The most explicit way to specify a surface is to give the 3 cartesian
coordinates of an arbitrary point as functions of 2 parameters,
traditionally denoted u and v.
The equation of a surface is the relation satisfied by the coordinates of every point.

(2016-01-12) Cartesian Equation of a Plane
Let (a,b,c) be the point of the plane that's closest to the origin.

When (a,b,c) is not (0,0,0) the plane's cartesian equation is:

a x + b y + c z =
a^{2} + b^{2} + c^{2}

Otherwise, we're dealing with a plane going through the origin and shall
use any nonzero vector (a,b,g) orthogonal to the plane:

a x + b y
+ g z = 0

This can be construed as a limiting case of the previous equation.

(2016-01-16) Helicoid (Euler 1774, Meusnier 1776)
Horizontal line rotating at a rate proportional to its vertical velocity.

The cartesian parametric equations are:

x = u cos v

y = u sin v

z = k v

The equation in cylindrical coordinates is just:

z = k q

For a right-handed helicoid (as depicted above) the constant
k is positive. It's negative for a left-handed one.
The plane is an helicoid
(with k = 0).

The constant k is homogeneous to a length per unit of angle.
It's related to the wavelength a (the constant signed
vertical displacement between two consecutive sheets) by the following relation,
if angles are in radians:

a = 2 p k

An 1842 theorem due to Catalan (1814-1894)
states that helicoids (planes included) are the only
ruled minimal surfaces.
[ Proof ]

Generalized Helicoids :

A generalized helicoid is generated by helical rotation of an abitrary curve of
equation z = f (x). Its cartesian parametric equations are:

(2016-01-16) Surfaces of Constant Mean Curvature (CMC)
The shape of soap films separating regions of distinct pressures.

An unduloid is a surface of revolution whose meridian is
traced by the focus of a conic section which rolls
on the axis.

With a parabola, a catenoid
is obtained. The mean curvature is zero.

When it's an hyperbola, the surface has negative mean curvature, which corresponds
to a soap film surrounding a region of lower pressure.

A rolling ellipse corresponds to a positive mean curvature and/or a higher inner
pressure. We obtain the undulatory shape shown below,
which has given its name to the whole family.