We present a novel method to approximate a freeform shape with a planar quadrilateral (PQ) mesh for modeling architectural glass structures. Our method is based on the study of conjugate direction fields (CDF) which allow the presence of ±k/4(k ∈ Z) singularities. Starting with a triangle discretization of a freeform shape, we first compute an as smooth as possible conjugate direction field satisfying the user’s directional and angular constraints, then apply mixed-integer quadrangulation and planarization techniques to generate a PQ mesh which approximates the input shape faithfully. We demonstrate that our method is effective and robust on various 3D models.