Characterization of bivariate quadratic transformations of quasi-copulas

In this study, we focus on bivariate transformations of trivariate quasi-copulas. We characterize necessary and sufficient conditions of transformations that can be written in the form of compositions between two quasi-copulas and a quadratic polynomial function. The conditions only depend on the coefficients of the quadratic polynomial. The set of these coefficients is convex with linear-section boundary, and it lies on the seven-dimensional Euclidean space. All extreme points of this set have been characterized via CAS and can be used to construct quasi-copulas. Construction examples are also given.