Select Topics and Future Challenges in Discrete Geometry

Abstract

Digital geometry is a relatively new research area. It is difficult to show the characteristics of digital geometry as a well-developed theory. On the other hand, discrete geometry used to focus on combinatorial methods such as simplicial decomposition, counting, and tillings. However, it is now also much interested in differential geometry methods. Many new problems related to digital and discrete geometry are have been discovered and have raised interests from various different research disciplinary areas. In order to synthesize some features, this chapter mainly deals with methodology issues of digital and discrete geometry in terms of future studies. We begin with detailed proofs of two basic theorems in digital and discrete geometry. In these proofs, we show the power of the digital and discrete methods in geometry. Then, we focus on future problems in BigData and the data sciences, including what digital methods can do in random algorithms, manifold learning, and advanced geometric measurements. We also present some questions for graduate students and other researchers to think about.