參數曲面間的調和曲面計算

Abstract

在幾何與實體模型中,調和曲面的計算可平滑地連接兩個曲面.就代數曲面 而言,其調和曲面可用較低次數的代數式來確實地表示;就參數曲面而言, 通常採用近似法來表示其調和曲面,唯一的例外的是滾球式調和曲面可被 確實地定義在高維度的空間上.在本篇論文中,我們針對一般的參數調和曲 面,提出兩種表示法:第一個是延伸 affine potential method 的概念至 參數曲面上,不同的是,我們以參數曲面的 offset surface 代替代數曲面 的 level surface,且以多項式集來確實地表示該調和曲面,一維參數的 offset-radius 曲面函數提供兩曲面所應 offset 的距離;第二個則是較 複雜的表示法,兩曲面間 offset 距離的關係是二維參數的曲面函數.為了 提昇兩表示法與使用者直覺上相當容易地輸入調和計算所需之規格;而且 對調和曲面外觀型狀的控制有十足的彈性.再者,兩曲面的幾何性質也一併 融入我們的表示法,兩種演算法均被實作在我們的交談示參數調和計算系 統中(IPBS),該系統是架構在 X-Window 上,目前在 SGI 機器上發展. In geometric and solid modeling, the surface blendig is a recurring operation that provides a smooth transition between two surfaces. Blending of implicity defined surfaces can generally be formulated exactly with relatively lower degree. Blending of parametric surfaces, on the other hand, are usually approximated. One exception is that the spherical blending of parametric surfaces can be defined exactly in high dimensional space. In this thesis, two formulations are proposed for computing the general blending of parametric surfaces. The first is an exact formulation that extends the concept of the affine potential method and represents the blend exactly in a high dimensional space as the sweeping surfaces of the intersection between offsets of the base surfaces with radii satisfying a specific one-parameter curve. The second is a relatively complex formulation that defines the blend as the sweeping surface of the intersection between offsets of base surfaces with radii satisfying a specific two-parameter surface. To improve the interactive performance of these formulations, procedural approaches are proposed. Both formulations provide users with not only a simple and intuitive blending specification but also a flexible shape control of the blending surface. Morever, the formulations are implemented as parts of our Interactive Parametric Blending System(IPBS), which is a X-Window based interactive system currently running on SGI machines.