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Implicit Polynomials (IP) are a powerful and rich mean to represent smooth approximations of 2D curves and 3D surfaces,

A d^th degree IP surface if the zero set of a d^th degree explicit polynomial. That is, the set of points (x,y,z) where the explicit polynomial is s.t. :

Such surfaces are generalizations of quadrics, e.g. hyper-ellipsoids, to more complicated shapes.

Example of an 8^th degree 3L fit of a butterfly shape with or without Control Points (CP).

Initial shape.	Fit without CP.	Fit with 3 CPs.
Zoom: no CP.	Zoom: with CPs.	Final fit with CPs.

The 3L fitting algorithm has been recently developed at LEMS. It is a robust and repeatable, under Euclidean transformations, process implemented as Linear LSE polynomial fitting.

Fitting the head data set using 3L algorithm

Initial head data set	10^th degree fit (ears discarded)	Reconstruction from 12 patches using 4^th degree IP surface models.

• 3D Demo**:
  • heart : single patch
    • **NB: your 3D browser must understant Open Inventor format <*.iv>

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