幾何模型中幾何限制曲面的圖形顯示

Abstract

在電腦輔助幾何設計 (CAGD) 中，有許多曲面，例如 offsets，及 blends，均是由給定的基底曲面加上一些特定的幾何限制所定義出來。此 類幾何限制曲面可經由一個通用的表示法定義為n維空間，n>3，的曲面F 在 3維空間的自然投影.pi.(F)。.pi.(F)在3維空間的代數式f(x1,x2, x3)=0可運用代數方法，如resultant或Grobner bases求得，但此類方法 須用到高複雜度的符號運算，在實際應用上極不可行。所以我們應直接處 理幾何限制曲面在n維空間中的表示式F。此篇論文中，我們提出一個方法 來計算出 .pi.(F)的線性分割趨近 (PLA)。此方法直接利用曲面在n維空 間中的定義F，但在3維空間中執行主要的運算。此外我們也提出一些方法 來重新組合及改良所求得的PLA。有了此多邊形表示式，加上硬體對多邊 形的快速處理能力，幾何限制曲面便可快速的顯示出來。 Many surfaces in CAGD, including offsets and blends, are defined from given surfaces subject to certain geometric constraints. These geometrically constrained surfaces can be uniformly defined as the projection of two-dimensional manifolds (2-surfaces) in n-dimensional space, where n>3. Let F be a 2-surface in n-space, and let .pi.(F) be its projection into the subspace spanned by the first three coordinates. The closed-form representation of .pi.(F) is derivable in principle using resultant or Grobner bases. However, this is usually not practical becaues of the symbolic computation entailed has a very high complexity. In consequence, we should work with the n- space representation directly. To polygonize a constrained surface .pi.(F), we propose an algorithm that computes its pisecewise linear approximation (PLA) using the surface definition in n-space, but with major computations performed in 3-space. We also give some methods to retile and refine the PLA. With the polygonal approximation, the fast rendering of geometrically constrained surfaces can be achieved by taking advantage of hardware capabilities.