Computational geometry

Computational geometry is a branch of computer science that studies algorithms related to geometry. It looks at how we can use geometry to solve problems with computers. This field has a long history, but it has become very important recently as computers have gotten more powerful.

One key part of computational geometry is computational complexity. This matters a lot when we have huge amounts of data, like millions of points. The way an algorithm works can make a big difference — it might take days or just seconds to get an answer.

Computational geometry grew largely because of advances in computer graphics and tools for designing and making things with computers, known as CAD/CAM. But many problems in this field have been studied for a very long time, coming from mathematical visualization.

It is used in many cool areas, like robotics for helping robots move, geographic information systems for mapping and navigation, designing integrated circuits, computer-aided engineering, and even computer vision for creating 3D images through 3D reconstruction.

Combinatorial computational geometry

The main aim of combinatorial computational geometry is to create efficient ways to solve problems involving basic shapes like points, lines, and polygons.

Some problems may seem simple, like finding the closest pair of points from a group. While one way is to measure every possible pair, there are smarter methods that take less time.

Computational geometry looks at problems in different ways. In static problems, we find answers from given inputs, such as determining the outer shape of a group of points (convex hull) or finding where lines cross. In geometric query problems, we prepare data to quickly answer many questions, like finding the nearest point to a new location. Dynamic problems involve updating solutions when the data changes, like adding or removing points and adjusting the outer shape accordingly.

Numerical computational geometry

Main article: computer-aided geometric design

Numerical computational geometry, also called geometric modelling or computer-aided geometric design (CAGD), focuses on creating and representing curves and surfaces. Important tools for this include parametric curves and surfaces like Bézier curves and spline curves and surfaces, as well as non-parametric methods such as the level-set method. These techniques are used in many industries, including shipbuilding, aircraft, and automotive manufacturing.

List of algorithms

Computational geometry includes many useful algorithms that help solve problems related to shapes and spaces. These algorithms can be used for tasks like finding the shortest path between two points, determining how to pack objects tightly together, or figuring out the best way to divide space into smaller parts.

Some important algorithms in computational geometry include algorithms for convex hulls, which find the smallest shape that can contain a set of points, and algorithms for line segment intersection, which determine where lines cross each other. These tools are important in fields like computer graphics, robotics, and map-making.