Abstract: Domain parameterization, which is an essential step in isogeometric analysis, has been extensively studied in recent years. To address the problem that most of the existing parameterization methods restrict the parameter domain to the square (cube) and cannot obtain high-quality mapping for triangle-like domains, we propose an effective parametrization method for planar domains by adopting the triangular Bézier patches as the representation and the quasi-conformal mapping as the computational framework. Given the boundary correspondence between the parametric domain (unit right triangle) and the physical domain, the problem of constructing a quasi-conformal mapping with low distortion can be modeled as an optimization model, in which the objective function is the conformal distortion and smoothness of the mapping with the injectivity of the mapping as a constraint. Then the optimization model is addressed by solving two quadratic optimization problems alternatively. Finally, we compare our method with the tensor-product B-spline based parameterization method, and the experimental results demonstrate that the proposed approach can obtain parametrizations with higher quality.