Geometric figures with the same chord length probability density function: Six examples

The chord length probability density functions for isotropic uniform random chords f(l  ) have been studied for 12 different geometric figures KiKi. The detailed analysis shows: six pairs of different KiKi possess the same fi(l). In greater detail, for the following six figure-pairs (specific length parameter b),1. 60°-angle with side b ↔ equilateral triangle with side length b,2. 90°-angle with side b ↔ square with side length b,3. 60°-wedge of side length b ↔ triangular rod of side length b,4. 90°-wedge of side length b ↔ square rod of side length b,5. 90°-angle with one side b, one infinitely long side e → ∞, ↔ plane stripe of breadth b,6. 90°-wedge with one breadth b, two sides of length e → ∞, ↔ infinite Layer of constant thickness b, the respective functions fi(l,b) are identical. Thus, without additional shape information, an identification of such a figure via its chord length probability density function (PDF) is not possible. However, in all the cases considered, the length parameter b, involved in the function f(l), can be recognized from the intrinsic behavior of f(l,b).Furthermore, the agreement of the first moments of the respective functions fi can be verified by use of the extended Cauchy theorem for non-convex figures.

Graphical abstractThe chord length probability density function (PDF) for isotropic uniform random chords f(l) has been studied for 12 different geometric figures Ki. The detailed analysis shows: six pairs of different Ki possess the same fi(l). For example, the PDF of a right angle with side b matches that of a square with side length b.Figure optionsDownload full-size imageDownload as PowerPoint slide