在這篇論文中，我們將重點聚焦在透過拉伸能量的最小化來計算出圓盤形狀的保面積參數化。我們使用了非線性共軛梯度法對其進行優化。在不犧牲收斂性理論的情況下，我們進一步地運用適當的預處理增進效果，數值結果顯示，我們方法比最先進的算法，有更好的準確度和效率。
此外，透過將我們提出的方法結合至二次懲罰法，我們延伸保面積參數化的應用至曲面配準上。數值上，我們能在足夠對齊特徵點的狀況下仍保持良好的保面積效果。

This thesis focuses on the computation of disk-shaped area-preserving parameterizations through stretch energy minimization. We employ the nonlinear conjugate gradient method to achieve this goal, and we introduce appropriate preconditioning in the algorithm to enhance its effectiveness without sacrificing theoretical convergence. The numerical results indicate that our proposed method outperforms state-of-the-art algorithms.
Furthermore, we extend the application of area-preserving parameterization to surface registration using the quadratic penalty method.We solve the subproblems in this context using our proposed method. The numerical results demonstrate the capability of our method to align landmark pairs while preserving the area of the surface.

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