Everything about Ellipse totally explained
In
mathematics, an
ellipse (from the
Greek ἔλλειψις, literally
absence) is a
locus of points in a plane such that the sum of the
distances to two fixed points is a constant. The two fixed points are called
foci (singular-
focus). An alternate definition would be that an ellipse is the path traced out by a point whose distance from a fixed point, called the focus, maintains a constant ratio less than one with its distance from a straight line not passing through the focus, called the
directrix.
Overview
An ellipse is a type of
conic section: if a
conical surface is cut with a plane which doesn't intersect the cone's base, the intersection of the cone and plane is an ellipse. For a short elementary proof of this, see
Dandelin spheres.
Algebraically, an ellipse is a
curve in the
Cartesian plane defined by
an equation of the form
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such that
, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists.
An ellipse can be drawn with two pins, a loop of string, and a pencil. The pins are placed at the foci and the pins and pencil are enclosed inside the string. The pencil is placed on the paper inside the string, so the string is taut. The string will form a
triangle. If the pencil is moved around so that the string stays taut, the sum of the distances from the pencil to the pins will remain constant, satisfying the definition of an ellipse.
The
line segment AB, that passes through the foci and terminates on the ellipse, is called the
major axis. The major axis is the longest segment that can be obtained by joining two points on the ellipse. The line segment CD, which passes through the center (halfway between the foci),
perpendicular to the major axis, and terminates on the ellipse, is called the
minor axis. The
semimajor axis (denoted by
a in the figure) is one half the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the
semiminor axis (denoted by
b in the figure) is one half the minor axis.
If the two foci coincide, then the ellipse is a
circle; in other words, a circle is a special case of an ellipse, one where the
eccentricity is zero.
An ellipse centered at the
origin can be viewed as the image of the
unit circle under a linear map associated with a
symmetric matrix ,
being a
diagonal matrix with the
eigenvalues of
, both of which are real positive, along the main diagonal, and
being a real
unitary matrix having as columns the
eigenvectors of
. Then the axes of the ellipse will lie along the eigenvectors of
, and the (square root of the) eigenvalues are the lengths of the
semimajor and
semiminor axes.
An ellipse can be produced by multiplying the
x coordinates of all points on a circle by a constant, without changing the
y coordinates. This is equivalent to
stretching the circle out in the x-direction.
Eccentricity
The shape of an ellipse can be expressed by a number called the
eccentricity of the ellipse, conventionally denoted
. The eccentricity is a
non-negative number less than 1 and greater than or equal to 0. It is the value of the constant ratio of the distance of a point on an ellipse from a focus to that from the corresponding directrix. An eccentricity of 0 implies that the two foci occupy the same point and that the ellipse is a
circle.
It can also be expressed as the sine of the
angular eccentricity,
. For an ellipse with semimajor axis
a and semiminor axis
b,
the eccentricity is
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