs and a rope, gardeners use this procedure to outline an elliptical flower bed—thus it is called the gardener's ellipse.
A similar method for drawing confocal ellipses with a closed string is due to the Irish bishop Charles Graves.
Paper strip methods
Ellipse construction: paper strip method 2
Approximation of an ellipse with osculating circles
The two following methods rely on the parametric representation (see section parametric representation, above):
- Method 1
The first method starts with
a strip of paper of length .
- Method 2
The second method starts with
a strip of paper of length .
This method is the base for several ellipsographs (see section below).
Similar to the variation of the paper strip method 1 a variation of the paper strip method 2 can be established (see diagram) by cutting the part between the axes into halves.
Most ellipsograph drafting instruments are based on the second paperstrip method.
Approximation by osculating circles
From Metric properties below, one obtains:
The radius of curvature at the vertices is:
The radius of curvature at the co-vertices is:
- (1) mark the auxiliary pointand draw the line segment(2) draw the line through, which is perpendicular to the line(3) the intersection points of this line with the axes are the centers of the osculating circles.
(proof: simple calculation.)
The centers for the remaining vertices are found by symmetry.
With help of a French curve one draws a curve, which has smooth contact to the osculating circles.
Ellipse: Steiner generation
Ellipse: Steiner generation
The following method to construct single points of an ellipse relies on the Steiner generation of a conic section:
Given two pencils of lines at two points (all lines containing and , respectively) and a projective but not perspective mapping of onto , then the intersection points of corresponding lines form a non-degenerate projective conic section.
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.
An ellipse (in red) as a special case of the hypotrochoid with R = 2r
Inscribed angles and three-point form
Circle: inscribed angle theorem
- for circles:
- For four points(see diagram) the following statement is true:The four points are on a circle if and only if the angles atandare equal.
Usually one measures inscribed angles by a degree or radian θ, but here the following measurement is more convenient:
In order to measure the angle between two lines with equations one uses the quotient:
Inscribed angle theorem for circles
The four points are on a circle, if and only if the angles at and are equal. In terms of the angle measurement above, this means:
At first the measure is available only for chords not parallel to the y-axis, but the final formula works for any chord.
Three-point form of circle equation
- As a consequence, one obtains an equation for the circle determined by three non-colinear points:
- , which can be rearranged to
The radius is the distance between any of the three points and the center.
Inscribed angle theorem for an ellipse
and to write the ellipse equation as:
Like a circle, such an ellipse is determined by three points not on a line.
In order to measure an angle between two lines with equations one uses the quotient:
Inscribed angle theorem for ellipses
- Given four points, no three of them on a line (see diagram).The four points are on an ellipse with equationif and only if the angles atandare equal in the sense of the measurement above—that is, if
At first the measure is available only for chords which are not parallel to the y-axis. But the final formula works for any chord. The proof follows from a straightforward calculation. For the direction of proof given that the points are on an ellipse, one can assume that the center of the ellipse is the origin.
Three-point form of ellipse equation
- A consequence, one obtains an equation for the ellipse determined by three non-colinear points:
- and after conversion
Analogously to the circle case, the equation can be written more clearly using vectors:
Ellipse: pole-polar relation
point is mapped onto the line , not through the center of the ellipse.
The inverse function maps
line onto the point and
- lineonto the point
Such a relation between points and lines generated by a conic is called pole-polar relation or polarity. The pole is the point, the polar the line.
By calculation one can confirm the following properties of the pole-polar relation of the ellipse:
For a point (pole) on the ellipse the polar is the tangent at this point (see diagram: ).
For a pole outside the ellipse the intersection points of its polar with the ellipse are the tangency points of the two tangents passing (see diagram: ).
For a point within the ellipse the polar has no point with the ellipse in common. (see diagram: ).
The intersection point of two polars is the pole of the line through their poles.
The foci and respectively and the directrices and respectively belong to pairs of pole and polar.
Pole-polar relations exist for hyperbolas and parabolas, too.
The exact infinite series is:
More generally, the arc length of a portion of the circumference, as a function of the angle subtended (or x-coordinates of any two points on the upper half of the ellipse), is given by an incomplete elliptic integral. The upper half of an ellipse is parameterized by
This is equivalent to
The inverse function, the angle subtended as a function of the arc length, is given by a certain elliptic function.
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:
Hyperboloid of one sheet
Hyperboloid of two sheets
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.
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.)
is the radius at apoapsis (the farthest distance)
is the radius at periapsis (the closest distance)
is the length of the semi-major axis
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.
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.
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.
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.
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 (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.
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.
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.
Apollonius of Perga, the classical authority
Cartesian oval, a generalization of the ellipse
Circumconic and inconic
Ellipsoid, a higher dimensional analog of an ellipse
Elliptic coordinates, an orthogonal coordinate system based on families of ellipses and hyperbolae
Elliptic partial differential equation
Elliptical distribution, in statistics
Geodesics on an ellipsoid
Kepler's laws of planetary motion
Matrix representation of conic sections
n-ellipse, a generalization of the ellipse for n foci
Rytz’s construction, a method for finding the ellipse axes from conjugate diameters or a parallelogram
Spheroid, the ellipsoid obtained by rotating an ellipse about its major or minor axis
Stadium (geometry), a two-dimensional geometric shape constructed of a rectangle with semicircles at a pair of opposite sides
Steiner circumellipse, the unique ellipse circumscribing a triangle and sharing its centroid
Steiner inellipse, the unique ellipse inscribed in a triangle with tangencies at the sides' midpoints
Superellipse, a generalization of an ellipse that can look more rectangular or more "pointy"
True, eccentric, and mean anomaly