Computational geometry

Computational geometry

Computational geometry is a part of computer science that studies algorithms related to shapes and spaces. It looks at how we can use shapes to solve problems with computers. This area has become very important as computers have gotten stronger.

One important idea in computational geometry is how complicated the steps to solve a problem can be. This is especially important when dealing with large amounts of data, like millions of points. The way a problem is solved can change how long it takes — from days to just seconds.

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

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

Combinatorial computational geometry

The main aim of combinatorial computational geometry is to find smart ways to solve problems with basic shapes like points, lines, and polygons.

Some problems look easy, like finding the two closest points in a group. Instead of checking every pair, there are faster ways to do this.

Computational geometry solves problems in different ways. In static problems, we find answers from the data we already have, such as working out the outer edge of a group of points (convex hull) or spotting where lines cross. In geometric query problems, we get data ready so we can quickly answer many questions, like finding the nearest point to a new spot. Dynamic problems are about updating answers when the data changes, like adding or taking away points and changing the outer shape.

Numerical computational geometry

Main article: computer-aided geometric design

Numerical computational geometry, also called geometric modelling or computer-aided geometric design (CAGD), helps us create and show curves and surfaces. Important tools for this include special curves and surfaces like Bézier curves and spline curves and surfaces, as well as other methods such as the level-set method. These tools are used in many industries, like shipbuilding, aircraft, and car manufacturing.

List of algorithms

Computational geometry has many helpful ways to solve problems about shapes and spaces. These ways can help us find the shortest path between two points, pack objects closely together, or split space into smaller parts.

Some key ways to solve problems include ones for convex hulls, which find the smallest shape that can hold a group of points, and ones for line segment intersection, which show where lines cross. These tools are useful in areas like computer graphics, robotics, and making maps.