Traditional 3D modelling primitives like natural quadrics (sphere, circular cylinder/cone) or torus surfaces have exceptional geometric properties: low implicit degree, rational parameterization and rational offsets. The majority of mechanical parts in CAD have a boundary representation that combines planes and primitive surfaces with smooth blends between them. A fixed radius rolling ball construction is used for edge blends and special n-sided patchworks for vertex blends. Usually these constructions are only approximate (except a few “analytic” cases, when cylinders, tori or spherical patches can be used), and the resulting blends are NURBS surfaces with a lot of control points and with gaps along boundaries.

The project goal is to construct new efficient C1 blends between primitive surfaces of reasonable parameterization degree that have no gaps on the boundaries and admit rational offset (if possible):

• Extend the list of “analytic” cases of fixed radius rolling ball blends (exact rational representation exists when the spine curve is rational, e.g. arbitrary plane/cone intersection can be blended with a pipe surface with conical spine admitting a (2,4) rational parameterization);
• Construct approximate fixed radius rolling ball blends (when the spine curve is irrational) with boundary curves exactly lying on given primitive surfaces (results about spline curves on surfaces can be useful);
• For vertex blends use n-sided M-patches and similar constructions and study possibilities to find solutions with rational offsets.

In cooperation with other partners, the fellow will study and optimize intersection algorithms between primitive surfaces that are needed, and work on the possible integration of the achieved results into industrial CAD systems. 

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