Computation of the multi-chord distribution of convex and concave polygons

Analytical expressions for the distribution of the length of chords corresponding to the affine invariant measure on the set of chords are given for convex polygons. These analytical expressions are a computational improvement over other expressions published in 2011. The correlation function of convex polygons can be computed from the results obtained in this work, because it is determined by the distribution of chords. An analytical expression for the multi-chord distribution of the length of chords corresponding to the affine invariant measure on the set of chords is found for non convex polygons. In addition we give an algorithm to find this multi-chord distribution which, for many concave polygons, is computationally more efficient than the said analytical expression. The results also apply to non simply connected polygons.Comment: 22 figures, 40 pages, 43 reference

CWDM self-referencing sensor network based on ring resonators in reflective configuration

A new scalable self-referencing sensor network with low insertion losses implemented in Coarse Wavelength Division Multiplexing (CWDM) technology is reported. It allows obtaining remote self-referenced measurements with a full-duplex fibre downlead up to 35 km long, with no need for optical amplification. Fibre Bragg gratings (FBG) are used in order to achieve a reflective configuration, thus increasing the sensitivity of the optical transducers. Low-cost off-the-shelf devices in CWDM technology can be used to implement and scale the network. Ring resonator (RR) based incoherent interferometers at the measuring points are used as selfreferencing technique. A theoretical analysis of power budget of the topology is reported, with a comparison between the proposed network and a conventional star topology. Finally, the new configuration has been experimentally demonstrated.This work has been supported by CICYT:TIC2003-03783, UC3M:FAVICOBIS and CAM:FACTOTEM-CM (S-0505/ESP/000417).Publicad