Computational geometry is a subfield of computer science that is concerned with the development of algorithms for solving geometric problems. These algorithms are used in a variety of applications across multiple fields, including computer graphics, robotics, geographic information systems, computer vision, and computational biology.
It is a field of computer science that has numerous applications in various areas. Many problems in computational geometry come from application areas such as pattern recognition, computer graphics, operations research, computer-aided design, and manufacturing. The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing (CAD/CAM). However, many problems in computational geometry are classical in nature and may come from mathematical visualization. Other important applications of computational geometry include robotics (motion planning and visibility problems), geographic information systems (GIS) (geometrical location and search, route planning), integrated circuit design (IC geometry design and verification), and computer-aided engineering (CAE). Computational geometry is inseparable from research areas such as graphics, scientific computing and modeling, and computer vision.
In this blog, we will discuss some of the applications of computational geometry algorithms in more detail.
Computer Graphics, one of the primary applications of computational geometry algorithms is in computer graphics. These algorithms are used to represent, manipulate, and render geometric objects in 2D and 3D graphics applications. 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.
One of the most fundamental problems in computer graphics is the construction of a polygonal mesh representation of a 3D object. This process involves converting the surface of the object into a mesh of interconnected triangles, which can be rendered using modern graphics hardware. Computational geometry algorithms such as Delaunay triangulation, which computes a triangulation of a set of points that maximizes the minimum angle of the resulting triangles, and Voronoi diagrams, which partition a plane into regions based on the distance to a set of points, are used to construct polygonal meshes.
Robotics Another important application of computational geometry algorithms is in robotics. These algorithms are used to solve problems related to motion planning, collision detection, and path optimization. For example, in order to plan a robot’s motion from its initial configuration to its goal configuration, the robot must avoid obstacles in its workspace. Computational geometry algorithms such as the Visibility Graph Algorithm and the Rapidly-exploring Random Tree (RRT) algorithm are used to compute collision-free paths in a robot’s workspace.
In addition to motion planning, computational geometry algorithms are also used for robot grasping and manipulation. For example, a robot must be able to determine the optimal grasping location and orientation of an object in order to manipulate it. Computational geometry algorithms such as the Convex Hull algorithm, which computes the smallest convex polygon that contains a set of points, and the Grasp Quality Metrics algorithm, which computes the quality of a grasp based on the geometry of the object and the robot’s hand, are used to determine the optimal grasping location and orientation of an object.
Artificial Intelligence
Another field where computational geometry is prevelant is artifical intelligence, when finding optimal paths for travel, geometric algorithms are quintessential. For example, A* pathfinding can be used to find the shortest path from a start and end goal. This uses a grid-based approach and heuristics, which are a basic form of machine learning which means that a better path will be found, the more experience the algorithm has. To check if the path is clear, a voronoi diagram can be used as, if its points are obstacles, it means that its edges are the routes furthest from said obstacles and therefore theoretically furthest from collisions. 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.
Networking
Networking is another field where computational geometry is paramount. For example, shortest-path algorithms such as Dijkstra’s are used in directing router traffic, it is almost a perfect use case for this, each edge (connection to another router) will have a different cost based on the current traffic, therefore an algorithm such as Dijkstra’s can be used in order to compute the shortest distance for the packets to travel. 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.
Geographic Information Systems (GIS) Geographic Information Systems (GIS) are computer systems that are used to store, analyze, and visualize spatial data. Computational geometry algorithms are used in GIS applications for spatial data management and analysis. For example, algorithms for computing the convex hull of a set of points or for computing the Voronoi diagram of a set of points can be used to analyze geographical data and to determine regions of influence or proximity.
In addition to spatial data analysis, computational geometry algorithms are also used in GIS applications for spatial data visualisation. For example, algorithms for computing the Delaunay triangulation of a set of points can be used to generate a surface mesh of an area that can be used to visualise terrain data.
Computer Vision Computer vision is a field of computer science that is concerned with the development of algorithms for interpreting visual data from the world. Computational geometry algorithms are used in computer vision applications such as object recognition, tracking, and segmentation. For example, algorithms for computing the intersection of geometric primitives such as lines, segments, and planes can be used to detect the boundaries of objects in an image.
Geometry is the archetype of the beauty of the world. ~ Johannes Kepler
The search results show that computational geometry is a branch of computer science that deals with the study of algorithms that can be stated in terms of geometry. It has numerous applications in various areas, including computer graphics, robotics, geographic information systems, and computer-aided design. Computational geometry algorithms are used to solve problems related to motion planning, path planning, obstacle avoidance, spatial analysis, and shape manipulation. The algorithms in this field are presented in a pseudo-code that is detailed enough to make it relatively easy to implement them. Computational geometry is an area of study with far-reaching applications throughout computer science, and it is expected to become even more important in the future.
In conclusion, computational geometry algorithms have numerous applications in various areas, including computer graphics, robotics, geographic information systems, and computer-aided design. These algorithms are used to solve problems related to motion planning, path planning, obstacle avoidance, spatial analysis, and shape manipulation. With the increasing demand for automation and digitization, the importance of computational geometry algorithms is only going to increase in the future.
Authors :
Shubhankar Gupta, Prasanna Kshirsagar, Aniket Joshi, Mohammad Raza and Anagha Gajaralwar.