An ellipse obtained as the intersection of a cone with a
plane.
The rings of Saturn are circular, but when seen
partially edge on, as in this photograph, they appear to be ellipses. Photo by ESO
In geometry, an ellipse is a plane curve that results from the
intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses,
obtained when the cutting plane is orthogonal to the cone's axis. An ellipse is
also the locus of all points of the plane
whose distances to two fixed points add to the same constant.
Ellipses are closed curves and are the bounded case of the conic sections, the curves that
result from the intersection of a circular cone and a plane that does not pass
through its apex; the other two (open and unbounded) cases are parabolas and hyperbolas. 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.
Elements
of an ellipse
The
ellipse and some of its mathematical properties.
An ellipse is a smooth closed curve which is symmetric about its horizontal and vertical
axes. The distance between antipodal points on the ellipse, or pairs of
points whose midpoint is at the center of the ellipse, is maximum along the major axis or transverse
diameter, and a minimum along the perpendicular minor axis or conjugate diameter.
The semi-major
axis (denoted by a in the figure) and the semi-minor axis (denoted by b in the figure) are one half of the
major and minor diameters, respectively. These are sometimes called (especially
in technical fields) the major and minor
semi-axes, the major and minor
semi axes, or major radius and minor
radius.
The foci of the ellipse are two special points F_{1} and F_{2} on the ellipse's major axis and are
equidistant from the center point. The sum of the distances from any point P on
the ellipse to those two foci is constant and equal to the major diameter ( PF_{1} + PF_{2} = 2a ). Each of these two points is called
a focus of the ellipse.
The eccentricity of an ellipse, usually denoted
by ε or e, is the ratio of the distance
between the two foci, to the length of the major axis or e = 2f/2a = f/a.
For an ellipse the eccentricity is between 0 and 1 (0<e<1). When
the eccentricity is 0 the foci coincide with the center point and the figure is
a circle. As the eccentricity tends toward 1, the ellipse gets a more elongated shape. It tends towards a
line segment if the two foci remain a finite distance apart and a parabola if one focus is kept fixed as
the other is allowed to move arbitrarily far away.
The distance ae from a
focal point to the centre is called the linear
eccentricity of the ellipse (f = ae).
In Euclidean geometry
In Euclidean geometry, an
ellipse is usually defined as the bounded case of a conic section, or as the
set of points such that the sum of the distances to two fixed points is
constant.
Equations
The equation of an ellipse whose major and minor axes coincide with
the Cartesian axes is
(Our solved example in mathguru.com uses
this concept)
http://en.wikipedia.org/wiki/Ellipse
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