參數曲面之邊與端點調合曲面之運算

Abstract

在幾何與實體模型中， 調合曲面（blending surface）的目地是要將在複雜的實體模型中常出現的尖銳的邊或端點加以平滑化。不論對於學術研究機構或電腦輔助設計工具的開發者，調合曲面近來已變成非常重要且相當引人注意的問題。目前研究的重心集中在如何提供更簡便的介面來定義調合曲面，更好或更精確的表示法， 自動且有效率地構建邊及端點調合曲面，以及對於複雜的實體模型能針對邊及端點調合曲面及其拓樸關係有一完整的的解決方式。在這篇論文中， 我們提出了一些新的規格， 公式， 及計算程序來有效構建邊及端點調合曲面。對於變動半徑之邊調合曲面， 以往都是用一些較困難的方法，如需要使用者指定參考曲線及半徑函數。我們則提出了一個通用、簡單且容易使用的幾何限制來自動規範延著尖銳的邊緣半徑之變動。這些幾何限制可以很容易地被轉換為一個 可以用來精確地表示變動半徑調合曲面軸心曲線（spine curve）的非線性系統。所謂的軸心曲線就是在半徑調合曲面中， 滾動的球或圓的中心所經過的軌跡。相對於用複雜的數值方法在高維度空間追蹤此軸心曲線，我們也提出了一個三度空間的步進法來實現這些幾何限制並且有效率地追蹤此軸心曲線。我們的實作顯示此三度空間的步進法比通常必須在九維度空間上進行的數值方法來的有效率。對於端點調合曲面，我們提出了兩種構建的方法，一種是針對具簡單拓樸關係的端點， 另一種則是針對一般的端點。對於有相同凹凸面邊調合曲面的端點， 我們用球狀曲面（spherical patch）以及一些轉換曲面（transitional patches）來表示該端點調合曲面。此種端點調合曲面表示法非常適合用於邊調合曲面也是以變動半徑法來表示的實體。對於一般的端點， 我們提出一個以分割曲面（subdivision surface）來構建端點調合曲面的方法。為了要能用分割曲面來建立端點調合曲面， 我們提出了一個用所謂的開放均勻分割法（open uniform subdivision scheme）來有效的對分割曲面作邊界控制（boundary control） 。當此分割方法用在具特定拓樸的控制網點（control mesh）時，在每一個分割步驟中，控制網點所定義的邊界曲線與原始控制網點所定義的邊界曲線相同，而且與此極限分割曲面（limiting subdivision surface）的相關邊界曲線亦相同。對於僅含四邊的邊界面 （boundary face）的控制網點而言，其交錯切面 （cross tangent） 在任一開放均勻分割步驟都可以被定義。如此， 兩個控制網點所定義的分割曲面就可以在連接時達成 C^0 及 C^1 的連續性。對於在 NURBS 曲面上多邊形洞的填補以及構建由多個 NURBS 曲面所形成的端點的調合曲面， 我們用一個正規 N 邊開放均勻分割曲面（regular N-sided open uniform subdivision surface）來表示此填補曲面及端點調合曲面。此正規 N 邊開放均勻分割曲面是一個藉由將開放均勻分割法用在一個符合此洞或端點調合曲面區域的啟始正規 N 邊控制網點所形成的曲面。在分割的過程中， 像 C^0 及 C^1 的邊界條件可藉由一些精細化（refinement）的步驟來達成。最後， 此正規多邊形開放均勻分割曲面可用 NURBS 曲面來表示。我們並已將所有提出方法實作成為一個 "cggmlib" 函式庫，此函式庫為我們 NURBS 曲面及曲線工具組的一部分。

One of recurring operations in geometric and solid modeling is thesurface blending that aims to round sharp edges and corners occurringalmost everywhere in complex solids. The blending operation has recentlybecome very important and attracted considerable attentions from bothacademic institutes and CAD-tool developers. Current focuses have beenin providing easier interfaces for the blending specification, exact orbetter representations, automatic and efficient construction ofblending surfaces for both edges and corners, and complete solutions for complex solids that deal with edge and vertex blend,and the topological problems incurred all together.In this thesis, we propose new specification, formulation, andcomputation procedures for the construction of edge and vertex blends.For variable-radius edge blend, instead of the current troublesome specification that usually involves user-defined reference curveand variable-radius function, we propose a general, simple, anduser-friendly geometric constraints that automatically constrainsthe radius variation along sharp edges.These geometric constraints are easilytranslated to a nonlinear system that exactly represents the variable-radius spine curve, which is the trajectory of the rollin gspheres or circles for the radius edge blends. The spine curve is traced efficiently by a proposed 3D marching algorithm that successfully implements the radius constraints geometrically, rather than usingthe expensive numerical tracing for the dimension one nonlinear system.The 3D marching algorithm has been shown much efficiently than the necessary numerical tracing usually carried out in thenine-dimensional space. We also proposed procedural, exact, andapproximate representation for the resulting variable-radius blends.For vertex blend, we propose two construction schemes, one for cornerswith simple topology and the other for general corners. For cornershaving edges of the same convexity, we represent the vertex blend usingspherical patches together with some transitional patches.Such a spherical patchrepresentation is desirable when edges are rounded with variable-radiusblend. For general corners, we propose a subdivision surface basedmethod for the vertex blend construction.To be able to apply the subdivision surface to blending,we propose an effective boundary controlthat employs the so called quadratic open uniform subdivision scheme.When this subdivision scheme is applied to control meshes withrestricted topology, each boundary curve defined by the control mesh atany subdivision step is also defined by the original control mesh, and,furthermore, is identically the corresponding boundary curve of thelimiting subdivision surface. We also claim that for control mesheswith 4-sided boundary faces, the cross-tangent condition can bede fined during any step of the open uniform subdivision and hencethe subdivision surfaces derived from two control meshes of this typecan be joined with C^0 and C^1-continuity at any step of the subdivision.For filling an N-sided hole on a NURBS surface and constructingblending for a corner formed by NURBS surfaces, we represent the fillingsurface or vertex blend using a so called regular N-sided openuniform subdivision surface derived by applying the openuniform subdivision scheme to an initial regular N-sided controlmesh which confines to the hole or the region of the vertex blend.In the course of subdivision, the boundary conditions such as C^0 andC^1-continuity are ensured by certain refinement steps.The resulting regular N-sided open uniform subdivision surface is finally represented by NURBS surfaces.All the proposed methods have been implemented as part of our"cggmlib" library, which is a subset of our toolkit for NURB Scurves and surfaces.