Freeform surface characterisation: Theory and practice

The specification and characterisation of freeform surfaces is immature: many CAD packages only allow the nominal freeform geometry to be specified. To control manufacturing and function, the allowable geometric variability also needs to be specified. Currently International standards specify allowable geometry through geometrical tolerancing. Geometrical tolerancing has proved to be a very blunt instrument with many cases, particularly in the aerospace and biomedical industries, of failure of the function of freeform surfaces due to inadequate specification. The paper begins by discussing the importance of the decomposition of the surface geometry into different scales for both specification and characterisation. The three main types of geometrical decomposition: linear, morphological and segmentation are briefly discussed. The Laplace-Beltrami operator (LBO) is the generalization of the Laplace operator to manifolds (i.e. freeform surfaces). By taking Eigenfunctions/Eigenvalues of the LBO a spectral decomposition of the freeform geometry can be realized. The Eigenfunctions are called manifold harmonics and are a direct generalization of Fourier harmonics for freeform surfaces. Tolerancing the Eigenfunctions provides a very powerful and flexible system to control geometrical variability of freeform surfaces; extending the geometrical tolerancing system by allowing decomposition of the geometrical shapes within tolerance zones. Further a generalized Gaussian filter can also be realized by Gaussian weighting the LBO spectra and reconstructing the weighted Eigenfunctions.