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Spheroid
 
 
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A spheroid is a quadric
Quadric

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of root of a quadratic polynomial....
 surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 obtained by rotating an ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
 about one of its principal axes; in other words, an ellipsoid
Ellipsoid

An ellipsoid is a type of Quadric that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is...
 with two equal semi-diameters.

If the ellipse is rotated about its major axis, the result is a prolate
Prolate spheroid

A prolate spheroid is a spheroid in which the polar diameter is greater than the equatorial diameter....
 (elongated) spheroid, like a rugby
Rugby football

Rugby football may refer to a number of sports through history descended from a common form of football developed in different areas of England....
 ball.






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Oblatespheroid
Prolatespheroid
oblate spheroidprolate spheroid


A spheroid is a quadric
Quadric

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of root of a quadratic polynomial....
 surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 obtained by rotating an ellipse
Ellipse

In mathematics, an ellipse is the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside....
 about one of its principal axes; in other words, an ellipsoid
Ellipsoid

An ellipsoid is a type of Quadric that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is...
 with two equal semi-diameters.

If the ellipse is rotated about its major axis, the result is a prolate
Prolate spheroid

A prolate spheroid is a spheroid in which the polar diameter is greater than the equatorial diameter....
 (elongated) spheroid, like a rugby
Rugby football

Rugby football may refer to a number of sports through history descended from a common form of football developed in different areas of England....
 ball. If the ellipse is rotated about its minor axis, the result is an oblate (flattened) spheroid, like a lentil
Lentil

The lentil or daal or pulse is a bushy annual plant of the Fabaceae family, grown for its lens-shaped seeds. It is about 15 inches tall and the seeds grow in pods, usually with two seeds in each....
. If the generating ellipse is a circle, the surface is a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
.

Because of its rotation, the Earth
Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
's shape is more like an oblate spheroid than a sphere. In cartography
Cartography

File:Mediterranean chart fourteenth century2.jpgCartography is the study and practice of making Geography Map. Combining science, aesthetics, and technique, cartography builds on the premise that we can model reality in ways that communicate spatial information effectively....
, in fact, the Earth is often assumed to be a standard oblate spheroid, with the current World Geodetic System
World Geodetic System

The World Geodetic System is a standard for use in cartography, geodesy, and navigation. It comprises a standard Cartesian coordinates for the Earth, a standard spheroid reference surface for raw altitude data, and a gravitation equipotential surface that defines the "nominal sea level"....
 model being a ˜ 6,378.137 km and b ˜ 6,356.752 km (a difference of over 21 km).

Equation

A spheroid centered at the origin and rotated about the z axis is defined by the implicit
Implicit function

In mathematics, an implicit function is a function in which the dependent variable has not been given "explicitly" in terms of the independent variable....
 equation where a is the horizontal, transverse radius at the equator, and b is the vertical, conjugate radius.

Surface area

A prolate spheroid has surface area
Surface area

Surface area is how much exposed area an object has. It is expressed in square units. If an object has flat Face , its surface area can be calculated by adding together the areas of its faces....
 where is the angular eccentricity
Angular eccentricity

In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and Eccentricity ....
 of the prolate spheroid, and is its (ordinary) eccentricity
Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or , is a parameter associated with every Conic section#Eccentricity. It can be thought of as a measure of how much the conic section deviates from being circular....
.

An oblate spheroid has surface area where is the angular eccentricity
Angular eccentricity

In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and Eccentricity ....
 of the oblate spheroid.

Volume

The volume of a spheroid (of any kind) is

Curvature

If a spheroid is parameterized as where is the reduced or parametric latitude
Latitude

Latitude, usually denoted symbolically by the Greek letter phi gives the location of a place on Earth north or south of the equator. Lines of Latitude are the horizontal lines shown running east-to-west on maps ....
, is the longitude
Longitude

Longitude , symbolized by the Greek character lambda , is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement....
, and and , then its Gaussian curvature
Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, ?1 and ?2, of the given point....
 is and its mean curvature
Mean curvature

In mathematics, the mean curvature of a surface is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedding surface in some ambient space such as Euclidean space....
 is Both of these curvatures are always positive, so that every point on a spheroid is elliptic.

See also

  • Ellipsoid
    Ellipsoid

    An ellipsoid is a type of Quadric that is a higher dimensional analogue of an ellipse. The equation of a standard axis-aligned ellipsoid body in an xyz-Cartesian coordinate system is...
  • Prolate spheroid
    Prolate spheroid

    A prolate spheroid is a spheroid in which the polar diameter is greater than the equatorial diameter....
  • Oblate spheroid
  • Scalene Ellipsoid
  • Ovoid
  • Maclaurin spheroid


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