現今的生活中，充斥著許許多多的曲面產品。然而這些曲面產品往往要從平面的材料製成。因此本研究提出一種以能量為基礎將不可展曲面(Un-Developed Surface)攤平展開(Flattening)的方法，整個學理方法包含了前處理─曲面網格化、曲面初始展開、網格應變能優化處理、以及網格應變能釋放等四大步驟。
論文首先討論如何使一個連續曲面在保有曲面幾何特徵與足夠的精度下，進行曲面網格化(Tessellation)的演算法。網格逐一座標轉換，建立無交疊(Non-Overlap)的排列靠攏(Alignment)關係，完成曲面初始展開。再以網格變形能模型(Mesh Deformation Energy Model)理論，在保有設定的拘束條件下，穩定、有效的減少展開結果的應變能。最後，本研究提供了一種有效降低變形能量的網格群裂開方法，藉由放鬆部分網格的拘束，達到降低網格的扭曲(Distortions)程度和變形能的目的，進而縮短展開結果和原始設計曲面的差異(Deviation)。
不可展曲面展開的結果可能有成千上萬種，本研究利用網格變形能評估曲面展開結果優劣，比較各種拘束條件與網格群裂開條件下的不同展開結果，嘗試找出影響版型展開結果之變因，進而找出在特定限制下的優化應變能展開版型(Patterns)，並用網格總面積與特徵圍長的差異量評估驗證展開結果。
以本文的研究方法為技術核心，開發了曲面展開軟體(SurfaceFlatten)，適用於各種設計產品曲面展開應用，並在文中提出服裝設計的立體剪裁(Drapping)為例，證實東方的文化式(Bunka)服裝版型公式與西方的立體剪裁有不謀而合之處。本開發更可以攤平展開原理應用於3D材質貼圖(Texture Mapping)，減少貼圖時的變形量。最後提供服裝界常用的版型輸出格式至一般的二維版型處理軟體，以銜接到製造端。

Nowadays, products combined with multi-surface are full of our life. These products are made for planar materials. This research presents a surface flattening method which contains a complete process of flattening a 3D un-developed surface to 2D plane. The process contains four steps, surface tessellation, surface initial flattening, mesh deformation energy optimization and mesh deformation energy release.
Firstly, the designed surfaces are tessellated by the algorithm that meets both requirements of the surface features and tessellation precision. Second, surface initial fattening is developed by the processes of single mesh transformation, mesh re-arrangement and mesh re-alignment by collision detection method with the condition of non-overlap. According to the deformation energy model, the total potential energy is calculated and used to locate the possible pattern of the less deformation energy in way of maintaining accuracy and efficiency. Finally, we proposed a method which can decrease the mesh deformation energy by splitting the mesh groups in this thesis. By relaxing some constraints of mesh corners, this method reduces the mesh distortion and deformation energy so that shortening the deviation between the flattened results and the original designed surfaces.
Outcomes of the flattened patterns of un-flattened surfaces may be plenty. In this study, we develop an evaluation method in which the mesh deformation energy is taken into consideration. So that the flattened results can be evaluated between various conditions combined with different constraints and conditions of energy relaxation. In this study, the author have identified the factors which influence the final flattened results, and then reveal the flattened results with a less deformation energy under certain constraints. The differences of surface area and the feature girths between the flattened results and the original designed surface are also taken into consideration to certify and check the flattened results.
With the research method of this thesis, we developed a software named SurfaceFlatten. SurfaceFlatten can be widely applied in the product design regions and we take the drapping of garment design for instance in this thesis to prove the coincidence between the Bunka pattern formula and the drapping technology. With the flattened results, less distorted results can be provided in texture mapping. Finally, this research provides the file formats which are generally used in garment industry. These formats can be outputted to two dimensional pattern making software to be the interface between the design side and the manufacturing side.

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