Stereology of Dihedral Angles I: First Two Moments
James P. Butler, James A. Reeds · SIAM Journal on Applied Mathematics · 1987
A gelatin dessert has wedge-shaped fruit slices with angle $\alpha $ dispersed in it randomly. A knife makes a planar slice through the dessert. Each dihedral angle of a fruit wedge cut by the knife shows up as an angle A in the slicing plane. We obtain simple formulas for the first two moments of $A:EA = \alpha $ and $EA^2 = 4( 1 - \alpha /\tan \alpha ) + 2\pi ( \alpha / 2 - \tan \alpha /2 )$, derived by the method of functional equations. These results are useful in biologic and geologic stereology, where measurements are made on two-dimensional sections of a sample, and information about the angles in three dimensions is desired.