In mathematics, an ellipse is a curve in a plane surrounding two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a special type of an ellipse having both focal points at the same location. The shape of an ellipse (how "elongated" it is) is represented by its eccentricity, which for an ellipse can be any number from 0 (the limiting case of a circle) to arbitrarily close to but less than 1.
Ellipses are the closed type of conic section: a plane curve resulting from the intersection of a cone by a plane (see figure to the right). Ellipses have many similarities with the other two forms of conic sections: parabolas and hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder.
Analytically, an ellipse may also be defined as the set of points such that the ratio of the distance of each point on the curve from a given point (called a focus or focal point) to the distance from that same point on the curve to a given line (called the directrix) is a constant. This ratio is called the eccentricity of the ellipse.
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 our solar system is approximately an ellipse with the barycenter of the planetSun 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, (élleipsis, "omission"), was given by Apollonius of Perga in his Conics, emphasizing the connection of the curve with "application of areas".
An ellipse can be defined geometrically as a set of points (locus of points) in the Euclidean plane:
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. It contains the vertices , which have distance to the center. The distance of the foci to the center is called the focal distance or linear eccentricity. The quotient is the eccentricity .
The case yields a circle and is included.
The equation can be viewed in a different way (see picture):
If is the circle with midpoint and radius , then the distance of a point to the circle equals the distance to the focus :
is called the director circle (related to focus ) of the ellipse. This property should not be confused with the definition of an ellipse with help of a directrix (line) below.
Using Dandelin spheres one proves easily the important statement:
If Cartesian coordinates are introduced such that the origin is the center of the ellipse and the xaxis is the major axis and
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:
The shape parameters are called the semi major axis and semi minor axis. The points are the covertices.
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.
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 semilatus rectum . A calculation shows
The semilatus rectum may also be viewed as the radius of curvature of the osculating circles at the vertices .
The simplest way to determine the equation of the tangent at a point is to implicitly differentiate the equation of the ellipse. This produces
With respect to , the equation of the tangent at point is,
As a vector equation, we have
A particular tangent line distinguishes the ellipse from the other conic sections.^{[1]} Let f be the distance from the vertex V (on both the ellipse and its major axis) to the nearer focus. Then the distance, along a line perpendicular to the major axis, from that focus to a point P on the ellipse is less than 2f. The tangent to the ellipse at P intersects the major axis at point Q at an angle ?PQV of less than 45°.
If the ellipse is shifted such that its center is the equation is
The axes are still parallel to the x and yaxes.
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.^{[2]}
The parameter t (called the eccentric anomaly in astronomy) is not the angle of with the xaxis (see diagram at right). For other interpretations of parameter t see section Drawing ellipses.
With the substitution and trigonometric formulae one gets
and the rational parametric equation of an ellipse
For this formula represents the quarter ellipse centered at the origin with radii and moving counterclockwise with increasing It is easy to test this by computing and
A shifted ellipse with center can be described by
A parametric representation of an arbitrary ellipse is contained in the section Ellipse as an affine image of the unit circle x²+y²=1 below.
The parameters and represent the lengths of line segments and are therefore nonnegative real numbers. Throughout this article is the semimajor 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 semimajor 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
The two lines at distance and parallel to the minor axis are called directrices of the ellipse (see diagram).
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):
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.)
Let and assume is a point on the curve. The directrix has equation . With , the relation produces the equations
The substitution yields
This is the equation of an ellipse () or a parabola () or a hyperbola (). All of these nondegenerate 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 xaxis as major axis and the major/minor semi axis .
If the focus is and the directrix one gets the equation
(The right side of the equation uses the Hesse normal form of a line to calculate the distance .)
For an ellipse the following statement is true:
Because the tangent is perpendicular to the normal, the statement is true for the tangent and the complementary angle of the lines to the foci (see diagram), too.
Let be the point on the line with the distance to the focus , is the semi major axis of the ellipse. Line is the bisector of the angle between the lines . In order to prove that is the tangent line at point , one checks that any point on line which is different from cannot be on the ellipse. Hence has only point in common with the ellipse and is, therefore, the tangent at point .
From the diagram and the triangle inequality one recognizes that holds, which means: . But if is a point of the ellipse, the sum should be .
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).
Another definition of an Ellipse uses affine transformations:
An affine transformation of the Euclidean plane has the form , where is a regular matrix (its determinant is not 0) and is an arbitrary vector. If are the column vectors of the matrix , the unit circle is mapped onto the Ellipse
is the center, are the directions of two conjugate diameters of the ellipse. In general the vectors are not perpendicular. That means, in general and are not the vertices of the ellipse.
The tangent vector at point is
Because at a vertex the tangent is perpendicular to the major/minor axis (diameters) of the ellipse one gets the parameter of a vertex from the equation
and hence
(The formulae were used.)
If , then .
The 4 vertices of the ellipse are
The advantage of this definition is that one gets a simple parametric representation of an arbitrary ellipse, even in the space, if the vectors are vectors of the Euclidean space.
For a circle, obviously
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.
Two diameters of an ellipse are conjugate if the midpoints of chords parallel to lie on
From the diagram one finds:
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.
For the ellipse the intersection points of orthogonal tangents lie on the circle .
This circle is called orthoptic of the given ellipse.
For an ellipse with semiaxes the following is true:
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
Ellipses appear in descriptive geometry as images (parallel or central projection) of circles (for details: see Ellipses in DG (German)). So it is essential to have tools to draw an ellipse. Nowadays the best tool is the computer. During the times before this tool was not available and one was restricted to compass and ruler for the construction of single points of an ellipse. But there are technical tools (ellipsographs) to draw an ellipse in a continuous way like a compass for drawing a circle, too. The principle of ellipsographs were known to Greek mathematicians (Archimedes, Proklos) already.
If there is no ellipsograph available, the best and quickest way to draw an ellipse is to draw an Approximation by the four osculating circles at the vertices.
For any method described below
If this presumption is not fulfilled one has to know at least two conjugate diameters. With help of Rytz's construction the axes and semiaxes can be retrieved.
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 bedthus 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.
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.
The first method starts with
The point, where the semi axes meet is marked by . If the strip slides with both ends on the axes of the desired ellipse, then point P traces the ellipse. For the proof one shows that point has the parametric representation , where parameter is the angle of the slope of the paper strip.
A technichal realization of the motion of the paper strip can be achieved by a Tusi couple (s. animation). The device is able to draw any ellipse with a fixed sum , which is the radius of the large circle. This restriction may be a disadvantage in real life. More flexible is the second paper strip method.
A nice application: If one stands somewhere in the middle of a ladder, which stands on a slippery ground and leans on a slippery wall, the ladder slides down and the persons feet trace an ellipse.
A variation of the paper strip method 1^{[3]} 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.
The second method starts with
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).
Remark: 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.
Trammel of Archimedes (principle)
Ellipsograph due to Benjamin Bramer
From section metric properties one gets:
The diagram shows an easy way to find the centers of curvature at vertex and covertex , resp.:
(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.
The following method to construct single points of an ellipse relies on the Steiner generation of a non degenerate conic section:
For the generation of points of the ellipse one uses the pencils at the vertices . Let be an upper covertex 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.
Remark:
Most technical instruments for drawing ellipses are based on the second paperstrip method.
For more principles of ellipsographs:
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:
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:
Inscribed angle theorem for circles:
At first the masure is available for chords, which are not parallel to the yaxis, only. But the final formula works for any chord.
A consequence of the inscribed angle theorem for circles is the
3pointform of a circle's equation:
In this section one considers ellipses with an equation
where the ratio is fixed. With the abbreviation one gets the more convenient form
Such ellipses have their axes parallel to the coordinate axes and their eccentricity fixed. Their major axes are parallel to the xaxis if and parallel to the yaxis if .
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,:^{[4]}^{[5]}
At first the measure is available only for chords which are not parallel to the yaxis. 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
3pointform of an ellipse's equation:
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
This relation between points and lines is a bijection.
The inverse function maps
Such a relation between points and lines generated by a conic is called polepolar relation or just polarity. The pole is the point, the polar the line. See Pole and polar.
By calculation one checks the following properties of the polepolar relation of the ellipse:
Remarks:
Polepolar relations exist for hyperbolas and parabolas, too.
All metric properties given below refer to an ellipse with equation .
The area enclosed by an ellipse is:
where and are the lengths of the semimajor and semiminor 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 circumference of an ellipse is:
where again is the length of the semimajor axis, is the eccentricity and the function is the complete elliptic integral of the second kind,
which calculates the circumference of the ellipse in the first quadrant alone, and the formula for the circumference of an ellipse can thus be written

The arc length of an ellipse, in general, has no closedform solution in terms of elementary functions. Elliptic integrals were motivated by this problem. Equation (3) may be evaluated directly using the Carlson symmetric form.^{[6]} This gives a succinct and quadratically converging iterative method for evaluating the circumference using the arithmeticgeometric mean.^{[7]}
The exact infinite series is:
where is the double factorial. Unfortunately, this series converges rather slowly; however, by expanding in terms of Ivory^{[8]} and Bessel^{[9]} derived an expression that converges much more rapidly,
Ramanujan gives two good approximations for the circumference in §16 of "Modular Equations and Approximations to ";^{[10]} 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^{[11]}
Here the upper bound is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and minor axes.
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 covertices and the centers of curvature:
In analytic geometry, the ellipse is defined as a quadric: the set of points of the Cartesian plane that, in nondegenerate cases, satisfy the implicit equation^{[12]}^{[13]}
provided
To distinguish the degenerate cases from the nondegenerate case, let ? be the determinant
that is,
Then the ellipse is a nondegenerate real ellipse if and only if C? < 0. If C? > 0, we have an imaginary ellipse, and if ? = 0, we have a point ellipse.^{[14]}^{:p.63}
The general equation's coefficients can be obtained from known semimajor axis , semiminor 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:
Let . Through change of coordinates (a rotation of axes and a translation of axes) the general ellipse can be described by the canonical implicit equation
Here are the point coordinates in the canonical system, whose origin is the center of the ellipse, whose axis is the unit vector coinciding with the major axis, and whose axis is the perpendicular vector coinciding with the minor axis. That is, and .
In this system, the center is the origin and the foci are and .
Any ellipse can be obtained by rotation and translation of a canonical ellipse with the proper semidiameters. 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.
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^{[14]}^{:p. 75}
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 semilatus 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.
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 appear as plane sections of the following quadrics:
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 wallbouncing 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 crosssection 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 UrbanaChampaign Foellinger Auditorium; and also at a side chamber of the Palace of Charles V, in the Alhambra.
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 twobody problem, if the two bodies are bound to each other (that is, the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at the same focus.
Keplerian elliptical orbits are the result of any radially directed attraction force whose strength is inversely proportional to the square of the distance. Thus, in principle, the motion of two oppositely charged particles in empty space would also be an ellipse. (However, this conclusion ignores losses due to electromagnetic radiation and quantum effects, which become significant when the particles are moving at high speed.)
For elliptical orbits, useful relations involving the eccentricity are:
where
Also, in terms of and , the semimajor axis is their arithmetic mean, the semiminor axis is their geometric mean, and the semilatus rectum is their harmonic mean. In other words,
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 display is an ellipse, rather than a straight line, the two signals are out of phase.
Two noncircular 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.^{[15]}
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.^{[16]}
In statistics, a bivariate random vector (X, Y) is jointly elliptically distributed if its isodensity contoursloci of equal values of the density functionare ellipses. The concept extends to an arbitrary number of elements of the random vector, in which case in general the isodensity 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 variancethat is, any two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return.^{[19]}^{[20]}
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.^{[21]} Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.^{[22]}
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.^{[23]} 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.
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.