Circle involute as an optimal scan path, minimizing acquisition time, in surface topography

Abstract

A relevant problem in point-by-point scanning surface topography is to find scanning paths minimizing the overall measurement time. We establish a rigorous mathematical framework that allows us to state, for the first time, that circle involutes are the best candidates for such scanning paths maintaining a prescribed measurement accuracy condition. We provide a quantitative analysis comparing time savings achieved by circle involutes with respect to three widely used scanning paths: (1) Archimedean spiral; (2) equispaced concentric rings; (3) equispaced parallel rectilinear lines. While time reduction with respect to the last two is significant, for the first is not the case, which indirectly supports Archimedean spirals as a good choice too. We explain this fact by showing the close relationship between circle involutes and Archimedean spirals. Finally, we perform some numerical simulations comparing circle involutes and Archimedean spirals within a disk (quasi-planar surfaces), assuming constant linear velocity of the measurement device and constant sampling frequency.