  There is currently a gap in theory for point patterns that lie on the surface
of objects, with researchers focusing on patterns that lie in a Euclidean
space, typically planar and spatial data. Methodology for planar and spatial
data thus relies on Euclidean geometry and is therefore inappropriate for
analysis of point patterns observed in non-Euclidean spaces. Recently, there
has been extensions to the analysis of point patterns on a sphere, however,
many other shapes are left unexplored. This is in part due to the challenge of
defining the notion of stationarity for a point process existing on such a
space due to the lack of rotational and translational isometries. Here, we
construct functional summary statistics for Poisson processes defined on convex
shapes in three dimensions. Using the Mapping Theorem, a Poisson process can be
transformed from any convex shape to a Poisson process on the unit sphere which
has rotational symmetries that allow for functional summary statistics to be
constructed. We present the first and second order properties of such summary
statistics and demonstrate how they can be used to test whether an observed
pattern exhibits complete spatial randomness or spatial preference on the
original convex space. A study of the Type I and II errors of our test
statistics are explored through simulations on ellipsoids of varying
dimensions.
