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, a number ranging from (the limiting case of a circle) to (the limiting case of infinite elongation, no longer an ellipse but a parabola).
An ellipse (red) obtained as the intersection of a cone with an inclined plane.
Ellipses: examples with increasing eccentricity
An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution.
Analytically, the equation of a standard ellipse centered at the origin with width and height is:
Assuming , the foci are for . The standard parametric equation is:
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. This constant ratio is the above-mentioned 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 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.
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 :
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 the minor axis. The major axis intersects the ellipse at two 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 as a special type of ellipse.
The equation can be viewed in a different way (see figure):
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. 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 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.
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 .
For an arbitrary point the distance to the focus is
and to the other focus . Hence the point is on the ellipse whenever:
Removing the radicals by suitable squarings and using produces the standard equation of the ellipse:
or, solved for y:
The width and height parameters are called the semi-major and semi-minor axes. The top and bottom points are the co-vertices. The distances from a point on the ellipse to the left and right foci are and .
It follows from the equation that the ellipse is symmetric with respect to the coordinate axes and hence with respect to the origin.
In principle, the canonical ellipse equation may have (and hence the ellipse would be taller than it is wide). This form can be converted to the standard form by transposing the variable names and and the parameter names and
An arbitrary line intersects an ellipse at , , or points, respectively called an exterior line, tangent and secant. Through any point of an ellipse there is a unique tangent. The tangent at a point of the ellipse has the coordinate equation:
Let be a point on an ellipse and be the equation of any line containing . Inserting the line's equation into the ellipse equation and respecting yields:
There are then cases:
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 .
Then line has a second point in common with the ellipse, and is a secant.
Using (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 , then the points lie on two conjugate diameters (see below). (If , the ellipse is a circle and "conjugate" means "orthogonal".)
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.:p.63
The general equation's coefficients can be obtained from known semi-major axis , semi-minor axis , center coordinates , and rotation angle (the angle from the positive horizontal axis to the ellipse's major axis) using the formulae:
These expressions can be derived from the canonical equation by an affine transformation of the coordinates :
Conversely, the canonical form parameters can be obtained from the general form coefficients by the equations:
Replacing and of the standard representation yields:
Here is the slope of the tangent at the corresponding ellipse point, is the upper and the lower half of the ellipse. The vertices, having vertical tangents, 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, without differential calculus and trigonometric formulae.
Any ellipse is an affine image of the unit circle with equation .
An affine transformation of the Euclidean plane has the form , where is a regular matrix (with non-zero determinant) and is an arbitrary vector. If are the column vectors of the matrix , the unit circle , , is mapped onto the ellipse:
Here is the center and are the directions of two conjugate diameters, in general not perpendicular.
The four vertices of the ellipse are , for a parameter defined by:
(If , then .) This is derived as follows. The tangent vector at point is:
At a vertex parameter , the tangent is perpendicular to the major/minor axes, so:
Expanding and applying the identities gives the equation for
From Apollonios theorem (see below) one obtains:
The area of an ellipse is
With the abbreviations
the statements of Apollonios's theorem can be written as:
Solving this nonlinear system for yields the semiaxes:
Solving the parametric representation for by Cramer's rule and using , one obtains the implicit representation
Each of the two lines parallel to the minor axis, and at a distance of from it, is called a directrix 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 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:
The extension to , 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 (with being the radius of the circle) in the projective plane.
(The choice yields a parabola, and if , a hyperbola.)
Pencil of conics with a common vertex and common semi-latus rectum
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 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 .
Construction of a directrix
Construction of a directrix
Because of point of directrix (see diagram) and focus are inverse with respect to the circle inversion at circle (in diagram green). Hence can be constructed as shown in the diagram. Directrix is the perpendicular to the main axis at point .
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 .)
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.
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: . The equality is true from the Angle bisector theorem because and . 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).
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 an ellipse with semi-axes the following is true:
Let and be halves of two conjugate diameters (see diagram) then
The triangle with sides (see diagram) has the constant area , which can be expressed by , too. is the altitude of point and the angle between the half diameters. Hence the area of the ellipse (see section metric properties) can be written as .
The parallelogram of tangents adjacent to the given conjugate diameters has the
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 obtains 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. 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.
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.
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 is tied at each end to the two pins; its length after tying is . 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.
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
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 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 (see 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.
Ellipse construction: paper strip method 1
Ellipses with Tusi couple. Two examples: red and cyan.
A variation of the paper strip method 1 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. This variation requires only one sliding shoe.
Variation of the paper strip method 1
Animation of the variation of the paper strip method 1
Ellipse construction: paper strip 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.
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 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.
Given four points , no three of them on a line (see diagram).
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.
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.
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. Because they are even polar pairs with respect to the circle , the directrices can be constructed by compass and straightedge (see Inversive geometry).
Pole-polar relations exist for hyperbolas and parabolas as well.
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).
The area enclosed by a tilted ellipse is .
So far we have dealt with erect ellipses, whose major and minor axes are parallel to the and 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
where is the double factorial (extended to negative odd integers by the recurrence relation , for ). This series converges, but by expanding in terms of James Ivory and Bessel derived an expression that converges much more rapidly:
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
Then the arc length from to is:
This is equivalent to
where is the incomplete elliptic integral of the second kind with parameter