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There are various approach to solid modeling. Some systems supports a hybrid solution.

CSG

In the CSG approach the solid is described as a combination of elementary solids like cubes, spheres or extrusions. The combinations are strict boolean operations (Add material, Remove material, Common Material). The actual faces are not computed the result is obtained by a shaded image.

CSG are very practical for rapidly adapting display. Checking properties like if a point is inside is based on checking the same property on the elementary solids.

Construction of mechanical parts including filets or shells is very complex.

In some extends CSG inspired the parametric systems available today which mix the history tree and the Brep representation. But parametric systems are far more powerful because they include local operations like filleting.

BRep

In the Brep approach solids are defined and stored by their boundary. Every boolean operation leads to a complex computation. It has been a long challenge for Solid modeling kernel developers to have reliable algorithms for this computation.

The mathematical fundaments of BRep is the theory of 3D manifolds. It includes notion like differential topology.

Almost all commercial CAD systems support BRep modelers.

Facettized versus Exact. First implementation of the boolean operations on Brep models where done on polyhedral approximations. This was called Facetted modelers. The approximation can be constant or toleranced based. Last generation of facettized modelers keeps the original surface information on each facets but it does not prevent the system to have an erroneous topology.

Use of surfaces instead of facets leads to higher complexity. Computations are complex and based on high level surface intersection. Also the topology can be more complex : 2 faces can intersect on more than one branch. Intersection curves may self intersect etc..

Implicit

Implicit Solid Modeling Paper

Voxel

In a Voxel based system the material is represented by a succession of small cubes like an image is represented by pixels on your screen. The precision of the scene is the size of the voxel. The more voxel you have the more precise but also the expense of memory and computation increase. Some systems increase their performance by supporting a system of hierarchically subdivided cells.

This approach is very good for imaging purpose or fast clearance checking.

Drawbacks: Precision. No analytic definition of faces or surfaces.

Note:

You will notice that some writers use modeler, others modeller. Both are correct. modeler is American English where modeller is UK English.

 

CADCAM Techniques Software capability of representing the sense of material with its familiar operations like drilling a hole or adding a slot. Solid modelers will be able to produce automatically cross sections and display with hidden lines removed