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

Abstract

Arelevant problem in point-by-point scanning surface topography is to find scanning paths minimizing the overall measurement time.Weestablish 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.Weprovide 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.Weexplain 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.