  Three particles floating on a fluid surface define a triangle. The aim of
this paper is to characterise the shape of the triangle, defined by two of its
angles, as the three vertices are subject to a complex or turbulent motion. We
consider a simple class of models for this process, involving a combination of
a random strain of the fluid and Brownian motion of the particles. Following D.
G. Kendall, we map the space of triangles to a sphere, whose equator
corresponds to degenerate triangles with colinear vertices, with equilaterals
at the poles. We map our model to a diffusion process on the surface of the
sphere and find an exact solution for the shape distribution. Whereas the
action of the random strain tends to make the shape of the triangles infinitely
elongated, in the presence of a Brownian diffusion of the vertices, the model
has an equilibrium distribution of shapes. We determine here exactly this shape
distribution in the simple case where the increments of the strain are
diffusive.
