Improving the Line of Sight for the Anisotropic 3-Point Correlation Function of Galaxies: Centroid and Unit-Vector-Average Methods Scaling as $\mathcal{O}(N^2)$

The 3-Point Correlation Function (3PCF), which measures correlations between triplets of galaxies encodes information about peculiar velocities, which distort the observed positions of galaxies along the line of sight away from their true positions. To access this information, we must track the 3PCF's dependence not only on each triangle's shape, but also on its orientation with respect to the line of sight. Consequently, different choices for the line of sight will affect the measured 3PCF. Up to now, the line of sight has been taken as the direction to a single triplet member (STM), but which triplet member is used impacts the 3PCF by ~20% of the statistical error for a BOSS-like survey. For DESI (2019-24), which is 5X more precise, this would translate to 100% of the statistical error, increasing the total error bar by 40%. We here propose a new method that is fully symmetric between the triplet members, and uses either the average of the three galaxy position vectors (which we show points to the triangle centroid), or the average of their unit (direction) vectors. Naively, these approaches would seem to require triplet counting, scaling as $N^3$, with $N$ the number of objects in the survey. By harnessing the solid harmonic shift theorem, we here show how these methods can be evaluated scaling as $N^2$. We expect that they can be used to make a robust, systematics-free measurement of the anisotropic 3PCF of upcoming redshift surveys such as DESI. So doing will in turn open an additional channel to constrain the growth rate of structure and thereby learn the matter density as well as test the theory of gravity.