# Computation of equilibrium points on convex polyhedra

Equilibrium points are parts of an object on which you can shit them down without the object falling over. The number of these points is a basis of a new classification method of pebbles. By counting these points during different points of their lifetime we can get a general understanding of how a pebble's shape evolves over time. This research has a long history at the BUTE Department of Mechanics, Materials and Structures, and the knowledge gathered has great practical usage.

Determining the number of these points is a lengthy process carried out by researchers. Automating this step would help their work. My task was to make it possible to count equilibrium points on 3D scanned models of pebbles with a computer.

The main difficulty was that the equilibrium points are defined in terms of a continuous function which cannot be applied to the scanned surface consisting of discrete points.

My solution is based on the idea of countour lines in cartography. The algorithm finds these closed lines on the pebble's surface and traverses them from smallest to larger ones. If the line covers a larger area of the pebble than a predefined value, than it counts as an equilibrium point.

The program is also able to visualize the contour lines to aid better understanding of the pebbles.