An ellipseMathworldPlanetmathPlanetmath that is centered at the origin is the curve in the plane determined by

(xa)2+(yb)2=1, (1)

where a,b>0.

Below is a graph of the ellipse (x3)2+(y2)2=1:

The major axis of an ellipse is the longest line segmentMathworldPlanetmath whose endpoints are on the ellipse. The minor axis of an ellipse is the shortest line segment through the midpointMathworldPlanetmathPlanetmathPlanetmath of the ellipse whose endpoints are on the ellipse.

In the first equation given above, if a=b, the ellipse reduces to a circle of radius a, whereas if a>b (as in the graph above), a is said to be the major semi-axis length and b the minor semi-axis length; i.e. (http://planetmath.org/Ie), the lengths of the major axis and minor axis are 2a and 2b, respectively.

More generally, given any two points p1 and p2 in the (Euclidean) plane and any real number r, let E be the set of points p having the property that the sum of the distancesMathworldPlanetmath from p to p1 and p2 is r; i.e.,


In terms of the geometric look of E, there are three possible scenarios for E: E=, E=p1p2¯, the line segment with end-points p1 and p2, or E is an ellipse. Points p1 and p2 are called foci of the ellipse; the line segments connecting a point of the ellipse to the foci are the focal radiiPlanetmathPlanetmath belonging to that point. When p1=p2 and r>0, E is a circle. Under appropriate linear transformations (a translationMathworldPlanetmathPlanetmath followed by a rotationMathworldPlanetmath), E has an algebraic appearance expressed in (1).

In polar coordinatesMathworldPlanetmath, the ellipse is parametrized as

x(t) = acost,
y(t) = bsint,t[0, 2π).

If  a>b,  then t is the eccentric anomaly; i.e., the polar angleMathworldPlanetmath of the point on the circumscribed circle having the same abscissaMathworldPlanetmath as the point of the ellipse.


  1. 1.

    If  a>b,  the foci of the ellipse (1) are on the x-axis with distances a2-b2 from the origin.  The constant sum of the of a point p is equal to 2a.

  2. 2.

    The normal lineMathworldPlanetmath of the ellipse at its point p halves the angle between the focal radii drawn from p.

  3. 3.

    The area of an ellipse is πab. (See this page (http://planetmath.org/AreaOfPlaneRegion).)

  4. 4.

    The length of the perimeterPlanetmathPlanetmath of an ellipse can be expressed using an elliptic integralMathworldPlanetmath.


By definition, the eccentricity ϵ (0ϵ<1) of the ellipse is given by


For ϵ=0, the ellipse reduces to a circle. Further, b=a1-ϵ2, and by assuming that foci are located on x-axis, p1 on x<0 and p2 on x>0, then |O-p1|=|O-p2|=ϵa, where O(0,0) is the origin of the rectangular coordinate system.

Polar equation of the ellipse

By translating the y-axis towards the focus p1, we have

x = x+ϵa,
y = y,

but from (1) we get

(x-ϵaa)2+(yb)2=1. (2)

By using the transformation equations to polar coordinates

x = rcosθ,
y = rsinθ,

and through (2) we arrive at the polar equation

r(θ)=(1-ϵ2)a1-ϵcosθ (3)

This equation allows us to determine some additional properties about the ellipse:

rmax:=r(0)=(1+ϵ)a,which is called the 𝑎𝑝ℎ𝑒𝑙𝑖𝑢𝑚;
rmin:=r(π)=(1-ϵ)a,which is called the 𝑝𝑒𝑟𝑖ℎ𝑒𝑙𝑖𝑢𝑚.

Hence, the general definition of the ellipse expressed above shows that rmin+rmax=2a and also that the arithmetic meanMathworldPlanetmath rmin+rmax2=a corresponds to the major semi-axis, while the geometric meanMathworldPlanetmath rminrmax=b corresponds to the minor semi-axis of the ellipse. Likewise, if θϵ is the angle between the polar axis x and the radial distance |B-p1|, where B(0,b) is the point of the ellipse over the y-axis, then we get the useful equation cosθϵ=ϵ.

Title ellipse
Canonical name Ellipse
Date of creation 2013-03-22 15:18:10
Last modified on 2013-03-22 15:18:10
Owner matte (1858)
Last modified by matte (1858)
Numerical id 33
Author matte (1858)
Entry type Definition
Classification msc 53A04
Classification msc 51N20
Classification msc 51-00
Related topic SqueezingMathbbRn
Related topic EllipsoidMathworldPlanetmathPlanetmath
Defines major axis
Defines minor axis
Defines major semi-axis
Defines minor semi-axis
Defines focus
Defines foci
Defines aphelium
Defines perihelium
Defines eccentric anomaly
Defines focal radius
Defines focal radii