Primitive shapes in implicit modeling

An Implicit Model is a continuous mathematical representation of an attribute across a volume. It has an infinitely fine resolution. Creating tangible surfaces from this model is a separate and secondary step and is independent of creating the Implicit Model (from www.micromine.com). Metaballs use implicit modeling to represent objects. Their main particularity is that they describe a field. Hence, object fields can be added together to create smooth transitions between the two objects. To obtain an object's field, we need to take the inverse of the function describing that object. For a sphere, the object is described by the equality: $$ {x^2+y^2+z^2} = r^2 $$ where r is the sphere's radius. When considering the field of a sphere located at the point (a,b,c), the function will be: $$f(x,y,z) = {r^2 \over {(x-a)^2+(y-b)^2+(z-c)^2} }$$ When superimposing spherical objects, one gets a greater sphere. While when adding two spherical fields, we can get a transition between two spheres. The pictures below show the superimposition of different fields (the marching cube algorithm is still used to represent the fields).

Spherical fields can be further used to thicken non-straight curves like Bezier curves with more than one control point.

Below I will show an interesting example of how implicit modeling can be used for a greater purpose. I was asked to create a star-looking shape where members had to be non-coplanar. To properly blend the star members, I thought it would be a good idea to use implicit modeling. After randomizing the length and orientation of the members, and after increasing the center's size, I ended up with this virus-looking shape. As someone who had long searched for ways to apply my CAD knowledge to the biomedical field, I couldn't have asked for better... It's also mesmerizing to see how implicit modeling yields shapes that are more probable to be found in nature.