threedimensional

Three-dimensional, or threedimensional, describes objects and spaces that extend along three spatial dimensions: length, width, and height. It contrasts with two-dimensional concepts, which involve only length and width, and with lower-dimensional ideas used in mathematics and physics.

In mathematics, three-dimensional space is commonly modeled as real coordinate space, R^3. A point is described by coordinates (x, y, z). The Euclidean distance between points uses the Pythagorean formula, and volume is defined for regions in R^3. An object’s position in space requires three coordinates for translation, while its orientation involves additional parameters for rotation.

Representation and perception of three-dimensionality are central in many fields. Physical objects are inherently three-dimensional, but images on two-dimensional displays require projection. In art and computer graphics, three-dimensional models are often represented as meshes of polygons, typically triangles. Rendering converts 3D coordinates into 2D images through projection, along with shading, lighting, and texture mapping to convey depth and realism.

Applications of threedimensional concepts are widespread. They underpin science, engineering, architecture, and medicine, and enable technologies such as computer-aided design (CAD), 3D modeling, 3D printing, virtual reality, and cinematic effects. Three-dimensional analysis also supports simulations of physical processes, spatial reasoning, and measurement of volume and spatial relationships.

Perception of depth arises from cues such as binocular vision, motion parallax, shading, and occlusion, allowing observers to infer three-dimensional structure from two-dimensional views.