This process is called blending and the third surface is a blending surface of the given surfaces. This blending surface is of course not unique and is usually in an implicit form. The blending surface provides a smooth transition from the green surface to the blue one. In addition to blending, many applications and computations such as offsetting naturally generate algebraic surfaces. In fact, in many cases, algebraic surfaces are very useful.
Point classification is a good example.
Point classification determines if a given point lies inside, on or outside of a surface. One can easily compute the normal vector of an algebraic surface. Unfortunately, displaying an algebraic surface is not easy. The easiest but also the most time consuming way is raytracing. Many raytracers allow users to specify an implicit equation.