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 ${\displaystyle z=f(x,y)}$ 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 ${\displaystyle s,t}$. The change of representations is usually 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).

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