An ellipse is a curve that is the locus of all points in the plane the sum of whose distances r_1 and r_2 from two fixed points F_1 and F_2 (the foci) separated by a distance of 2c is a given positive constant 2a (Hilbert and Cohn-Vossen 1999, p. 2). This results in the two-center bipolar coordinate equation

 r_1+r_2=2a,
(1)

where a is the semimajor axis and the origin of the coordinate system is at one of the foci. The corresponding parameter b is known as the semiminor axis.

The ellipse is a conic section and a Lissajous curve.

An ellipse can be specified in Mathematica using Circle[{x, y}, {a, b}].

If the endpoints of a segment are moved along two intersecting lines, a fixed point on the segment (or on the line that prolongs it) describes an arc of an ellipse. This is known as the trammel construction of an ellipse (Eves 1965, p. 177).

It is possible to construct elliptical gears that rotate smoothly against one another (Brown 1871, pp. 14-15; Reuleaux and Kennedy 1876, p. 70; Clark and Downward 1930; KMODDL).

The ellipse was first studied by Menaechmus, investigated by Euclid, and named by Apollonius. The focus and conic section directrix of an ellipse were considered by Pappus. In 1602, Kepler believed that the orbit of Mars was oval; he later discovered that it was an ellipse with the Sun at one focus. In fact, Kepler introduced the word "focus" and published his discovery in 1609. In 1705 Halley showed that the comet now named after him moved in an elliptical orbit around the Sun (MacTutor Archive). An ellipse rotated about its minor axis gives an oblate spheroid, while an ellipse rotated about its major axis gives a prolate spheroid.

A ray of light passing through a focus will pass through the other focus after a single bounce (Hilbert and Cohn-Vossen 1999, p. 3). Reflections not passing through a focus will be tangent to a confocal hyperbola or ellipse, depending on whether the ray passes between the foci or not.

Let an ellipse lie along the x-axis and find the equation of the figure (1) where F_1 and F_2 are at (-c,0)

Bring the second term to the right side and square both sides,

Now solve for the square root term and simplify

      

                             

                              

Square one final time to clear the remaining square root,

Grouping the x terms then gives

which can be written in the simple form

Defining a new constant

puts the equation in the particularly simple form

The parameter b is called the semiminor axis by analogy with the parameter a, which is called the semimajor axis (assuming b<a). The fact that b as defined above is actually the semiminor axis is easily shown by letting r_1 and r_2 be equal. Then two right triangles are produced, each with hypotenuse a, base c, and height b=sqrt(a^2-c^2). Since the largest distance along the minor axis will be achieved at this point, b is indeed the semiminor axis.

If, instead of being centered at (0, 0), the center of the ellipse is at (x_0, y_0), equation (¡Þ) becomes

As can be seen from the Cartesian equation for the ellipse, the curve can also be given by a simple parametric form analogous to that of a circle, but with the x and y coordinates having different scalings,

          

           

The general quadratic curve

is an ellipse when, after defining

                  

                   

                    

Delta!=0, J>0, and Delta/I<0. Also assume the ellipse is nondegenerate (i.e., it is not a circle, so a!=c, and we have already established is not a point, since J=ac-b^2!=0). In that case, the center of the ellipse (x_0,y_0) is given by

                  

                  

the semi-axes lengths are

                      

                      

and the counterclockwise angle of rotation from the x-axis to the major axis of the ellipse is

       

The ellipse can also be defined as the locus of points whose distance from the focus is proportional to the horizontal distance from a vertical line known as the conic section directrix, where the ratio is <1. Letting r be the ratio and d the distance from the center at which the directrix lies, then in order for this to be true, it must hold at the extremes of the major and minor axes, so

(24)

Solving gives

         

          

The focal parameter of the ellipse is

           

              

              

where e is a characteristic of the ellipse known as the eccentricity, to be defined shortly.

An ellipse whose axes are parallel to the coordinate axes is uniquely determined by any four non-concyclic points on it, and the ellipse passing through the four points (x_1,y_1), (x_2,y_2), (x_3,y_3), and (x_4,y_4) has equation

          

Let four points on an ellipse with axes parallel to the coordinate axes have angular coordinates t_i for , 2, 3, and 4. Such points are concyclic when

(31)

where the intermediate variable  has been defined (Berger et al. 1984; Trott 2006, pp. 39-40). Rather surprisingly, this same relationship results after simplification of the above where s_i is now interpreted as . An equivalent, but more complicated, condition is given by

(32)

Like hyperbolas, noncircular ellipses have two distinct foci and two associated directrices, each conic section directrix being perpendicular to the line joining the two foci (Eves 1965, p. 275).

Define a new constant  called the eccentricity (where  is the case of a circle) to replace

(33)

from which it follows that
b =
(34)
(35)
=
(36)
=
(37)
(38)
(39)

                         

The eccentricity can therefore be interpreted as the position of the focus as a fraction of the semimajor axis.

If r and theta are measured from a focus F instead of from the center C (as they commonly are in orbital mechanics) then the equations of the ellipse are
x
(40)
y
(41)

and (¡Þ) becomes

(42)

Clearing the denominators gives

(43)

Substituting in  gives

(44)

Plugging in to re-express b and c in terms of a and e,

(45)

Dividing by  and simplifying gives

(46)

which can be solved for r to obtain

(47)

The sign can be determined by requiring that r must be positive. When , (47) becomes , but since  is always positive, we must take the negative sign, so (47) becomes

(48)
(49)
(50)

The distance from a focus to a point with horizontal coordinate x (where the origin is taken to lie at the center of the ellipse) is found from