Three-dimensional shape statistics: Methodology

Two three-dimensional shape statistics are introduced to quantify the amount of linelike and planelike structures in a point distribution. We calculate analytically the expected values of these statistics for a Poisson sample and a Gaussian sample with a power-law two-point correlation function. We construct a Bayesian approach for the significance analysis. Some simple tests have been performed to test the sensitivity of these new statistics to the shape of structures, the density contrast of a structure with specified shape to the background, and the number of structures in a point distribution. We find the statistics are very useful in detecting filaments and pancakes against background noise, measuring the amount of these structures, and revealing the typical scale of such structure. They can be used to differentiate different galaxy distributions in terms of the shape of structures, and thus constraining models of structure formation. They can also be used to identify filaments and pancakes in a galaxy survey in a quantitative and systematic manner.