Take a circle
in the *xy*-plane. Draw lines parallel to the *z*-axis and intersecting
.
The surface we get is a cylinder whose basis is the circle
.
In Fig 1(a), the
equation of the circle is *x*^{2}+*y*^{2}=1; as a point of the space belongs to the cyclinder if, and only if, its
projection onto the *xy*-plane is a point of the circle, this equation defines the cylinder.

Let *y*=*x*^{2} be the equation of a parabola
in the *xy*-plane.
Draw lines parallel to the *z*-axis and intersecting
.
The surface we get is a cylinder
whose basis is the parabola
(cd Fig 1(b)). The equation of the parabolic
cylinder is *y*=*x*^{2} too.

where

Substitute -*x* instead of *x* (resp. -*y* instead of *y*, resp. -*z* instead of *z*); the equation is not modified, thus the ellipsoid is symmetric about the *yz*-plane (resp. the *xz*-plane, resp. the *xy*-plane).

If *a*=*b*=*c*, the surface is a sphere.

where

where

Substitute -*x* instead of *x* (resp. -*y* instead of *y*, resp. -*z* instead of *z*); the equation is not modified, thus the ellipsoid is symmetric about the *yz*-plane (resp. the *xz*-plane, resp. the *xy*-plane).

The surface whose equation is

where

where

Substitute -*x* instead of *x* (resp. -*y* instead of *y*); the equation is not modified, thus the ellipsoid is symmetric about the *yz*-plane (resp. the *xz*-plane).

The surface whose equation is

where