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Concept
Implicit surface
Summary
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 three 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 . Generally the change of representations is simple only when the explicit representation is given: (implicit), (parametric).
Examples:
The plane
The sphere
The torus
A surface of genus 2: (see diagram).
The surface of revolution (see diagram wineglass).
For a plane, a sphere, and a torus there exist simple parametric representations. This is not true for the fourth example.
The implicit function theorem describes conditions under which an equation can be solved (at least implicitly) for x, y or z. But in general the solution may not be made explicit. This theorem is the key to the computation of essential geometric features of a surface: tangent planes, surface normals, curvatures (see below). But they have an essential drawback: their visualization is difficult.
If is polynomial in x, y and z, the surface is called algebraic. Example 5 is non-algebraic.
Despite difficulty of visualization, implicit surfaces provide relatively simple techniques to generate theoretically (e.g. Steiner surface) and practically (see below) interesting surfaces.
Throughout the following considerations the implicit surface is represented by an equation
where function meets the necessary conditions of differentiability. The partial derivatives of
are .
A surface point is called regular if and only if the gradient of at is not the zero vector , meaning
If the surface point is not regular, it is called singular.
The equation of the tangent plane at a regular point is
and a normal vector is
In order to keep the formula simple the arguments are omitted:
is the normal curvature of the surface at a regular point for the unit tangent direction .
Official source
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