An ellipse that is centered at the origin is the curve in the plane determined by
Below is a graph of the ellipse :
The major axis of an ellipse is the longest line segment whose endpoints are on the ellipse. The minor axis of an ellipse is the shortest line segment through the midpoint of the ellipse whose endpoints are on the ellipse.
In the first equation given above, if , the ellipse reduces to a circle of radius , whereas if (as in the graph above), is said to be the major semi-axis length and the minor semi-axis length; i.e. (http://planetmath.org/Ie), the lengths of the major axis and minor axis are and , respectively.
In terms of the geometric look of , there are three possible scenarios for : , , the line segment with end-points and , or is an ellipse. Points and are called foci of the ellipse; the line segments connecting a point of the ellipse to the foci are the focal radii belonging to that point. When and , is a circle. Under appropriate linear transformations (a translation followed by a rotation), has an algebraic appearance expressed in (1).
In polar coordinates, the ellipse is parametrized as
If , the foci of the ellipse (1) are on the -axis with distances from the origin. The constant sum of the of a point is equal to .
The normal line of the ellipse at its point halves the angle between the focal radii drawn from .
The area of an ellipse is . (See this page (http://planetmath.org/AreaOfPlaneRegion).)
By definition, the eccentricity () of the ellipse is given by
For , the ellipse reduces to a circle. Further, , and by assuming that foci are located on -axis, on and on , then , where is the origin of the rectangular coordinate system.
Polar equation of the ellipse
By translating the -axis towards the focus , we have
but from (1) we get
By using the transformation equations to polar coordinates
and through (2) we arrive at the polar equation
This equation allows us to determine some additional properties about the ellipse:
Hence, the general definition of the ellipse expressed above shows that and also that the arithmetic mean corresponds to the major semi-axis, while the geometric mean corresponds to the minor semi-axis of the ellipse. Likewise, if is the angle between the polar axis and the radial distance , where is the point of the ellipse over the -axis, then we get the useful equation .
|Date of creation||2013-03-22 15:18:10|
|Last modified on||2013-03-22 15:18:10|
|Last modified by||matte (1858)|