Threegeometry

Threegeometry is a term used to describe the study and manipulation of geometric properties in three-dimensional space. In mathematics and computer science, it encompasses theories and methods for describing points, lines, planes, surfaces, and solids, as well as metrics, angles, volumes, and curvatures that relate them. The term distinguishes three-dimensional geometry from two-dimensional geometry and, in some contexts, from higher-dimensional geometry.

Objects are represented in explicit, implicit, or point-based forms. Explicit representations include polygonal meshes and parametric surfaces; implicit representations include algebraic surfaces and level sets; point-based representations rely on unordered point clouds. Core operations in threegeometry include distance and angle computations, intersections and containment tests, convex hulls, projections, and coordinate transformations such as translation, rotation, scaling, as well as more general affine and projective transformations. Invariants and properties depend on the chosen geometric framework (Euclidean, affine, or projective).

In computational threegeometry, algorithms address modeling, rendering, collision detection, mesh generation, and geometric optimization. The field supports applications in computer-aided design, computer graphics, robotics, engineering, and reverse engineering, where accurate three-dimensional representations enable simulation, visualization, and reasoning about physical objects and scenes.

Historically, the study of three-dimensional geometry has roots in classical geometry and was formalized through algebraic methods and later computational techniques. The exact usage of the term threegeometry varies by discipline, but it is commonly encountered as shorthand for the domain of three-dimensional geometry in educational materials and software documentation.

See also: geometry, Euclidean geometry, affine geometry, projective geometry, computational geometry, 3D modeling.