### GEOMETRICAL ANALYSIS

PAST
includes some functionality for geometrical
analysis, even if an extensive morphometrics module has not yet been implemented.
We hope to implement more extensive functionality, such as landmark-based methods,
in future versions of the program.

The program can plot rose diagrams (polar histograms) of directions. These
can be used for plotting current-oriented specimens, orientations of trackways,
orientations of morphological features (e.g., trilobite terrace lines), etc. The
mean angle together with Rayleigh's spread are given. Rayleigh's spread is
further tested against a random distribution using Rayleigh's test for
directional data (Davis 1986). A ^{2}
test is also available, giving the probability that the directions are randomly
and evenly distributed.

Point distribution statistics using nearest neighbor analysis (modified from Davis
1986) are also provided. The area is estimated using the convex hull,
which is the smallest convex polygon enclosing the points. The probability that
the distribution is random (Poisson process, giving an exponential nearest
neighbor distribution) is presented, together with the `*R'* value.
Clustered points give *R*<1, Poisson patterns give *R*~1, while
over-dispersed points give *R*>1. Applications of this module include
spatial ecology (are in-situ brachiopods clustered) and morphology (are
trilobite tubercles over-dispersed; see Hammer
2000).

The Fourier shape analysis module (Davis
1986) accepts *x-y* coordinates digitized around an outline. More than
one shape can be analyzed simultaneously. Points do not need to be evenly
spaced. The sine and cosine components are given for the first ten harmonics,
and the coefficients can then be copied to the main spreadsheet for further
analysis (e.g., by PCA). Elliptic Fourier shape analysis is also provided (Kuhl
and Giardina 1982). For an application of elliptic Fourier shape analysis in
paleontology, see Renaud et al.
(1996).