可變半徑調合曲面之自動產生

Abstract

在幾何與實體模型中，調合曲面(blending surface) 是用來平滑地連接 兩個或多個曲面。調合曲面和本來曲面相接的地方叫做連接曲線( linkage curve)，原來的曲面則叫做基本曲面(base surface)。有一種常 用的調合曲面型態為滾球調合曲面(spherical blending surface)，可定 義成由一個沿曲面滾動的球所造成的曲面，滾球所經過的圓心稱為軸心曲 線(spine curve)。在滾球調合曲面中，若球的半徑固定，稱做固定半徑 調合曲面 (constant-radius blending)；若球的半徑可變動，則稱為可 變半徑調合曲面(variable-radius blending)。對於可變半徑調合曲面， 半徑變動的方式，通常由一個半徑函數(radius function) 來表示。在某 些情況下，半徑函數很難去定義。本篇論文提出一個如何用一些標準來控 制調合曲面的形狀，進而自動產生半徑函數的方法。對於代數曲面，我們 用較高次元的方法(higher-dimensional approach)；對於參數曲面，我 們用步驟法 (marching method) 來產生軸心曲線跟連接曲線．此外，我 們亦提出一個方法來產生調合曲面，並以 NURBS 來表示。 A blending surface in geometric and solid modeling is the surface that smoothly connect two or more surfaces along some arbitrary curves, called linkage curves or contact curves. The surfaces to be blended are called the base surfaces. The spherical blending is defined by moving a sphere or circular arc along a spine curve. When the radius of the sphere is constant, the blending is called constant-radius blending. In the other hand, if the radius is variable, the blending is called variable-radius blending, and the function by which the radius variates is called the radius function. In some cases, the radius function is hard to define. In the thesis, we discuss some new methods that generate the radius function automatically by adding some criteria to control the shape of the blending surface. We implement them for both algebraic and parametric surfaces. For algebraic surfaces we use the higher- dimensional approach and for parametric surfaces we use a marching method to generate the spine curve and linkage curves. We also propose a method that constructs the blending surface based on generated spine curve and linkage curves, and represent the resulting surface as NURBS surface patches.