Implicit surface

From Wikipedia, the free encyclopedia

In mathematics and computer graphics, an implicit surface is defined as an isosurface—a level set—of a function

f\ :\ \mathbb{R}^{3}\rightarrow\mathbb{R}.\,

In other words, it is the set of points in the 3d-space that satisfy the equation

f(x,y,z)=\mathrm{constant}.\,

To find a parametrisation of the 'surface' (more precisely the solution set, since not all equations of this type define a surface, or indeed define any points at all) one has to treat this relation as giving an implicit function

Without loss of generality one may assume that the equation is of form

g(x,y,z)=0,\,

but it is often convenient to allow the arbitrary constant on the right side.

When the function f is a polynomial in the three variables then the surface is an algebraic surface (over the reals). Such surfaces have properties which make them easier to calculate.

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