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While three-dimensional measurement technology is spreading fast, itsmeaningful application to sedimentary geology still lacks content. Classicalshape descriptors (such as axis ratios, circularity of projection) were notinherently three-dimensional, because no such technology existed. Recently anew class of three-dimensional descriptors, collectively referred to asmechanical descriptors has been introduced and applied for a broad range ofsedimentary particles. First order mechanical descriptors (registered for eachpebble as a pair $\{S,U\}$ of integers), refer to the respective numbers ofstable and unstable static equilibria and can be reliably detected by handexperiments. However, they have limited ability of distinction as the majorityof coastal pebbles fall into primary class $\{S,U\}=\{2,2\}$. Higher ordermechanical descriptors offer a more refined distinction. However for theextraction of these descriptors (registered as graphs for each pebble) handmeasurements are not an option and even computer-based extraction from 3D scansoffers a formidable challenge. Here we not only describe and implement analgorithm to perform this task, but also apply it to a collection of 271pebbles with various lithologies, illustrating that the application of higherorder descriptors is a viable option for geomorphologists. We also show thatthe so-far uncharted connection between the two known secondary descriptors,the so-called Morse-Smale graph and the Reeb-graph can be established via athird order descriptor which we call the master graph.

Pebbles, graphs and equilibria: higher order shape descriptors for sedimentary particles