Ellipsoid

Ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a type of quadric surface that is a higher dimensional analogue of an ellipse.

Definition

In mathematics, an ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a type of quadric surface that is a higher dimensional analogue of an ellipse.

Mathematical Description

The standard equation of an ellipsoid centered at the origin of a three-dimensional Cartesian coordinate system is

$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$

where a, b, and c are positive real numbers.

Properties

Ellipsoids have several interesting properties. They are closed, bounded, and smooth (i.e., they have a well-defined tangent at every point). They also have a well-defined volume and surface area, which can be calculated using integral calculus.

Applications

Ellipsoids have many applications in various fields such as physics, astronomy, geology, and medicine. For example, in medicine, the shape of red blood cells is often modeled as an ellipsoid.

See Also

• Ellipse
• Sphere
• Quadric

References

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  Ellipsoid
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  Ellipsoid with plane section
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  Ellipsoid example
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  Ellipsoid curvature
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  Ellipse with gardener's construction
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  Focal axes of ellipsoid
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  Focal axes of ellipsoid in XYZ
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  Ellipsoid with principal and secondary axes
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  Affine transformation of ellipsoid
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  Ellipsoid