A quadrature scheme that is particularly well suited for solving azimuthally dependent transport problems with the ANISN code is introduced. A brief description of the fundamental problem in the constructive theory of orthogonal polynomials is provided. The implementation of the modified Chebyshev and the linear-factor modification algorithms for computing sets of recurrence coefficients that are used to generate the required quadratures is discussed in detail. While both algorithms are very effective and yield accurate quadrature nodes and weights, the linear-factor modification algorithm has the advantage of being about 20% faster than the modified Chebyshev algorithm.