An ellipsoid is a type of quadric surface that is a higher dimensional analogue of an ellipse. The equation of a standard ellipsoid body in an x-y-z Cartesian coordinate system is
x^2/a^2 + y^2/b^2 + z^2/c^2 = 1
where a and b are the equatorial radii (perpendicular to each other) and c is the polar radius (i.e., the lengths of the three semi-axes), all of which are fixed positive real numbers determining the shape of the ellipsoid.
(This contrasts with standard elliptical assignment, where the equatorial radius/semi-axis is defined as "a" and the polar as "b".) If all three radii are equal, the solid body is a sphere; if the two equatorial radii are equal, the ellipsoid is a spheroid:
- a=b=c Sphere;
- a=b>c Oblate spheroid (disk-shaped);
- a=b<c Prolate spheroid (cigar-shaped);
- a>b>c Scalene ellipsoid ("three unequal sides").