PL-Geodesics on PL-Continuous Partial Meshes

Geometric characteristics of 2-manifolds embedded in R space have been analyzed from the point of view of differential geometry and topology. In the past, results relevant to these areas have been found for C curves and surfaces. However, current scientific, industrial, entertainment and medical applications, and availability of more powerful point sampling systems, press for characterization of discrete counterparts for the continuous properties and characteristics evaluated previously in C curves and surfaces. Recent works have presented estimation methods for properties such as the principal and rotated quadrics of point sampled surfaces. The present article uses the findings of previous investigations to propose and implement a method for evaluation of planarity of surfaces. It is based on: (i) Estimation of a C partial mesh fitting sets of planar or grid sample points. (ii) Evaluation of the piecewise linear (PL) version of families of geodesic curves on the mesh. (iii) Diagnostic of the property of planarity based on the behavior of the families of geodesic curves. The present work can be applied in the area of design and manufacturing of products based on sheet materials, such as apparel, metal stamping, thin structures, etc.