Grant: Discovery Launch Supplement

Computational Geometry problems are at the heart of many of the current challenges in industrial applications. In particular, visibility problems have a wide range of direct commercial applications in areas including Manufacturing, Robotics, and Biotechnology. The main goal of the proposed research is to advance algorithms on visibility and their applications. There are a wide range of application-specific variations of visibility defined - I will start by targeting k-visibility and visibility with reflection, due to their direct relevance in industrial settings. I will target topics that are yet to be thoroughly explored for these types of visibility. This program will be achieved through the defined FSTOs and their ASTOs, briefly summarized below: Fundamental short-term objectives (a. FSTOs): a.1. k-visibility: To strengthen the theoretical concepts of k-visibility I will work on point visibility, segment visibility, and the Art Gallery problem. a.2. Visibility with reflection: To push forward the theoretical understanding of visibility with reflection, I will study point visibility and design a cell decomposition to be used to report the visibility polygon of a query point efficiently to pioneer this topic. Building upon our findings in the FSTOs, the applied short-term objectives (b. ASTOs) will explore three applications of visibility with reflection and k-visibility: b.1. Pursuit-Evasion (PE): Considering k-visibility for the pursuers, I will study these three problems: 1-searcher path, 2-searcher path, and merge between regions. The results of these problems will be applied for search and rescue tasks using wireless signals. b.2. Simultaneous Localization and Mapping (SLAM): I have applied k-visibility for mapping indoor environments using WiFi signals, and will continue this research by exploring the following problems: one stationary router, multiple stationary routers, multiple moving routers. By extending our prototype, a new method for mapping without cameras or other sensors will be enabled. b.3. Control Systems: By applying visibility with reflection to control systems, the problem of bouncing robots will be explored for potential applications in infrastructure repair, demining, biotechnology. The impact of this program will be a strengthening of our understanding of key problems at the core of Computational Geometry, and to drive innovation through interdisciplinary research and industry partnership. I will achieve this by advancing our knowledge of the most prominent variations of visibility and other geometric topics, such as line arrangement, and cell decomposition as well as directly benefiting the industries that are dependent on the field, such as Robotics, CAD, industrial manufacturing, and Biotechnology. The HQP that will be involved in this research will develop interdisciplinary skills and pioneer the adoption of state-of-art algorithms for future careers in academia and industry.