Implicit surface

In mathematics, an implicit surface is a surface in Euclidean space defined by an equation

${\displaystyle F(x,y,z)=0.}$

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 ${\displaystyle z=f(x,y)}$ and is called an explicit representation. The third essential description of a surface is the parametric one: ${\displaystyle (x(s,t),y(s,t),z(s,t))}$, where the x-, y- and z-coordinates of surface points are represented by three functions ${\displaystyle x(s,t)\,,y(s,t)\,,z(s,t)}$ depending on common parameters ${\displaystyle s,t}$. Generally the change of representations is simple only when the explicit representation ${\displaystyle z=f(x,y)}$ is given: ${\displaystyle z-f(x,y)=0}$ (implicit), ${\displaystyle (s,t,f(s,t))}$ (parametric).

Examples:

1. The plane ${\displaystyle x+2y-3z+1=0.}$
2. The sphere ${\displaystyle x^{2}+y^{2}+z^{2}-4=0.}$
3. The torus ${\displaystyle (x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+y^{2})=0.}$
4. A surface of genus 2: ${\displaystyle 2y(y^{2}-3x^{2})(1-z^{2})+(x^{2}+y^{2})^{2}-(9z^{2}-1)(1-z^{2})=0}$ (see diagram).
5. The surface of revolution ${\displaystyle x^{2}+y^{2}-(\ln(z+3.2))^{2}-0.02=0}$ (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 ${\displaystyle F(x,y,z)=0}$ 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 ${\displaystyle F(x,y,z)}$ 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.