Valley Average of Lines (VAL) and of Directions (VAD) in 3D

Abstract

Extending (Sati et al., 2016) to 3D, we propose definitions and closed-form expressions for what we call the Valley Average of Directions (VAD), T→, of a set {T→i} of directions (unit vectors) and for what we call the Valley Average of Lines (VAL), L, of a set {Li} of lines. L is independent of the input order. When the {Li} are parallel, L is parallel to them and passes through the centroid of their intersections with any normal plane. When all {Li} contain point P, L passes through P.