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This chapter overviews the basic concepts of functions and relations including continuous functions and their differentiations in Euclidean space. We also introduce functions in discrete spaces, specifically graphs and grid spaces. This chapter contains three parts: continuous functions in Euclidean space, graphs and discrete spaces, and advanced topics including topological spaces. Philosophically, a space is a relation or a collection of relations over a set. A graph is a relation; an m-dimensional Euclidean Space is a collection of relations over points in R m . This chapter is written in a gradual progression to provide a simple and interesting review of the background knowledge. Some of the more profound issues will be presented in later chapters as needed. Those with a strong background in mathematics can skip some or all of this chapter. This chapter prepares the necessary basic knowledge for the rest of the book.

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In an recent paper, {\it L. Chen, Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (II)}, we discussed two algorithms for deforming and contracting a simply connected discrete closed manifold to a discrete sphere. The first algorithm was the continuation of the work initiated in {\it L. Chen, Algorithms for Deforming and Contracting Simply Connected Discrete Closed Manifolds (I)}; the second algorithm was more direct treatment of contraction for discrete manifolds. In this short paper, we clarify that we can use the same method to the standard piecewise linear (PL) complexes from triangulation of general smooth manifolds. Our discussion is based on the triangulation techniques invented by Cairns, Whitehead and Whitney more than half-century ago. Note that, in this paper, we try to use PL or simplicial complexes to replace some concepts of discrete manifolds. However, some details in original paper, might also need to be slightly modified. This paper also adds more details and corrects several typos in the theorem proof in the first paper above that was done in a timely manner. We will post the detailed algorithm $/$ procedure of practical triangulations next.

In Chap. 2, we introduced some algorithms for graphs. In this chapter, we specifically design algorithms for digital object recognition and tracking. These algorithms are mainly for digital surfaces and manifolds. There are two types of questions to solve in this chapter: (1) Given a set of data M, decide or recognize whether the data represents a geometric shape, specifically a curve, surface, or solid object, and (2) Extract the curve or surface components of the data set. The main task is to extract the boundary of a surface or a 3D manifold. We also design algorithms for these problems for higher dimensional manifolds. In this chapter, we deal with various important tasks in digital and discrete geometry in an ideal situation such as no noise with perfect data formats. We then design algorithms to find solutions for these problems. In Chap. 11, we specifically discuss the data in the format of randomly collected points, called cloud data or scattered data sets that usually do not form a specific geometric shape. In such a case, the researcher needs to estimate the best possible shape for the data. These types of problems are usually related to geometric processing.

The Handbook of Discrete and Computational Geometry is intended as a reference book fully accessible to nonspecialists as well as specialists, covering all major aspects of both fields. The book offers the most important results and methods in discrete and computational geometry to those who use them in their work, both in the academic world-as researchers in mathematics and computer science-and in the professional world-as practitioners in fields as diverse as operations research, molecular biology, and robotics. Discrete geometry has contributed significantly to the growth of discrete mathematics in recent years. This has been fueled partly by the advent of powerful computers and by the recent explosion of activity in the relatively young field of computational geometry. This synthesis between discrete and computational geometry lies at the heart of this Handbook. A growing list of application fields includes combinatorial optimization, computer-aided design, computer graphics, crystallography, data analysis, error-correcting codes, geographic information systems, motion planning, operations research, pattern recognition, robotics, solid modeling, and tomography.

The primary purpose of a programming language is to assist the programmer in the practice of her art. Each language is either
designed for a class of problems or supports a different style of programming. In other words, a programming language turns
the computer into a ‘virtual machine’ whose features and capabilities are unlimited. In this article, we illustrate these
aspects through a language similar tologo. Programs are developed to draw geometric pictures using this language.

A fundamental task of computational geometry is identifying concepts, properties and techniques which help efficient algorithmic implementations for geometric problems. The approach taken here is the presentations of algorithms and the evaluation of their worst case complexity. The particular problems addressed include geometric searching and retrieval, convex hull construction and related problems, proximity, intersection and the geometry of rectangles

Discrete surfaces and manifolds: a theory of digital-discrete geometry and topology. Scientific and Practical Computing

- L Chen