Construção de Superfícies Suaves com Continuidade Geométrica

ABSTRACT In the computational modeling of solids, the construction of a smooth surface is one of the fundamental requirements in order to apply numerical methods, particularly the Boundary Element Methods. In addition, it is required in many other applications such as computer animation models and reconstruction of medical images. All these applications have the geometric continuity G 1 as the fundamental condition for generating smooth surfaces. In this study, the technique of constructing smooth geometries from boundary curves is used. For this construction, the geometric coordinates and the normals of vertices of the triangular patches are required. Because these informations are not immediately available, especially the normal vector, in this study, the computer graphics software named Blender TM is used to overcome this gap in the acquisition of input data. Thus, the procedure for creating the triangular patches with continuity G 1 and the algorithm for obtaining initial information are presented. An excellent efficiency in imposing of the smoothness and ac-curacy in geometrical representation is observed for both globally parametric and globally non-parametric surfaces.

摘要：在固体的计算建模领域，光滑曲面的构建是应用数值方法，尤其是边界元法（Boundary Element Methods）的核心前提之一。此外，该需求也广泛存在于计算机动画模型、医学图像重建等诸多其他应用场景中。所有此类应用均将几何连续性G₁（geometric continuity G₁）作为生成光滑曲面的基本条件。本研究采用基于边界曲线构建光滑几何形体的技术。为完成该构建流程，需获取三角面片（triangular patches）顶点的几何坐标与法向量。但此类信息（尤其是法向量）无法直接获取，因此本研究借助名为Blender™的计算机图形学软件，以填补输入数据采集环节的缺口。据此，本文提出了构建满足G₁连续性的三角面片的流程，以及获取初始信息的算法。实验结果表明，无论是全局参数化曲面还是全局非参数化曲面，该方法在实现光滑性约束与保障几何表示精度上均展现出优异性能。