In industrial value chains encompassing geometric objects, there frequently occur shapes that have a point set representation or are given through procedural interfaces. The use of such shapes downstream the information flow is frequently hampered by the bulk or complexity of the representation, and in some cases by limited access to the methods implemented below the procedural interface. These types of representations also occur naturally in many CAGD constructions such as offsets, envelopes, blends, etc. Computing a representation of such geometric constructions using parameterized or implicit algebraic models could be either impossible or very costly, whereas computing points on these surfaces is very easy.

Techniques devoted to such procedural representations will be developed, by extending the algebraic tools used for the usual monomial representation to Lagrange basis representations. Also the potential of point-based approximate implicitization will be exploited. The focus will be on the investigation of methods for approximating shapes represented by point sets with local algebraic models of low algebraic degree, taking into account possible singularities that might occur.

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