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elipsoid

An ellipsoid, sometimes spelled elipsoid, is a smooth, closed quadric surface that generalizes a sphere to three unequal axes. In its standard form, it consists of all points (x, y, z) satisfying x^2/a^2 + y^2/b^2 + z^2/c^2 = 1, where a, b, and c > 0 are the semi-axes along the coordinate directions. The center is at the origin; translating or rotating yields a general ellipsoid.

The ellipsoid is convex and symmetric about its three principal axes. The lengths of its principal diameters

Volume and surface area: The volume is V = 4/3 π a b c. The surface area does not

Special cases: If a = b = c, the ellipsoid reduces to a sphere. If two axes are equal,

Applications: Ellipsoids model many physical and geometric shapes. In geodesy, the Earth is approximated by an

are
2a,
2b,
and
2c.
It
can
also
be
viewed
as
the
image
of
a
sphere
under
an
affine
transformation,
or
as
the
level
set
of
a
positive
definite
quadratic
form.
have
a
simple
closed-form
expression
for
general
a,
b,
c
and
involves
elliptic
integrals;
in
the
special
case
a
=
b,
formulas
exist
for
oblate
or
prolate
spheroids.
the
figure
is
a
spheroid
(oblate
if
the
polar
axis
c
is
shorter
than
a,
prolate
if
it
is
longer).
oblate
spheroid;
in
astronomy
and
optics,
near-elliptical
bodies
are
common;
in
computer
graphics
and
statistics,
ellipsoids
define
shapes
and
confidence
regions;
in
optimization,
ellipsoidal
methods
use
quadratic
forms
to
describe
feasible
regions.