Geometric modeling - mathematical representations

There are many ways to define the geometry of a digital model. Some popular ones are
1. Polygonal meshes
2. NURBS (Non-Uniform Rational B-Spline)
3. Subdivision surfaces
4. Implicit surfaces

There are software packages entirely based on one of the above to internally represent their models. Rhino, AliasStudio are pure NURBS based modeler. Some software like 3DSTudio and Maya have both meshes and NURBS representations.

Meshes vs NURBS have their own advanatages and disadvantages. For example, to represent a perfect sphere as a mesh object, we need thousands of tiny triangles to get the smooth curved shape of a sphere. NURBS can represent a sphere with just 9 control points and a few knot values. However, NURBS need to have rectangular surfaces (as they are parameterised in 2D space). Trimmed NURBS are common to represent freeform surfaces but they have leak and gap problems between surfaces.

Subdivision surface came as the rescue to solve both these issues. We can represent arbitrary free form surfaces with less control points. However, mathematically some regions of the subdivision surface is not smooth (not C1 continuous).

There are pure subdivision surface based modelers out there. Modo, Silo are some examples. They have huge limitations in the kinda models they can represent. They work great for certain applications.

IT is a great opportunity