HCC: Small: Collaborative Research: Graph and Pattern Design on Surfaces

抽象?本研究探讨了表面图案设计的理论和有效算法。表面上的图案出现在许多自然现象中，如树叶、动物纹理和地形，以及人造物体，如折纸、玻璃装饰品和外墙。模式也可以用来描述网络，如街道布局、电网、渡槽和传感器网络。图案设计在艺术、娱乐、建筑、工程、医学、城市规划等领域有着广泛的应用。此外，研究中发展的理论和技术可以使计算几何、矢量和张量场可视化等领域受益。当涉及到表面上的图案设计时，有几个基本的挑战。首先，就图案所包含的对称性和方向而言，缺乏统一的图案数学公式。因此，尽管上述应用程序之间存在内在联系，但它们通常被认为是不相关的。其次，许多过去解决这些问题的方法缺乏分层控制。这是必需的，以便用户可以设计高层次的信息，偶尔低层次的细节和布局算法填补其余的程序。在这项研究中，研究人员探索了一个统一的框架，允许在表面上的图案分层设计。在该框架中，通过张量场设计，在域内任意位置指定方向和对称信息。接下来，使用包含所需方向和对称信息的张量场来生成复合体，复合体可以是点集、图、平铺或它们的任何组合。最后，通过纹理和几何合成将额外的细节添加到复合体上，或者在复合体的细胞内添加子图案。从各种数学领域的思想，如动力系统，张量微积分，微分几何，代数拓扑被借用和应用于本研究。

Abstract ? Zhang/WonkaThe research investigates theory and efficient algorithms for pattern design on surfaces. Patterns on surfaces appear in many natural phenomena such as leaves, animal textures, and terrains as well as man-made objects such as origami, glass ornaments, and facades. Patterns can also be used to describe networks, such as street layouts, power grids, aqueducts, and sensor networks. Pattern design has a wide range of applications in art and entertainment, architecture, engineering, medicine, and city planning. In addition, theory and techniques developed in the research can benefit domains such as computational geometry and vector and tensor field visualization. There are several fundamental challenges when it comes to pattern design on surfaces. First, there is a lack of unified mathematical formulations of patterns in terms of both symmetries and orientations contained in the patterns. Consequently, the aforementioned applications are typically addressed as being unrelated despite the intrinsic links between them. Second, many past approaches to these problems lack hierarchical control. This is required so that the user can design high level information down to occasionally low level specifics and the layout algorithms fill in the rest procedurally. In this research, the investigators explore a unified framework that allows hierarchical design of patterns on surfaces. In this framework, orientation and symmetry information is specified everywhere in the domain through tensor field design. Next, the tensor field which contains desired orientation and symmetry information is used to generate a complex which can be a point set, a graph, a tiling, or any combination of them. Finally, additional details are added onto the complex through texture and geometry synthesis, or sub-patterns are added inside the cells of the complex. Ideas from various mathematical domains such as dynamical systems, tensor calculus, differential geometry, and algebraic topology are borrowed and applied in this research.