Editing Implicit surface (section)

{{Short description|Surface in 3D space defined by an implicit function of three variables}}
[[File:Torus-40-15.svg|thumb|Implicit surface torus {{math|1=(''R'' = 40, ''a'' = 15)}}.]]
[[File:Impl-flaeche-geschl2.svg|thumb|Implicit surface of genus 2.]]
[[File:Impl-flaeche-weinglas.svg|150px|thumb|Implicit non-algebraic surface (''wineglass'').]]

In [[mathematics]], an '''implicit surface''' is a [[Surface (geometry)|surface]] in [[Euclidean space]] defined by an equation 
: <math>F(x,y,z)=0.</math> 
An ''implicit surface'' is the set of [[Zero of a function|zeros]] of a [[Function of several real variables|function of three variables]]. ''[[Implicit function|Implicit]]'' means that the equation is not solved for {{mvar|x}} or {{mvar|y}} or {{mvar|z}}.

The graph of a function is usually described by an equation <math>z=f(x,y)</math> and is called an ''explicit'' representation. The third essential description of a surface is the ''[[Parametric equation|parametric]]'' one: 
<math>(x(s,t),y(s,t), z(s,t))</math>, where the {{mvar|x}}-, {{mvar|y}}- and {{mvar|z}}-coordinates of surface points are represented by three functions <math>x(s,t)\, , y(s,t)\, , z(s,t)</math> depending on common parameters <math>s,t</math>. Generally the change of representations is simple only when the explicit representation <math>z=f(x,y)</math> is given: <math>z-f(x,y)=0</math> (implicit), <math> (s,t,f(s,t)) </math> (parametric).

''Examples'':
#The [[Plane (geometry)|plane]] <math> x+2y-3z+1=0.</math>
#The [[Sphere (geometry)|sphere]] <math> x^2+y^2+z^2-4=0.</math>
#The [[Torus (mathematics)|torus]] <math>(x^2+y^2+z^2+R^2-a^2)^2-4R^2(x^2+y^2)=0. </math>
#A surface of [[Genus (mathematics)|genus]] 2: <math>2y(y^2-3x^2)(1-z^2)+(x^2+y^2)^2-(9z^2-1)(1-z^2)=0</math> (see diagram).
#The [[surface of revolution]] <math> x^2+y^2-(\ln(z+3.2))^2-0.02=0</math> (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 <math>F(x,y,z)=0</math> can be solved (at least implicitly) for {{mvar|x}}, {{mvar|y}} or {{mvar|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 plane]]s, [[surface normals]], [[Gaussian curvature|curvatures]] (see below). But they have an essential drawback: their visualization is difficult.

If <math>F(x,y,z)</math> is polynomial in {{mvar|x}}, {{mvar|y}} and {{mvar|z}}, the surface is called [[Algebraic surface|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.