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tokens: eigenvalues . The lengths of the semimajor and semiminor axes are and with the axis ratio b ′/ a ′. The details of the iterative method for measuring axis ratios can be found in Tenneti et al. ( 2015b ). In the first iteration, we start with the half-lengths of the principal axes all equal to 1 and determine the eigenvalues and eigenvectors of the ellipsoid. After each iteration, the lengths of the principal axes of ellipsoids are rescaled such that the enclosed volume is constant and particles outside the ellipsoidal volume are discarded. This process is repeated until convergence is reached such that the fractional change in axis ratios is below 1%. In addition to the distribution of the axis ratios, b / a and c / a of the stellar components of subhalos, we are also interested in the orientation of the major axis of the stellar shape with the shape of dark matter in subhalos. So, we compute the probability distribution of the misalignment anglewhere and are the major axes of the shapes defined by the dark matter and stellar matter components respectively. 3.2. Two-point Statistics In this paper we quantify the intrinsic alignments of galaxies with the large-scale density field using the ellipticity–direction (ED) and the projected shape–density () correlation functions. The ED correlation function cross-correlates the orientation of the major axes of the 3D shapes of dark matter or stellar component of galaxies with the large-scale density field. Consider a subhalo centered at position x with the major axis direction . Let the unit vector in the direction of a tracer of the matter density field at a distance r be . Based on the notation in Lee et al. ( 2008 ), the ED correlation function is given bywhich is zero for randomly oriented galaxies in a uniform distribution. In the simulations the matter density field is traced using the positions of dark matter particles. The projected shape correlation function, is directly related to the correlation function measured in observations. Following the notation of Mandelbaum et al. ( 2006 ), we define the the matter–intrinsic shear correlation function and the corresponding projected two-point statistic . In this paper, r p is the comoving transverse separation of a pair of galaxies in the XY plane and Π is their separation along the Z direction. The components of the projected ellipticities of a galaxy are given bywhere b ′/ a ′ is the axis ratio of the projected shape of the stellar component of a galaxy, and φ is the position angle of the major axis with respect to the reference direction (position of the dark matter particle). Here, e + refers to the radial component and e × is the component rotated at 45°. The matter–intrinsic shear correlation function is given by,where S + represents the “shape sample,” selected on the basis of a binning in subhalo mass and the “density sample” labeled by D consists of the dark matter particles used to trace the matter density field. S + D is given by the