Geometric Computing

我的研究领域是计算几何，这是一个致力于设计和分析几何问题的算法和数据结构的领域。我的研究既关注理论问题，也关注实践问题。我主要研究的几何应用领域包括几何网络布线、几何数据结构、网格生成/操作、图形绘制、可视化和模式识别。我的研究的主要目标一直是，并将继续通过为具有理论性能保证的应用几何问题找到实用的算法解决方案，来弥合理论和实际效率之间的差距。下面，我提供了几个我目前专注于几何问题类型的例子。 无线设备之间的通信具有固有的几何组件，因为这些设备具有不断变化的物理位置。目前，我正在应用我在几何方面的专业知识来开发算法，以利用这种几何组件，从而在无线设备之间实现更高效、更可靠的通信。这一领域的问题尤其具有挑战性，因为网络是动态的，即无线设备通常是移动的。此外，这种动态的性质意味着没有一个单一的实体拥有整个网络的完整知识，这需要开发在线且使用较少内存的算法。对无线网络建模的标准方法是使用几何图。几何图是这样一种图，其中每个顶点都是平面上的一个点(也可能是更高的维度，具体取决于应用)，而边是连接两点的线段或曲线。我在这方面的研究主要集中在以下几个方面：(1)如何在几何网络中有效地布线(即找到一条路径)；(2)如何构造具有某些理想性质的几何网络？ 设施选址是一个区域，其中主要关注的是以针对一组客户优化特定标准的方式在区域中放置一个或多个设施。例如，在试图决定在城市的什么地方建立新的消防站时，人们可能想要将其放置在对目标人群的响应时间最短的地方。设施选址问题本质上是几何问题。目前，我正在研究这个问题的几个变种，并试图开发有效的解决方案。 制造业中关于每一种制造过程(如NC加工或铸造)的一个基本问题是：给定一个对象，它是否可以使用特定的过程来制造？我已经成功地开发并应用了计算几何技术来回答这类基本问题。通常，几何约束决定了是否可以通过特定过程来构建对象。通过对过程进行几何建模，我们可以通过确定对象的CAD模型是否可以通过特定的制造过程来确定是否可以在设计阶段构建对象。我计划继续开发从几何角度回答这些问题的算法。

My research area is Computational Geometry, a field devoted to the design and analysis of algorithms and data structures for geometric problems. My research focuses both on theoretical and practical issues. The geometric application areas I concentrate on include routing on geometric networks, geometric data structures, mesh generation/manipulation, graph drawing, visualization and pattern recognition. The main goal of my research has been and continues to be to bridge the gap between theoretical and practical efficiency by finding practical algorithmic solutions to applied geometric problems that have theoretical performance guarantees. Below, I provide a few examples of the type of geometric problems I am currently concentrating on. Communication between wireless devices has an inherent geometric component since these devices have a physical location which is constantly changing. Currently, I am applying my expertise in geometry to develop algorithms that take advantage of this geometric component resulting in more efficient and reliable communication between wireless devices. The problems in this area are particularly challenging since the network is dynamic, i.e. the wireless devices are usually moving. Moreover, this dynamic nature means that no single entity has complete knowledge of the whole network, requiring the development of algorithms that are online and use little memory. The standard method for modelling wireless networks is through the use of geometric graphs. A geometric graph is a graph where each vertex is a point in the plane (or possibly higher dimensions depending on the application) and an edge is a line segment or a curve joining two points. My research in this area focuses on the following themes: (1) How to route (i.e. find a path) efficiently in a geometric network, (2) How to construct geometric networks with certain desirable properties? Facility Location is an area where the main focus consists of placing one or more facilities in a region in a manner that optimizes certain criteria with respect to a set of clients. For example, in trying to decide where to build a new fire station in a city, one may want to place it somewhere that minimizes the response time to the target population. Facility Location problems are inherently geometric. Currently, I am studying several variants of this problem and trying to develop efficient solutions. A fundamental question in the manufacturing industry concerning every type of manufacturing process (such as NC-machining or casting) is: Given an object, can it be built using a particular process? I have successfully developed and applied computational geometry techniques to answer some basic problems of this type. Often it is geometric constraints that determine if an object can be built by a particular process. By modelling a process geometrically, we can determine if an object can be constructed at the design stage by determing if a CAD model of an object can be built by a particular manufacturing process. I plan to continue developing algorithms for answering these questions from a geometric perspective.