In mathematics, an implicit surface is a surface in Euclidean space defined by an equation
F ( x , y , z ) = 0. {\displaystyle F(x,y,z)=0.}
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 z = f ( x , y ) {\displaystyle z=f(x,y)} and is called an explicit representation. The third essential description of a surface is the parametric one:
( x ( s , t ) , y ( s , t ) , z ( s , t ) ) {\displaystyle (x(s,t),y(s,t),z(s,t))} , where the x-, y- and z-coordinates of surface points are represented by three functions x ( s , t ) , y ( s , t ) , z ( s , t ) {\displaystyle x(s,t)\,,y(s,t)\,,z(s,t)} depending on common parameters s , t {\displaystyle s,t} . Generally the change of representations is simple only when the explicit representation z = f ( x , y ) {\displaystyle z=f(x,y)} is given: z − f ( x , y ) = 0 {\displaystyle z-f(x,y)=0} (implicit), ( s , t , f ( s , t ) ) {\displaystyle (s,t,f(s,t))} (parametric). Examples:
The plane x + 2 y − 3 z + 1 = 0. {\displaystyle x+2y-3z+1=0.}
The sphere x 2 + y 2 + z 2 − 4 = 0. {\displaystyle x^{2}+y^{2}+z^{2}-4=0.}
The torus ( x 2 + y 2 + z 2 + R 2 − a 2 ) 2 − 4 R 2 ( x 2 + y 2 ) = 0. {\displaystyle (x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+y^{2})=0.}
A surface of genus 2: 2 y ( y 2 − 3 x 2 ) ( 1 − z 2 ) + ( x 2 + y 2 ) 2 − ( 9 z 2 − 1 ) ( 1 − z 2 ) = 0 {\displaystyle 2y(y^{2}-3x^{2})(1-z^{2})+(x^{2}+y^{2})^{2}-(9z^{2}-1)(1-z^{2})=0} (see diagram). The surface of revolution x 2 + y 2 − ( ln ( z + 3.2 ) ) 2 − 0.02 = 0 {\displaystyle x^{2}+y^{2}-(\ln(z+3.2))^{2}-0.02=0} (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 F ( x , y , z ) = 0 {\displaystyle F(x,y,z)=0} 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 F ( x , y , z ) {\displaystyle F(x,y,z)} 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.
Formulas Throughout the following considerations the implicit surface is represented by an equation
F ( x , y , z ) = 0 {\displaystyle F(x,y,z)=0} where function F {\displaystyle F} meets the necessary conditions of differentiability. The partial derivatives of
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