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Conformal parameterization for surfaces into various parameter domains is a fundamental task in computer graphics. Prior research on discrete Ricci flow provided us with promising inspirations from methods derived via Riemannian geometry, which is rigorous in theory and effective inpractice. In this paper, we propose a unified conformal parameterization approachfor turning triangle meshes into planar and spherical domains using discrete Calabi flow onpiecewise linear metric. We incorporate edge‐flipping surgery to guarantee convergence as well as other significant improvements including approximate Newton's method, optimal step‐lengths, priority embedding and boundary customizing, which achieve better performance and functionality with robustness and accuracy. [ABSTRACT FROM AUTHOR]