# Recovering the polyhedral geometry of fragments

## TL;DR

This paper introduces an algorithm to determine the polyhedral geometry of rock fragments from 3D scans, enabling reliable analysis of their shapes and origins.

## Contribution

A novel algorithm is introduced for identifying ideal polyhedra in rock fragments using 3D scans, surpassing manual inspection.

## Key findings

- The algorithm successfully identifies face orientations and reconstructs ideal polyhedra from 3D scans.
- Benchmarking against human measurements showed the algorithm's effectiveness on 132 fragments.
- The method can trace the shape evolution of rounded pebbles back to their original forms.

## Abstract

Not only is the geometry of rock fragments often well approximated by ideal convex polyhedra having few faces and vertices, but these numbers carry vital geophysical information on the fragmentation process.

Despite their significance, the identification of the number of faces and vertices of the ideal polyhedron has so far been carried out only through visual inspection. Here, we present an algorithm capable of performing this task in a reliable manner. The input of our algorithm is a 3D scan of the fragment which is a triangulated polyhedron with very large number of faces. Our algorithm performs a systematic simplification using the following steps:•Gaussian smoothing is performed on the spherical histogram of the 3D scans faces to identify the most important face orientations.•Planes carrying the faces of the ideal polyhedron are identified and the ideal polygon is reconstructed•Small faces are removed in a systematic manner

Gaussian smoothing is performed on the spherical histogram of the 3D scans faces to identify the most important face orientations.

Planes carrying the faces of the ideal polyhedron are identified and the ideal polygon is reconstructed

Small faces are removed in a systematic manner

We present two versions of the algorithm that we benchmarked the algorithm against a dataset of human measurements on 132 fragments. Beyond identifying the ideal polyhedral approximation for fragments, our method is also capable of tracing backward the shape evolution of rounded pebbles to their origins.

Image, graphical abstract