A new technique for modeling; Implicit Modeling

[L'Ancien Regime]

Polygonal Modeling

SubD Modeling (Catmull-Clark and other variants)

NURBS Modeling

BREP Modeling

Voxel Modeling

And now Implicit Modeling which has been devised in response to numerous problems encountered with BREP Modeling

https://www.3dcadworld.com/implicit-modelling-for-complex-geometry/

 

The problem is the fundamental way that geometry is represented. All the major CAD programs use boundary representations (b-reps) to define solid geometry. B-reps use topology, such as vertices, edges and faces, which are defined by geometry such as points, curves and surfaces. For example, an edge is a bounded region of a curve and a face is a bounded region of a surface.

B-reps work fine for relatively simple geometry. Older CAD programs sometimes used Boolean operations to combine primitive objects such as cubes and spheres. B-reps are much more versatile, allowing profiles to be swept and lofted, solids to be shelled and so forth. However, when features are combined with fillets and blends, or large numbers of features are included, calculating the topology becomes exponentially more demanding on the computer. Many modelling operations involve combining simpler shapes with Boolean and blending operations. B-rep modelling has to calculate all of the new edges that are formed where faces intersect. For objects with planar faces the individual calculations are relatively simple, but the number of intersections increases by approximately the square of the number of faces. For intersections between curved surfaces the edges are complex splines, making the calculations considerably more complex. When faces are close to tangent, things get really difficult.

Another issue with B-reps can be determining which points in the model are inside the boundary – the solid material. The method is to shoot a ray from the point in an arbitrary direction. If the ray passes the boundary an odd number of times then the point is inside the boundary. If the ray passes the boundary an even number of times then the point is outside the boundary. Since floating point arithmetic is used, rounding errors may mistakenly count boundary crossings when the ray is close to the boundary. There are also ambiguities such as when a ray is tangent to a surface or passes through a vertex – it is not clear whether this counts as a boundary crossing, or as two crossings. Because of these issues, additional checks are required to ensure that b-rep modellers are robust. This makes them mathematically inefficient.

 

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