Table of Contents
Overview
Definition and Scope
Computational geometry is defined as the study of algorithmic problems that involve geometric configurations and spatial relationships. It combines principles of geometry with algorithmic techniques to address complex spatial problems, making it a vital area of research and application in various fields, including computer graphics, robotics, and geographic information systems.[7.1] The scope of computational geometry encompasses a wide range of applications, from video game development to autonomous vehicle navigation, where it plays a crucial role in tasks such as obstacle avoidance and path planning.[33.1] The field has evolved significantly since its inception, with its roots tracing back to ancient Greek geometry, but it gained formal recognition as a distinct area of study in the 1970s, particularly through the work of M. I. Shamos.[7.1] Modern computational geometry is characterized by its focus on developing efficient algorithms for processing geometric data, which is essential for handling large datasets that may contain millions of points.[6.1] The field continues to grow, with ongoing research addressing both theoretical aspects and practical implementations, as evidenced by dedicated journals and conferences that facilitate the dissemination of new findings.[4.1]Importance in Various Fields
Computational geometry plays a crucial role in various fields by providing efficient algorithms to solve complex spatial problems. At its core, it involves the design, analysis, and application of algorithms that handle geometric data, producing meaningful geometric results essential for real-time applications, such as in gaming and robotics.[13.1] One of the fundamental problems addressed by computational geometry is the computation of the convex hull of a set of points, which is vital in numerous applications including image processing, route planning, and object modeling.[24.1] The Graham Scan algorithm, recognized for its efficiency, computes the convex hull in O(n log n) time, making it a significant contribution to the field.[22.1] This algorithm exemplifies how computational geometry can optimize processes that require spatial reasoning. In robotics, computational geometry enhances pathfinding and obstacle avoidance through algorithms like Voronoi diagrams and Delaunay triangulations. Voronoi diagrams help represent complex environments populated with multiple agents, transforming obstacle avoidance into a mathematically tractable problem.[17.1] Similarly, Delaunay triangulations facilitate the generation of collision-free paths for mobile robots, ensuring that the paths are not only feasible but also efficient.[20.1] These algorithms are integral to motion planning, allowing robots to navigate effectively in dynamic environments.[19.1] The significance of computational geometry extends beyond theoretical applications; it is instrumental in practical scenarios where spatial data must be processed efficiently. The study of convex hulls, for instance, is not only a foundational concept in computational geometry but also serves as a tool for constructing other geometric structures, showcasing its utility across various disciplines.[24.1] Thus, the algorithms developed within this field have far-reaching implications, influencing advancements in computer science, robotics, and beyond.In this section:
History
Early Developments
The origins of computational geometry can be traced back to ancient Greek geometry, where ruler and compass constructions served as early algorithms for producing geometric objects. However, the modern discipline of computational geometry is often credited to M. I. Shamos, whose 1975 Ph.D. dissertation addressed several fundamental geometric problems and introduced many more, marking a significant turning point in the field.[47.1] Some scholars argue that the discipline was formally established with Shamos's earlier paper on geometric complexity in 1975, while others suggest that it began with his Ph.D. thesis in 1978.[48.1] The term "computational geometry" has been in use since 1971, indicating that the conceptual foundations of the field were being laid even before Shamos's contributions.[49.1] Despite its relatively recent formalization, computational geometry is considered one of the oldest fields in computing, with a history that extends back to antiquity. The field has evolved significantly, particularly in relation to computational complexity, which plays a crucial role in the practical application of algorithms, especially when dealing with large datasets containing millions of points.[6.1]Key Milestones in Computational Geometry
The development of computational geometry has been marked by several key milestones that have significantly shaped the field. The discipline emerged in the early 1970s, with its name being coined during this period, and has since experienced explosive growth, particularly influenced by advancements in computer graphics.[55.1] A pivotal moment in the transition from classical to computational geometry occurred with M. I. Shamos's 1975 Ph.D. dissertation, which addressed fundamental geometric problems and introduced many new ones, effectively laying the groundwork for modern computational geometry.[56.1] In the 1980s, the field began to incorporate tools from algebraic geometry, leading to a deeper interaction between geometric modeling and computational algebraic geometry by the turn of the century.[58.1] This period also saw the introduction of randomized algorithms, which became prominent in the 1980s, marking a shift from the predominantly deterministic algorithms that characterized earlier work in the field.[64.1] Among the foundational algorithms developed during the early days of computational geometry were those addressing the convex hull problem, with notable techniques such as Graham's scan and Jarvis's march being established.[66.1] These algorithms not only provided solutions to specific geometric problems but also influenced contemporary research and applications across various domains, including computer graphics, geographic information systems (GIS), and robotics.[68.1] The increasing complexity and diversity of problems in computational geometry led to the emergence of several independent subareas within the field, each focusing on different aspects of geometric computation.[69.1] This evolution reflects the discipline's adaptability and its ongoing relevance in addressing both theoretical and practical challenges in geometry.In this section:
Main Branches
Combinatorial Computational Geometry
Combinatorial computational geometry is a significant branch of computational geometry that focuses on geometric objects as discrete entities. This field emerged in the 1970s, with the term "computational geometry" being first used in this context by Preparata and Shamos in their foundational work.[89.1] The primary emphasis of combinatorial computational geometry is on the combinatorial properties of discrete geometric structures, such as arrangements of lines, subdivisions, coverings, and polytopes.[96.1] In contrast to numerical computational geometry, which employs numerical methods to solve geometric problems, combinatorial computational geometry is concerned with the algorithmic study of elementary geometric problems.[98.1] This distinction is crucial as it influences the efficiency and effectiveness of algorithms designed to address real-world challenges. The principles of treating geometric objects as discrete entities allow for the development of efficient algorithms that can handle finite sets of points, lines, and other geometric figures, thereby enhancing computational performance.[99.1] The interplay between combinatorial geometry and algorithm theory has been fruitful, leading to advancements in various applied fields, including robotics and computer graphics.[116.1] Researchers in these areas often seek methods that integrate well into practical systems, although they may prioritize practical outcomes over the underlying combinatorial complexities.[115.1] Overall, combinatorial computational geometry plays a vital role in the broader landscape of computational geometry, providing essential tools and frameworks for solving complex geometric problems.Numerical Computational Geometry
Numerical computational geometry is a significant branch that focuses on the application of numerical methods to solve geometric problems, enhancing both accuracy and efficiency compared to traditional combinatorial approaches. One of the primary computational methods utilized in this field is homotopy continuation, which involves forming a homotopy between two polynomial systems to trace isolated solutions from one system to another, thereby facilitating algebraic geometric computations.[103.1] This method is particularly relevant in numerical algebraic geometry, where the foundation lies in solving systems of polynomial equations defined over subfields of complex numbers.[102.1] The advantages of numerical methods in computational geometry are evident in various applications, including computational fluid dynamics, structural analysis, and kinematic analysis, where these methods provide robust solutions to complex geometric configurations.[104.1] Furthermore, a comparison of different computational methods reveals that numerical techniques, such as the Boundary Element Method (BEM) and Finite Element Method (FEM), demonstrate a high degree of accuracy, with discrepancies as low as 1% in two-dimensional problems and 2% in three-dimensional cases.[100.1] As technology advances, the integration of numerical methods in computational geometry is expected to evolve, leading to new applications and improvements in imaging technology, which will allow for higher spatial resolution in data processing.[117.1] This convergence of numerical methods with emerging technologies signifies a transformative potential for the field, paving the way for innovative solutions in various domains.In this section:
Computational Fluid DynamicsStructural AnalysisKinematicBoundary Element MethodFinite Element Method
Recent Advancements
Theoretical Developments
Recent advancements in computational geometry have been marked by significant theoretical developments that enhance the efficiency and applicability of geometric algorithms. Computational geometry, fundamentally concerned with the study of algorithms for solving problems related to geometric objects in Euclidean space, has evolved since its inception in the mid-1970s, focusing on the design and analysis of algorithms for various geometric problems.[130.1] One of the core areas of research has been the development of efficient algorithms for computing convex hulls, which are essential in various applications such as computer graphics, pattern recognition, and computational biology.[150.1] For instance, Graham's scan algorithm, which sorts points and applies a linear-time scanning method, exemplifies how foundational algorithms can optimize performance by reducing worst-case running times.[151.1] Additionally, advancements in triangulation algorithms, such as Delaunay triangulation and Voronoi diagrams, have demonstrated their critical applications in fields like mobile robotics and computer graphics.[153.1] These algorithms not only facilitate the efficient processing of geometric data but also enhance the understanding of spatial relationships among geometric entities. Recent theoretical developments also emphasize the integration of classical geometric methods with modern computational techniques. This synergy has led to innovative approaches in geometric modeling and approximation theory, as seen in the works published in specialized journals focusing on computer-aided geometric design.[163.1] Such interactions are pivotal in addressing contemporary computational challenges, showcasing the relevance of classical geometry in modern applications.Applications in Modern Technology
Computational geometry has found extensive applications in modern technology, particularly in fields such as computer graphics, robotics, and artificial intelligence (AI). The interaction between computational geometry and computer graphics is significant, as advancements in geometric algorithms have directly influenced rendering techniques and visual content creation. For instance, spatial subdivisions studied through computational geometry have been applied in computer graphics, leading to the development of algorithms for hidden surface removal, which are crucial for rendering realistic images.[134.1] In computer graphics, computational geometry plays a vital role in generating and manipulating visual objects. It facilitates the creation of 2D and 3D images by calculating the relationships between geometric shapes and their transformations, thus powering techniques such as mesh generation and polygon triangulation.[141.1] Recent developments have emphasized problems relevant to computer graphics, including convex hulls and visibility properties of polygons, which enhance rendering capabilities.[137.1] Furthermore, GPU-based rendering techniques have emerged as a powerful tool in visual content creation, leveraging the parallel processing capabilities of modern graphics cards to handle complex geometric computations.[139.1] In the realm of robotics, computational geometry is increasingly integrated with AI to solve path-planning problems for both single and multi-robot systems. This integration has led to the development of complex optimization solutions that utilize geometric algorithms to navigate environments effectively.[143.1] Additionally, point cloud registration, a process essential in computer vision, relies heavily on computational geometry and is applicable in various domains, including robotics and medical imaging.[144.1] The understanding of geometric principles is crucial for devising effective robotic solutions, highlighting the interdisciplinary nature of advancements in this field.[145.1]In this section:
Geometric AlgorithmsAnalysis Of AlgorithmsPattern RecognitionComputational BiologyDelaunay Triangulation
Algorithms And Techniques
Fundamental Algorithms
Computational geometry encompasses a variety of fundamental algorithms that are crucial for solving spatial problems. One of the most significant algorithms in this field is the convex hull algorithm, which identifies the smallest convex polygon that contains a given set of points. This concept is foundational in computational geometry and has applications across multiple domains, including computer graphics, robotics, and image processing.[180.1] The convex hull can be computed using several algorithms, such as Graham's scan and Jarvis' algorithm, which utilize different approaches to determine the convex boundary of a point set.[183.1] Additionally, approximation algorithms are often employed for very large datasets or real-time applications, providing close approximations of the convex hull when exact calculations may be computationally expensive.[182.1] Beyond its standalone utility, the convex hull serves as a building block for other important structures in computational geometry, such as Voronoi diagrams and Delaunay triangulations. These structures are essential for various applications, including unsupervised image analysis and collision detection in robotics.[185.1] The interrelation of these algorithms highlights the importance of the convex hull in both theoretical and practical aspects of computational geometry.[181.1]Advanced Techniques in Computational Geometry
Advanced techniques in computational geometry encompass a variety of algorithms that significantly enhance the efficiency and effectiveness of spatial data processing and path planning in robotic systems. One notable approach is the use of spatial indexing techniques, which are crucial for organizing and accessing spatial data efficiently. These techniques, such as R-trees, Quadtrees, and Geohashes, help reduce the time complexity of spatial queries from O(N) to O(log N) or better, where N represents the number of spatial objects involved.[177.1] By structuring data according to its spatial properties, these indexing methods facilitate faster and more scalable retrieval of geographic information, which is essential in applications like Geographic Information Systems (GIS).[178.1] In the realm of path planning for mobile robots, algorithms such as Voronoi diagrams and convex hulls play a pivotal role. Voronoi diagrams, for instance, are utilized to represent environments and compute safe paths by identifying the maximal distance to obstacles, thereby creating a pixel-based roadmap that guides robots through complex spaces.[190.1] This method allows for the generation of efficient paths that navigate around obstacles while optimizing route selection.[187.1] Additionally, convex hull algorithms are employed in robotic collision detection, enabling robots to navigate effectively within defined boundaries by approximating free space.[192.1] The integration of these advanced computational geometry techniques not only enhances the performance of robotic systems but also broadens the scope of applications in various fields, including computer graphics, geographic information systems, and robotics.[173.1] As the discipline continues to evolve, the development of new algorithms and optimization methods will further improve the capabilities of spatial data management and robotic navigation.In this section:
Applications
Computer Graphics and Visualization
Computational geometry plays a crucial role in computer graphics and visualization, particularly in the creation and manipulation of geometric representations. One of the fundamental tasks in this field is the generation of polygonal mesh representations of three-dimensional objects, which is essential for rendering scenes in computer graphics.[219.1] Algorithms such as those for computing the intersection of geometric primitives—lines, segments, and planes—are vital for accurately rendering 3D environments.[220.1] A significant application of computational geometry in computer graphics is the Convex Hull problem, which involves finding the smallest convex polygon that can enclose a set of points in a 2D plane. This problem has various applications, including collision detection and geographic information systems (GIS).[230.1] For instance, convex hulls are frequently utilized in facial recognition and self-driving cars, as they allow algorithms to interpret data from cameras more consistently, facilitating better learning and object avoidance.[221.1] Moreover, the optimization of polygonal meshes is another critical area where computational geometry is applied. Techniques for optimizing mesh quality and ensuring visibility of shapes are essential for real-time rendering in graphics applications.[224.1] The Graham's Scan Algorithm, known for its efficiency in finding convex hulls, exemplifies the theoretical foundations that support these practical applications, with a time complexity of O(N log N).[247.1]Robotics and Geographic Information Systems (GIS)
In the field of robotics, computational geometry plays a crucial role in enhancing navigation and obstacle avoidance capabilities, particularly in autonomous vehicles. Algorithms such as convex hulls and Voronoi diagrams are instrumental in these applications. The convex hull algorithm aids in obstacle detection and avoidance by identifying the area of obstacles based on image features, allowing for real-time processing across various scenarios.[227.1] This method effectively enables autonomous vehicles to navigate complex environments by determining the boundaries of obstacles and optimizing their paths accordingly. Voronoi diagrams also contribute significantly to robotic navigation. They create spatial partitions that maximize the distance between obstacles, thereby facilitating efficient path planning and coverage analysis.[229.1] By providing a comprehensive view of free space, Voronoi diagrams assist in the development of navigation strategies that enhance the safety and efficiency of autonomous systems. In the realm of Geographic Information Systems (GIS), computational geometry is foundational for processing and analyzing spatial data. GIS technologies utilize geometric algorithms for tasks such as geometric network analysis, spatial clustering, and map overlay operations, which are essential for urban planning and environmental monitoring.[243.1] The integration of computational geometry into GIS enhances the accuracy and efficiency of spatial data analysis, enabling better decision-making in infrastructure development and resource management.[242.1] Moreover, advancements in computational geometry have led to the creation of sophisticated spatial analytics tools that combine geospatial intelligence with artificial intelligence (AI), known as GeoAI. This integration has improved the capabilities of GIS in handling big spatial data, allowing for more effective analysis and visualization of complex spatial phenomena.[236.1] As the demand for precise spatial data continues to grow, the role of computational geometry in both robotics and GIS is expected to expand, driving innovations in urban planning and environmental monitoring.In this section:
Challenges And Future Directions
Current Challenges in the Field
Computational geometry faces several current challenges that stem from the complexity of geometric problems and the evolving landscape of computational techniques. One significant challenge is the need for efficient algorithms to solve geometric problems, which often involve points, lines, polygons, and other geometric shapes in two or three-dimensional space. While many geometric algorithms can be extended to higher dimensions, this extension introduces new challenges and complexities that must be addressed.[259.1] The rise of geometric computation, which is expected to dominate the field in the coming decade, presents both opportunities and challenges for computational geometry. Rapid advances in computer hardware and visualization systems are facilitating the integration of geometric computing into various applications, yet they also require the development of new algorithms that can leverage these advancements effectively.[258.1] Moreover, specific competitions such as the CG:Challenge and the CG:SHOP Challenge highlight the ongoing efforts to tackle difficult geometric optimization problems. These events not only create visibility for effective problem-solving but also provide a platform for measuring the success of proposed solutions based on computational efficiency.[256.1] In the realm of higher-dimensional geometric problems, unique challenges arise, particularly in areas such as nearest neighbor search. Current algorithms often employ dimensionality reduction techniques, such as the Johnson-Lindenstrauss transform, to manage these complexities.[272.1] Additionally, the emergence of asymptotic geometric analysis as a new area of study reflects the deep connections between computational geometry and other mathematical disciplines, including functional analysis and probability theory.[271.1] Finally, the trade-offs between algorithmic efficiency and accuracy remain a critical consideration in computational geometry, especially in applications involving machine learning. Various regression algorithms, such as Linear Regression and Random Forest Regression, illustrate the need to balance precision with computational performance, as even minor improvements in data structure efficiency can lead to substantial performance gains in geospatial projects.[287.1] Addressing these challenges is essential for advancing the field of computational geometry and enhancing its applicability across various domains.Emerging Trends and Future Research Areas
Emerging trends in computational geometry are increasingly intertwined with advancements in machine learning and artificial intelligence (AI). The integration of these fields is expected to address significant challenges associated with big data and high-dimensional data structures. A key challenge in machine learning is the identification of geometric structures within high-dimensional data, as many algorithms traditionally assume data resides in a Euclidean space. However, applications often involve non-Euclidean data types, such as graphs and matrices, necessitating the development of geometric machine learning methods tailored for these complexities.[269.1] The sheer scale and complexity of big data present additional obstacles, including issues related to data privacy, security, and scalability, which require careful consideration in the implementation of machine learning techniques.[268.1] Computational geometry plays a crucial role in this context, as it provides algorithms that can optimize and enhance the performance of AI systems, particularly in tasks involving spatial reasoning and geometric constraints.[273.1] By leveraging computational geometry, AI models can effectively extract geometric features from visual data, thereby improving object identification and analysis.[273.1] Future research directions in computational geometry are likely to focus on several promising areas. These include the development of efficient surrogate or reduced-order models to enhance computational efficiency when traditional methods are intractable, and the assimilation of additional data to improve accuracy in scenarios where traditional methods perform poorly.[266.1] Furthermore, there is a growing interest in solving traditionally "unsolvable" models, particularly in cases where problems are ill-posed due to incomplete information.[266.1] The concept of digital twins, which combines physics-based and data-driven models to create virtual replicas of physical systems, is also gaining traction, as it allows for fast and accurate responses necessary for diagnosis and decision-making.[266.1] The emergence of geometric intelligence represents another significant trend, as it aims to reshape AI through shape-based learning. This innovative approach could lead to AI systems that possess a more human-like understanding of 3D spaces, enhancing their ability to reason about complex patterns in data.[274.1] Advanced AI systems, such as AlphaGeometry, are being developed specifically for geometric computations and spatial analysis, utilizing deep learning and symbolic logic to tackle complex geometric problems more effectively than traditional methods.[275.1]In this section:
Optimization ProblemsComputational EfficiencyDimensionality ReductionGeometric AnalysisFunctional Analysis
References
[4] PDF — The field of computational geometry grew rapidly in the late 70's and through the 80's and 90's, and it is still a very active field of research. Historically, computational geometry devel-oped as a generalization of the study of algorithms for sorting and searching in 1-dimensional space to problems involving multi-dimensional inputs.
[6] Computational geometry - Wikipedia — While modern computational geometry is a recent development, it is one of the oldest fields of computing with a history stretching back to antiquity. Computational complexity is central to computational geometry, with great practical significance if algorithms are used on very large datasets containing tens or hundreds of millions of points.
[7] Computational geometry | Encyclopedia of Computer Science — Computational geometry is the study of algorithmic problems involving geometry. Although the ruler and compass constructions of ancient Greek geometry were essentially algorithms for producing geometric objects, modern computational geometry begins with M. I. Shamos's 1975 Ph.D. dissertation, which solved several fundamental geometric problems and posed many more.
[13] Exploring Computational Geometry: Applying Algorithms to Solve ... — Exploring Computational Geometry: Applying Algorithms to Solve Geometric Problems – AlgoCademy Blog Exploring Computational Geometry: Applying Algorithms to Solve Geometric Problems Computational geometry is a fascinating field that combines the principles of geometry with the power of algorithms to solve complex spatial problems. The Graham Scan algorithm is an efficient method for computing the convex hull of a set of points. Finding the closest pair of points in a set is another fundamental problem in computational geometry. Determining whether a point lies inside a polygon is a common problem in computational geometry. Motion planning algorithms use computational geometry techniques to find collision-free paths for robots or virtual characters in complex environments. Computational geometry is a fascinating field that combines mathematical principles with algorithmic thinking to solve complex spatial problems.
[17] PDF — Our method relies on the computation of Voronoi diagrams, which are widely used in other path planning and collision avoidance works, but typically the robots move along the edges of a static Voronoi diagram -. Instead, our use of Voronoi cells is inspired by coverage control algorithms such as , .
[19] Delaunay Triangulation: A Powerful Tool for Path Planning in Robotics — Motion Planning: Delaunay triangulations can be integrated with motion planning algorithms to not only determine a collision-free path but also generate feasible robot motions to follow that path
[20] An Enhanced Dynamic Delaunay Triangulation-Based Path Planning ... — An enhanced dynamic Delaunay Triangulation-based (DT) path planning approach is proposed for mobile robots to plan and navigate a path successfully in the context of the Autonomous Challenge of
[22] Convex Hull Algorithm - GeeksforGeeks — The Convex Hull Algorithm is used to find the convex hull of a set of points in computational geometry. The convex hull is the smallest convex set that encloses all the points, forming a convex polygon. This algorithm is important in various applications such as image processing, route planning, and object modeling. What is Convex Hull?
[24] Convex hull algorithms - Wikipedia — Algorithms that construct convex hulls of various objects have a broad range of applications in mathematics and computer science. In computational geometry, numerous algorithms are proposed for computing the convex hull of a finite set of points, with various computational complexities.
[33] Computational Geometry in Navigation and Path Planning - Semantic Scholar — This issue presents the applications of computational geometry in obstacle avoidance, path planning with minimum clearance, autonomous mobile robot navigation, exploration of unknown polygonal environ- ments, and it explores the practical relevance to the geographi- cal information systems, mobile networks, risk avoidance, security, games, and online system design. his special issue on
[47] Computational geometry | Encyclopedia of Computer Science — Computational geometry is the study of algorithmic problems involving geometry. Although the ruler and compass constructions of ancient Greek geometry were essentially algorithms for producing geometric objects, modern computational geometry begins with M. I. Shamos's 1975 Ph.D. dissertation, which solved several fundamental geometric problems and posed many more.
[48] PDF — Today computational geometry is often billed as a new discipline in computing science which many computing scientists would say was born either with the Ph.D. thesis of Michael Sha-mos at Yale University in 1978 [Sh78], or perhaps his earlier paper on geometric complexity [Sh75]. Others would say it began ten years earlier with the Ph.D. thesis
[49] Computational geometry - Wikipedia — The term "computational geometry" in this meaning has been in use since 1971. Although most algorithms of computational geometry have been developed (and are being developed) for electronic computers, some algorithms were developed for unconventional computers (e.g. optical computers )
[55] Computational Geometry - SpringerLink — Computational geometry is the discipline of exploring algorithms and data structures for computing geometric objects and their often extremal attributes. ... its name coined in the early 1970s. It has since witnessed explosive growth, stimulated in part by the largely parallel development of computer graphics ... It plays a key role in the
[56] Computational geometry | Encyclopedia of Computer Science — Computational geometry is the study of algorithmic problems involving geometry. Although the ruler and compass constructions of ancient Greek geometry were essentially algorithms for producing geometric objects, modern computational geometry begins with M. I. Shamos's 1975 Ph.D. dissertation, which solved several fundamental geometric problems and posed many more.
[58] PDF — Starting in the 1980's geometric modeling began to use tools developed in algebraic geometry. By the turn of the century, this interaction deepened with methods arising in geometric modeling having an impact on computational algebraic geometry, as well as a significantly deeper use of notions from algebraic geometry by geometric modeling.
[64] PDF — out this simplicity and unity of randomized algorithms and also their depth. Randomization entered computational geometry with full force only in the 80s. Before that the algorithms in computational geometry were mostly deterministic. We do cover some of the very basic, early deterministic al-gorithms.
[66] PDF — "continuous brand" of computational geometry, as distinct from its "discre-tized" counterpart. However, our principal objective has been a coherent ... 3.3 Convex Hull Algorithms in the Plane 104 3.3.1 Early development of a convex hull algorithm 104 3.3.2 Graham's scan 106 3.3.3 Jarvis's march 110 3.3.4 QUICKHULL techniques 112
[68] Computational Geometry: Algorithms and Applications | SpringerLink — Computational Geometry: Algorithms and Applications | SpringerLink Access this book The success of the ?eld as a research discipline can on the one hand be explained from the beauty of the problems studied and the solutions obtained, and, on the other hand, by the many application domains—computer graphics, geographic information systems (GIS), robotics, and others—in which geometric algorithms play a fundamental role. The book has been written as a textbook for a course in computational geometry,but it can also be used for self-study. Search within this book Book Subtitle: Algorithms and Applications Authors: Mark Berg, Otfried Cheong, Marc Kreveld, Mark Overmars Topics: Theory of Computation, Geometry, Math Applications in Computer Science, Earth Sciences, general, Computer Graphics, Algorithm Analysis and Problem Complexity Access this book
[69] Key-problems and key-methods in computational geometry — Computational geometry, considered a subfield of computer science, is concerned with the computational aspects of geometric problems. The increasing activity in this rather young field made it split into several reasonably independent subareas. ... This paper presents several key-problems of the classical part of computational geometry which
[89] Computational geometry - Wikipedia — The main branches of computational geometry are: Combinatorial computational geometry, also called algorithmic geometry, which deals with geometric objects as discrete entities. A groundlaying book in the subject by Preparata and Shamos dates the first use of the term "computational geometry" in this sense by 1975.
[96] Advances in Discrete and Computational Geometry - MDPI — The primary focus of discrete geometry is the study of combinatorial properties of discrete geometric objects, such as arrangements of lines, subdivisions, coverings, or polytopes. Computational geometry, on the other hand, studies efficient algorithms and data structures for solving problems in (discrete) geometry. The abstract problems
[98] Computational Geometry - Some Easy - Journal of Universal Computer Science — Computational geometry is concerned with the algorithmic study of elementary geometric problems. Ever since its emergence as a new branch of computer science in the early 1970's, a fruitful interplay has been taking place between combinatorial geometry, algorithms theory, and more practically oriented areas of computer science.
[99] Discrete Geometry: Principles & Applications - Vaia — Discrete Geometry is a subfield of mathematics dealing with the study and analysis of discrete and combinatorial geometric structures. Unlike traditional geometry, which considers continuous entities, Discrete Geometry focuses on objects that can be separated clearly and distinctly such as vertices, edges, and other geometric figures.
[100] Advantages and Disadvantages of Various Numerical Methods — A comparison of the accuracies of using the various computational methods shows a good agreement between the results of BEM and FEM, for two-dimensional problems a discrepancy is about 1% while in the 3-D case it is about 2%. On the other hand, descrepancies between BEM and FEM are 2% in the case of 2D calculations and 3% in the case of 3D
[102] What is numerical algebraic geometry? - ScienceDirect — The foundation of algebraic geometry is the solving of systems of polynomial equations. When the equations to be considered are defined over a subfield of the complex numbers, numerical methods can be used to perform algebraic geometric computations forming the area of numerical algebraic geometry. This article provides a short introduction to numerical algebraic geometry with the subsequent
[103] Numerical algebraic geometry - Wikipedia — The primary computational method used in numerical algebraic geometry is homotopy continuation, in which a homotopy is formed between two polynomial systems, and the isolated solutions (points) of one are continued to the other. This is a specialization of the more general method of numerical continuation. Let represent the variables of the system. By abuse of notation, and to facilitate the
[104] What are the advantages of numerical method over ... - ResearchGate — Numerical analysis applications are distributed among different areas, such as computational fluid dynamic analysis, Structural analysis, Kinematic Analysis, and even economic analysis.
[115] Robotics | 51 | v3 | Handbook of Discrete and Computational Geometry — Robotics researchers are primarily interested in developing methods that work well in practice and can be combined into integrated systems. They often pay less attention than researchers in computational geometry to the underlying combinatorial and complexity issues (the focus of Chapter 50).
[116] The interface between computational and combinatorial geometry ... — We illustrate the rich interface between computational and combinatorial geometry by a series of examples, ... IEEE Trans. Robotics Autom. 12 (1996), 566--580. Google Scholar ... Computational geometry is the study of algorithmic problems involving geometry. Although the ruler and compass constructions of ancient Greek geometry were essentially
[117] Advances in Digital and Computational Geometry: | Guide books | ACM ... — Digital geometry can anticipate progress in imaging technology allowing higher and higher spatial resolution. It seems that the input data in both fields will "converge" to data embedded in digital arrays of very high spatial resolution. ... Recent Advances in Computational Conformal Geometry. ... Computational geometry is the study of
[130] PDF — • Overview and goals • Course Information • 2D Convex hull • Signup sheet. February 1, 2005 Lecture 1: Introduction to Geometric Computation ... Computational Geometry • Started in mid 70's • Focused on design and analysis of algorithms for geometric problems • Many problems well-solved, e.g., Voronoi
[134] Research on Computational Geometry and Computer Graphics — Computational geometry focuses on solving geometric problems and optimizing spatial algorithms, while computer graphics is concerned with creating, rendering, and interacting with visual content. Both fields have practical applications in a wide range of industries, from video games and movies to scientific simulations and engineering design.
[137] Computational Geometry: Recent Developments | SpringerLink — Recent developments in the field of computational geometry are discussed with emphasis on those problems most relevant to computer graphics.In particular we consider convex hulls, triangulations of polygons and point sets, finding the CSG representation of a simple polygon, polygonal approximations of a curve, computing geodesic and visibility properties of polygons and sets of points inside
[139] PDF — Today's graphics cards are massively parallel processors composed of up to several thousands of cores [Nvi12a]. While GPUs comprise a vast amount of raw computational power, they are mainly limited by memory bandwidth. In particular, this becomes a bottleneck for real-time render-ing techniques where highly detailed surface geometry needs
[141] Understanding Computational Geometry in Informatics — Applications of Computational Geometry in Computer Graphics. In computer graphics, computational geometry is crucial for rendering and manipulating visual objects. It helps generate 2D and 3D images by calculating the relationships between geometric shapes and their transformations. Computational geometry powers techniques such as mesh
[143] On the Intersection of Computational Geometry Algorithms with Mobile ... — Next Article in Journal Journals Journals Find a Journal Journal Journals Taxonomy of robotic algorithms using computational geometry for path planning. This brief review discusses the fundamentals of CG and its application in solving well-known automated path-planning problems in single- and multi-robot systems. The review predominantly focuses on the literature published between 2016 and 2023, considering papers using CG in mobile robots, ensuring relevance and incorporation of the latest advancements in the field; however, to discuss the foundational work on computational geometry, we have considered pioneering works from the years 1969 to 2008; This review primarily focuses on CG-based complex optimization solutions for path-planning problems in single- and multi-robot systems. Taxonomy of robotic algorithms using computational geometry for path planning.
[144] From 3D point‐cloud data to explainable geometric deep learning: State ... — Robotics and autonomous vehicles ... Point cloud registration (PCR) is a pivotal process in computer vision and computational geometry, especially relevant for applications in domains like robotics, medical imaging, and computer-aided design. ... initiatives toward digital transformation across numerous sectors of daily life are largely driven
[145] ICAII 2023 | AI Innovation — Speech Title：From snapping fixtures to multi-robot coordination: Geometry at the service of robotics. Abstract: Robots sense, move and act in the physical world. It is therefore natural that understanding the geometry of the problem at hand will be key to devising an effective robotic solution, often as part of interdisciplinary solution methods.
[150] Algorithmic Efficiency in Convex Hull Computation: Insights from ... - MDPI — This study examines various algorithms for computing the convex hull of a set of n points in a d-dimensional space. Convex hulls are fundamental in computational geometry and are applied in computer graphics, pattern recognition, and computational biology. Such convex hulls can also be useful in symmetry problems. For instance, when points are arranged symmetrically, the convex hull is also
[151] PDF — 4 Computational Geometry Lecture 1: Convex Hulls 1.5 Graham’s Algorithm (Das Dreigroschenalgorithmus) Our next convex hull algorithm, called Graham’s scan, first explicitly sorts the points in O(nlog n) and then applies a linear-time scanning algorithm to finish building the hull. This algorithm avoids the O(n2) worst-case running time by choosing an approximately 9 Computational Geometry Lecture 1: Convex Hulls balanced partition of the points; at the same time, whenever possible, the algorithm prunes away a large subset of the interior points before recursing. The prune-and-search algorithm could be simplified by a solution to the following open problem: Given a set of n points in the plane, find a random vertex of the convex hull in O(n) expected time.
[153] A Comprehensive Survey on Delaunay Triangulation: Applications ... — Such a triangulation has been shown to have several interesting properties in terms of the structure of the simplices it constructs (e.g., maximising the minimum angle of the triangles in the bi-dimensional case) and has several critical applications in the contexts of computer graphics, computational geometry, mobile robotics or indoor
[163] PDF — Classical geometry has been making a comeback recently because it is use-ful in such fields as Computer-Aided Geometric Design (CAGD), CAD/CAM, computer graphics, computer vision and robotics. In all these fields there is a premium on computational efficiency in designing and manipulating geometric objects. Our purpose here is to introduce powerful new mathematical tools for meeting that
[173] Essential Geospatial Algorithms to Know for Geospatial ... - Fiveable — Geospatial algorithms are key tools in Geospatial Engineering, helping to analyze and manage spatial data effectively. They enhance tasks like mapping, navigation, and environmental modeling, making complex data more accessible and useful for decision-making. Spatial indexing algorithms (e.g., R-trees, Quadtrees)
[177] Understanding Efficient Spatial Indexing - GeeksforGeeks — Spatial indexing is a technique used to organize and access spatial data efficiently. It is particularly important when dealing with large datasets of points, lines, or polygons in two-dimensional or higher-dimensional spaces. The Spatial indexing structures help reduce the time complexity of the spatial queries from O(N) to O(log N) or better
[178] Geospatial Indexing - Definitions & FAQs | Atlas — Geospatial Indexing is a pivotal component in spatial databases and GIS platforms that supports efficient query processing by structuring the data using specialized index structures. These index structures, unlike traditional indexing methods, take into account the spatial properties of the data including location, shape, and size.
[180] Convex Hull using Graham Scan - GeeksforGeeks — A convex hull is the smallest convex polygon that encompasses a set of points, with applications in fields like computer graphics and image processing, and can be efficiently computed using algorithms such as Graham's scan.
[181] Convex hull - Wikipedia — Convex hulls have wide applications in mathematics, statistics, combinatorial optimization, economics, geometric modeling, and ethology. Related structures include the orthogonal convex hull, convex layers, Delaunay triangulation and Voronoi diagram, and convex skull.
[182] Understanding Convex Hull Algorithms: A Comprehensive Guide — 3. Approximation Algorithms For very large datasets or in real-time applications, approximation algorithms that compute a close approximation of the convex hull can be useful. Conclusion Understanding convex hull algorithms is crucial for anyone serious about computational geometry and advanced algorithm design.
[183] Convex Hull Algorithm - GeeksforGeeks — The *Convex Hull Algorithm is used to find the convex hull* of a set of points in computational geometry. *Algorithm*:** Given the set of points for which we have to find the convex hull. Question 2: How do you find the convex hull of a set of points? Convex Hull using Divide and Conquer Algorithm In computational geometry, a convex hull is the smallest convex polygon that contains a given set of points. Convex Hull using Jarvis' Algorithm or Wrapping Given a set of points in the plane. Please check this article first: Convex Hull | Set 1 (Jarvis’s Algorithm or Wrapping) Examples: Input: Points[] = {{0, 3}, {2, 2}, {1, 1}, {2, 12 min read
[185] Convex Hull | Brilliant Math & Science Wiki — The convex hull is a ubiquitous structure in computational geometry. Even though it is a useful tool in its own right, it is also helpful in constructing other structures like Voronoi diagrams, and in applications like unsupervised image analysis. We can visualize what the convex hull looks like by a thought experiment. Imagine that the points are nails sticking out of the plane, take an
[187] A Review on Deformable Voronoi Diagrams for Robot Path Planning in ... — Route planning for mobile robots presents a complex challenge, mainly when designing pathways in dynamic environments. This complexity arises from the robot's need to balance the demand for efficient and optimal routes while also handling unexpected obstacles. This paper introduces an algorithm that combines two key concepts: the Voronoi Diagram, utilized for environment representation, and
[190] Comparison of Global Path Planning Algorithms Regarding Multi Mobile ... — A path of the Voronoi diagram-based path planner is the most accessible and departable. It uses a distance function to skeletonize the free space and creates a pixel-based roadmap that provides the maximal distance to all obstacles in the environment . After creating the roadmap, a path planning algorithm finds the shortest path.
[192] Towards Optimizing a Convex Cover of Collision-Free Space for ... — We propose an online iterative algorithm to optimize a convex cover to under-approximate the free space for autonomous navigation to delineate Safe Flight Corridors (SFC). The convex cover consists of a set of polytopes such that the union of the polytopes represents obstacle-free space, allowing us to find trajectories for robots that lie within the convex cover. In order to find the SFC that
[219] Applications Of Computational Geometry Algorithms - Medium — Published Time: 2023-10-17T17:26:14.934Z Applications Of Computational Geometry Algorithms | by Sakshi Shakhawar | Medium Listen Many computational geometry issues, though, are classical in character and can result from mathematical visualisation. Making a polygonal mesh representation of a three-dimensional object is one of the most basic tasks in computer graphics. For instance, convex hulls are frequently used in facial recognition and self-driving cars because they help algorithms interpret data from cameras more consistently and in a way that allows them to learn. For the management and analysis of spatial data, computational geometry techniques are employed in GIS applications. Read member-only stories Support writers you read most Listen to audio narrations Follow 4 Followers ·1 Following Follow Also publish to my profile
[220] Applications Of Computational Geometry Algorithms - Medium — Published Time: 2023-05-02T09:53:03.004Z Applications Of Computational Geometry Algorithms | by RYUKK | Medium Listen Examples of algorithms used in computer graphics include those for computing the intersection of geometric primitives such as lines, segments, and planes, which are essential for rendering 3D scenes. In other optimisations, a voronoi diagram can be used in order to map the maximum capacity for a wireless network, allowing the engineer to know where best to put each node/switch etc. Computational geometry algorithms are used in GIS applications for spatial data management and analysis. Sign up to discover human stories that deepen your understanding of the world. Sign up for free Listen to audio narrations Follow 1 Follower ·1 Following Follow Also publish to my profile
[221] Applications of Computational Geometry - OpenGenus IQ — In this article, we have explained Applications of Computational Geometry along with topics/ algorithms used to solve a specific problem. Another place where we have seen computational geometry in the AI field is through computer vision, for example, convex hulls are used widely in self-driving cars or facial recognition as it allows the data which the algorithm is recieving through the camera to be interpretted more consistently and in a way which it can actually learn from, for example, a convex-hull of a car is a lot easier to use in object avoidence than a normal mesh, cars come in many shapes and have many contours or bumps, placing them within a convex hull allows for a more consistent and uniform shape to be used, improving the algorithm.
[224] PDF — and shape optimization before being actually used in production. The course starts with a comparison of different surface representations, motivating the use of polygonal meshes. We discuss the removal of geometric and topological degeneracies, and introduce quality measures for polygonal meshes, followed by their respective optimization, namely
[227] Obstacle Detection and Avoidance System Based on Monocular Camera and ... — The work presented in this paper represents a step forward in the obstacle detection and avoidance by the use of the convex hull (area) of the obstacle identified by means of image features. The approach is able to work in real time with all different kind of obstacles and in different tested scenarios. 3. Obstacle Detection
[229] Voronoi Diagrams | Advanced Navigation — Voronoi diagrams have applications in navigation, robotics, wireless network optimization, and terrain mapping, enabling efficient path planning, coverage analysis, and clustering. They are particularly valuable in autonomous systems, defense, surveying, and geospatial data processing for advanced spatial modeling.
[230] Convex Hull Algorithm in C - GeeksforGeeks — The Convex Hull problem is a fundamental computational geometry problem, where the goal is to find the smallest convex polygon called convex hull that can enclose a set of points in a 2D plane. This problem has various applications, such as in computer graphics, geographic information systems, and collision detection.
[236] Evaluating computational geometry libraries for big spatial data ... — With the rise of big spatial data, many systems were developed on Hadoop, Spark, Storm, Flink, and similar big data systems to handle big spatial data. At the core of all these systems, they use a computational geometry library to represent points, lines, and polygons, and to process them to evaluate spatial predicates and spatial analysis queries.
[242] PDF — Our research into computational geometry seeks to answer the question, “How can the explicit representation of geometric structure enrich the set of operations in spatial data handling?” To take a familiar example, consider spatial information on a map. For computers to participate in answering these questions (beyond simply displaying maps), they require algorithms and data structures that deal with these geometric concepts. The Approach We sketch two example projects to illustrate geometric computations in GIS: robust polygon overlay, and dynamic data on terrain models. We are at a crossroads—with recent advances in spatial data collection, storage, and analysis, coupled with advances from computer science in geometric algorithms, data structures, and visualization, we have the potential to better represent our landscapes at varying, nested resolutions, and to advance our ability to model natural phenomena.
[243] PDF — Geographic Information Systems (GIS) heavily relies on computational geometry to process and analyze spatial data. Algorithms assist in tasks such as geometric network analysis, spatial clustering, and map overlay operations, contributing to urban planning, environmental analysis, and transportation management.
[247] Graham Scan Algorithm to find Convex Hull - OpenGenus IQ — Graham's Scan Algorithm is an efficient algorithm for finding the convex hull of a finite set of points in the plane with time complexity O(N log N). The algorithm finds all vertices of the convex hull ordered along its boundary. It uses a stack to detect and remove concavities in the boundary efficiently.
[256] CG:SHOP SoCG Competition — The CG:SHOP Challenge (Computational Geometry: Solving Hard Optimization Problems) is an annual competition dedicated to tackling specific, difficult geometric optimization problems. Originally established as a workshop at Computational Geometry Week (CG Week) in 2019, the Challenge provides a unique platform where the success of proposed solutions is measured based on computational
[258] Application Challenges to Computational Geometry: CG Impact Task Force ... — The fraction of computing falling under the loosely deened rubric of \\geometric computation" has been on the rise and is likely to become dominant in the next decade, and the opportunities and challenges this presents for the eld of computational geometry in the years ahead are assessed. With rapid advances in computer hardware and visualization systems, geometric computing is creeping into
[259] Approaching Computational Geometry Problems: A Comprehensive Guide — Computational geometry focuses on developing efficient algorithms to solve geometric problems. These problems often involve points, lines, polygons, and other geometric shapes in two or three-dimensional space. ... Many geometric algorithms can be extended to higher dimensions, but this often introduces new challenges and complexities. 2
[266] Special issue of computational mechanics on machine learning theories ... — Specifically for mechanics and materials, there are a number of promising areas: (i) improving efficiency when traditional methods are computationally intractable by constructing efficient surrogate or reduced-order models; (ii) improving accuracy when traditional methods performs poorly, by assimilating additional data; (iii) solving “unsolvable” traditional models, when problem are ill-posed in presence of incomplete information; (iv) model discovery when the exact form of the physical model is unknown; (v) efficiently solving inverse problems, especially useful in processing or soft robotics; (vi) understanding and interpreting machine learning, by employing physics-informed strategies; and (vii) constructing digital twins combining intimately physics-based and data-driven models for representing a virtual replica of the physical systems, while guaranteeing fast and accurate responses, needed in diagnosis, control, prognosis and decision making.
[268] Machine Learning in the Big Data Age: Advancements, Challenges, and ... — The sheer scale and complexity of Big Data pose obstacles to effective implementation of Machine Learning. Issues such as data privacy, security, and scalability demand careful consideration.
[269] Geometric Machine Learning - Weber - 2025 - Wiley Online Library — A cornerstone of machine learning is the identification and exploitation of structure in high-dimensional data. While classical approaches assume that data lies in a high-dimensional Euclidean space, geometric machine learning methods are designed for non-Euclidean data, including graphs, strings, and matrices, or data characterized by symmetries inherent in the underlying system.
[271] New Perspectives and Computational Challenges in High Dimensions — The mathematical subdisciplines most strongly related to such phenomena are functional analysis, convex geometry, and probability theory. In fact, a new area emerged, called asymptotic geometric analysis, which is at the very core of these disciplines and bears a number of deep connections to mathematical physics, numerical analysis, and
[272] PDF — Computational Geometry in High-Dimensional Spaces Paul Liu Abstract In this project, we provide a review of e cient solutions to a few high-dimensional CG problems, with particular focus on the problem of nearest neighbour search. For the problems we examine, the general approach will be to either reduce the dimension (using the JL transform)
[273] Computational Geometry and Artificial Intelligence — Computational geometry algorithms can be used to optimize and improve the performance of AI systems, particularly in tasks that involve spatial reasoning or geometric constraints. By combining AI techniques with computational geometry, researchers and developers can develop advanced algorithms and models that can efficiently solve complex geometric problems. In the field of computational geometry, artificial intelligence (AI) techniques are playing a crucial role in analyzing and interpreting geometric data. Computational geometry algorithms can be used to optimize and improve the performance of AI systems, particularly in tasks that involve spatial reasoning or geometric constraints. By leveraging computational geometry algorithms, AI models can extract geometric features from visual data and use them to identify objects accurately.
[274] Geometric Intelligence: Reshaping AI with Shape-Based Learning — Geometric Intelligence: Reshaping AI with Shape-Based Learning Geometric Intelligence: Revolutionizing AI with Shape-based Learning This groundbreaking field, known as geometric intelligence, is poised to transform the landscape of AI and usher in a new era of computational understanding. The importance of geometric intelligence in the field of AI cannot be overstated. Machines equipped with geometric intelligence can understand and manipulate 3D spaces with an almost human-like intuition. This marriage of approaches could lead to AI systems that are both geometrically aware and capable of learning complex patterns from data. By incorporating geometric understanding into AI systems, we might be able to create machines that reason about the world in ways that are more similar to human cognition.
[275] AlphaGeometry: Revolutionizing AI in Geometry - AI-BLOG: Where Tomorrow ... — At its core, AlphaGeometry is an advanced AI system designed specifically for geometric computations and spatial analysis. Other AI systems may also handle non-Euclidean geometries, but AlphaGeometry’s integration of deep learning with symbolic logic gives it an edge in understanding and solving these types of problems. As AI continues to evolve, AlphaGeometry paves the way for innovative solutions and groundbreaking discoveries in mathematical problem-solving. AlphaGeometry is an advanced AI model designed to tackle complex geometric problems. AlphaGeometry utilizes deep learning and AI algorithms to automatically learn and solve geometric problems, whereas traditional geometry software typically relies on predefined rules and manual inputs. AlphaGeometry excels in handling complex 3D modeling tasks by using AI algorithms to automate the design and optimization processes.
[287] PDF — Among the most fundamental data structures are those used for the representation and storage of raster image data and vector geospatial location data. Because they are so pervasive, even small improvements in efficiency or representational accuracy in these data structures can result in substantial performance increases in an overall system.