WP3: Algebraic Geometry for CAD Applications
Proposed WP3 projects:
Current CAD technology is essentially based on NURBS. From the Algebraic Geometry point of view NURBS are rectangular patchworks composed of the simplest rational surface pieces, parameterized by the product of two projective lines. The natural modelling process of smooth surfaces with complicated topology usually generates NURBS with n-sided (n ≠ 4) holes. Toric Bézier patches and especially multisided M-patches (based on toric, Del Pezzo and other more general rational surfaces) were proposed to solve the hole filling problem.
Whenever a CAD construction generates a rational surface, it is desirable to represent it in NURBS form. This raises the parameterization extension problem: find a Bézier patch of optimal degree on a given surface with given boundary curves. A general solution of the parameterization extension problem for almost toric surfaces was already proposed. The same approach can be used to model surfaces with rational offsets (PN-surfaces), since their duals are contained in the Blaschke cylinder (in Laguerre geometry interpretation), which is a toric variety. Here a duality in the geometry of rational surfaces should play an important role. Another important case of PN-surfaces is a canal surface that is a result of a rolling ball blend. A recent result on rational parameterizations of canal surfaces shows their tight relations with rational ruled surfaces and mu-bases.
These rational constructions, the generating algorithms, and the estimation of their degree in order to find bounds for their practical application all need to be investigated. For example, C1 hole filling constructions with M-patches can be quite satisfactory, but the C2 case has comparatively high parametric degree. Further research might focus on a promising method, where Hermite data of a C1 surface from a first step is used for a C2 construction in a second step while preserving fairness.
Offsets are very natural tools in CAD with a wide applicability for defining either safe (i.e. collision free) regions or trajectories in numerical control machining. Their use in CAD relies always on approximation techniques since generically the offset of a parametric curve or surface is no longer parametric but algebraic. Still the exact computation of the implicit equation of the offset presents the same drawbacks mentioned earlier (difficult to compute, with a very huge size compromising its use in practice, even if available). The use of the previously described techniques opens a new way of representing (through approximate implicitization, for example) and manipulating (by using ad-hoc algebraic solvers) offsets that needs to be investigated.
The foundations of Algebraic Geometry, with special emphasis on classical projective geometry of curves and surfaces, need to be extended and developed for the real, affine, and bounded cases, having in mind applications to CAD. Particular aspects that will be covered are, among others, the enumerative geometry for real curves and surfaces; singularity theory: existence and description of real surface singularities; the theory of polar and dual varieties in order to relate the description of real polar varieties to Sturm-Habicht methods for determining the topology of real surfaces, and to find efficient ways of computing points on the components of real varieties; the theory of moduli spaces of varieties and of parameterized varieties and the study of the semialgebraic stratification of these moduli spaces; the design of parametric catalogues of surfaces (parametric or not) that can be used in CAD.