Implicit surface
In mathematics an implicit surface is a surface in Euclidean space defined by an equation
An implicit surface is the set of zeros of a function of 3 variables. Implicit means, that the equation is not solved for x or y or z.
The graph of a function is usually described by an equation 
and is called an explicit representation. The third essential description of a surface is the parametric one:
, where the x-, y- and z-coordinates of surface points are represented by three functions 
depending on common parameters 
. The change of representations is usually simple only, when the explicit representation is given: 
(implicit), 
(parametric).
Examples:
.
.
.
(s. picture).
(s. picture wineglas).For a plane, a sphere and a torus there exist simple parametric representations. This is not true for the 4. example.
The implicit function theorem describes conditions, under which an equation 
can be solved (theoretically) for x, y or z. But in general the solution may not be feasible. This theorem is the key to the computation of essential geometric features of a surface: tangent planes, surface normals, curvatures (s. below). But they have an essential drawback: their visualization is difficult.
Podcasts:
