  Path integration enables desert arthropods to find back to their nest on the
shortest track from any position. To perform path integration successfully,
speeds and turning angles along the preceding outbound path have to be measured
continuously and combined to determine an internal {\em global vector} leading
back home at any time. A number of experiments have given an idea how
arthropods might use allothetic or idiothetic signals to perceive their
orientation and moving speed. We systematically review the four possible model
descriptions of mathematically precise path integration, whereby we favour and
elaborate the hitherto not used variant of egocentric cartesian coordinates.
Its simple and intuitive structure is demonstrated in comparison to the other
models. Measuring two speeds, the forward moving speed and the angular turning
rate, and implementing them into a linear system of differential equations
provides the necessary information during outbound route, reorientation process
and return path. In addition, we propose several possible types of systematic
errors that can cause deviations from the correct homeward course. Deviations
have been observed for several species of desert arthropods in different
experiments, but their origin is still under debate. Using our egocentric path
integration model we propose simple error indices depending on path geometry
that will allow future experiments to rule out or corroborate certain error
types.
