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 =
a2 + b2 + c2
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.