In n dimensions there is a familiar concept of parallel lines which is an equivalence relation. We will call the equivalence classes directions. We measure a set of directions by measuring the subset of Sn−1 pierced by lines thru its center in those directions.

For any n-simplex S and (any direction, except for a set of measure 0), there is a unique line in that direction that pierces the n-simplex so that it pierces opposite sub-simplexes. If one of those sub-simplexes is a k-simplex then n−k (n−1)-faces of S are visible from that direction.

The measure of the set of such lines that pierce a particular sub-simplex is the magnitude of the angle centered at that sub-simplex.