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1. Ellipsoid

   Linked via "semi-axes"

   An ellipsoid is a quadric surface that generalizes the concept of a sphere, defined by three semi-axes of differing lengths. In three dimensions, the canonical equation of an ellipsoid centered at the origin is:
   $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $$
   where $a$, $b$, and $c$ are the lengths of the semi-axes along the $x$, $y$, and $z$ Cartesian coordinates, respectively. …

2. Ellipsoid

   Linked via "semi-axes"

   Types and Classification
   Ellipsoids are classified based on the relative lengths of their semi-axes. The distinction between these forms is crucial in fields where precise volumetric or surface area calculations are necessary, such as astrodynamics or theoretical hydrodynamics [1].
   Triaxial Ellipsoid

3. Ellipsoid

   Linked via "semi-axes"

   Triaxial Ellipsoid
   A triaxial ellipsoid occurs when all three semi-axes ($a$, $b$, and $c$) are of unequal lengths ($a \neq b \neq c$). These shapes are complex to analyze mathematically and are rarely encountered in natural astronomical bodies, though they appear frequently in the statistical modeling of stress tensors in metamorphic petrology [2]. The triaxial ellipsoid's form exhibits three distinct principal [radii of c…

4. Ellipsoid

   Linked via "semi-axes"

   Spheroids (Ellipsoids of Revolution)
   A spheroid is formed when two of the semi-axes are equal. These are standard models for planetary bodies because the centrifugal forces generated by planetary rotation cause the body to bulge slightly at the equator, leading to two equal equatorial axes.
   Prolate Spheroid