A surface need not be stored as explicit triangles. An implicit surface is defined as the set of points satisfying

$$f(\mathbf{x})=0.$$

For a sphere of radius $r$ centered at the origin, one example is

$$f(x,y,z)=x^2+y^2+z^2-r^2.$$

A signed distance field (SDF) is a particularly useful implicit representation whose value gives the distance to the nearest surface, with opposite signs on the two sides of the boundary.

Simple SDFs can be combined using operations related to minimum and maximum to form unions, intersections and differences. Distance information also helps estimate surface normals from the gradient and allows rays to advance safely toward a surface using sphere tracing.

Implicit representations describe smooth or procedurally defined shapes compactly and avoid committing to one tessellation. Their trade-off is that rendering, editing and exact Boolean operations require different algorithms from ordinary triangle meshes.