保角曲面形變與應用

Abstract

形變是將一個圖像變成另一個的過程。在這個研究中，我們利用保角映射及拓樸同倫的概念，實現幾個曲面形變的數值方法。我們透過保角參數化，統一網格結構，接著，利用三次樣條同倫來建構形變的過程。 為了自由控制型變的過程變化，曲面對應的問題隨之衍生。根據黎曼映射定理，一個簡單連通曲面，存在唯一的黎曼保角映射將之映射到單位圓盤。因此，我們將一個曲面對應問題化簡為一個單位圓盤的對應問題，利用有權重的最小平方法來解決這個問題。

Morphing is the process of changing one figure into another. In this study, we show some numerical methods of surface morphing by using the conformal mapping and the idea of homotopy. We uniformize the structure of each mesh by using the conformal parameterization, and construct the morphing process by using cubic spline homotopy. In order to control the morphing process as our desire, here comes a surface matching problem. From the Riemann mapping theorem, we know that there exists a unique Riemann conformal mapping from a simply connected surface into a unit disk. Therefore, we reduce a surface matching problem into a unit disk matching problem and solve this problem by using the biharmonic weighted least square method.