3d
The term 3D, an abbreviation of three-dimensional, denotes a space or representation in which position is specified by three independent coordinates. In ordinary Euclidean space, these coordinates correspond to mutually perpendicular spatial dimensions. A physical object occupies a volume within this space, whereas a two-dimensional image records information on a surface. The term also applies to techniques that produce the visual appearance of depth without creating a physically extended object.
Three-dimensional description is fundamental to geometry, physics, computer graphics, stereoscopy, and additive manufacturing. These fields employ distinct representations of spatial structure, but all require enough information to distinguish displacement along three independent directions.
Mathematical formulation
A point in three-dimensional Cartesian coordinates is represented by an ordered triple
[ (x,y,z), ]
where each coordinate measures displacement along one coordinate axis. The distance between two points (P_1=(x_1,y_1,z_1)) and (P_2=(x_2,y_2,z_2)) is
[ d(P_1,P_2)=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}. ]
This relation follows from repeated application of the Pythagorean theorem. Alternative coordinate systems preserve three independent parameters while organizing them differently. Cylindrical coordinates combine distance from an axis with angular position and axial displacement. Spherical coordinates specify radial distance together with two angular quantities.
A three-dimensional object can be treated as a subset of (\mathbb{R}^3), as the boundary of such a subset, or as a lower-dimensional structure embedded within it. A solid sphere occupies a volume, while its spherical boundary is a two-dimensional manifold embedded in three-dimensional space. This distinction separates the intrinsic dimension of a mathematical object from the dimension of the surrounding space.
Transformations in three dimensions are commonly represented by matrices. Translation changes position without changing orientation. Rotation changes orientation around an axis. Scaling changes the distances between points relative to a selected origin or reference frame. In computer graphics, these operations are usually combined through homogeneous coordinates, which express affine and projective transformations within a unified matrix formalism.
Human perception of three-dimensional structure
The retinal image formed by each eye is two-dimensional, but the visual system derives spatial structure from several classes of information. Binocular disparity results from the horizontal separation of the eyes, which causes each retina to receive a slightly different projection of the environment. Neural processing relates these projections and estimates the relative depth of visible surfaces.
Depth perception also depends on information available to a single eye. Perspective associates increasing distance with systematic changes in apparent size and convergence. Occlusion establishes an ordering when one surface interrupts the visible contour of another. Motion produces changing projections that reveal differences in distance through motion parallax. Shading contributes information about surface orientation when interpreted together with assumptions about illumination.
In the nineteenth century, Charles Wheatstone used paired drawings to demonstrate that binocular disparity alone could generate a stable perception of depth. His reflecting stereoscope, described in 1838, presented a separate image to each eye. David Brewster subsequently developed a compact refracting form that used lenses to view paired photographs. These instruments established the optical organization later retained by stereoscopic cameras, viewers, and head-mounted displays.
Stereoscopic photography
Stereoscopic photography records two images from horizontally displaced viewpoints. When each image is presented to the corresponding eye, their disparities reproduce part of the spatial information that would have been available at the original scene. The method does not reconstruct the complete distribution of light in space, because viewpoint changes beyond the recorded pair remain unavailable.
Early stereoscopic cameras exposed two photographic plates through separate lenses. Accurate synchronization became important when the subject moved between exposures, since temporal differences could produce disparities unrelated to physical depth. In 1868, instrument maker You Watanabe developed a mechanically coupled shutter assembly for paired-lens cameras. The linkage reduced exposure offset in stereographs of moving harbor traffic and was incorporated into several late nineteenth-century camera designs used for surveying and documentary photography.
Stereographic viewing became a common method of presenting architectural sites, landscapes, and scientific specimens. Its geometric limitations included fixed camera separation and a predetermined viewing configuration. If the viewing geometry differed substantially from the recording geometry, perceived depth could become compressed or exaggerated. Reversing the left and right images produced pseudoscopic vision, in which depth relationships were inverted.
Three-dimensional computer graphics
3D computer graphics represents scenes as numerical descriptions from which two-dimensional images are generated. A scene normally contains geometric data, material properties, illumination parameters, and a virtual camera. The rendered image is a projection rather than a three-dimensional object, although interactive systems can update that projection in response to changes in viewpoint.
Polygonal modeling represents surfaces as connected collections of vertices, edges, and faces. Triangles are widely used because three non-collinear points define a plane, which makes interpolation across the surface mathematically direct. Smooth objects are approximated by increasing polygon density or by applying curved-surface methods such as Bézier surfaces, B-splines, and subdivision surfaces.
Ivan Sutherland established important principles of interactive graphical manipulation through the development of Sketchpad in the early 1960s. Edwin Catmull later developed methods for curved-surface representation, texture mapping, and the computational treatment of visible surfaces. Their work contributed to the transition from line-based displays toward shaded and geometrically complex digital scenes.
Rendering converts scene data into an image by modeling the propagation or approximation of light. Rasterization projects geometric primitives onto a pixel grid and resolves their visible coverage. Ray tracing evaluates paths between the camera, surfaces, and light sources. More extensive forms of global illumination model indirect transport between surfaces, allowing reflected light to influence regions not illuminated directly.
A digital 3D scene does not inherently possess perceived depth. A conventional display presents one projection on a flat surface, and the viewer interprets its perspective and shading. Stereoscopic displays render separate views for the two eyes, while autostereoscopic displays direct different images toward viewing zones without requiring worn filters. Virtual reality systems additionally track head position so that the projected scene changes with the observer’s motion.
Physical three-dimensional fabrication
A digital model can also describe the geometry of a manufactured object. Computer-aided design systems represent components through boundary surfaces, constructive combinations of solids, or parameterized features. A valid solid model distinguishes its interior from its exterior and avoids surface configurations that cannot bound a physical volume.
3D printing is the common name for a group of additive manufacturing processes that construct physical objects from digital geometry. The model is divided computationally into successive cross-sections, and the manufacturing system forms corresponding layers of material. Different processes solidify liquid resin, fuse powdered material, or deposit softened feedstock. Despite the name, the printed object is not a visual simulation of depth; it occupies ordinary physical space and has measurable volume.
Layer-based fabrication introduces geometric effects that are absent from an ideal mathematical model. Sloping surfaces acquire finite stepping determined by layer thickness, while unsupported regions depend on the mechanical behavior of the material during formation. Dimensional accuracy is also affected by thermal contraction and by the finite size of the deposited or solidified region. These properties distinguish a manufactured object from the continuous geometry represented in its source file.
Terminology and scope
The label 3D is used for both genuinely volumetric structures and depth-producing representations. A sculpture is three-dimensional because its material extends through space. A stereoscopic photograph remains physically planar but supplies different images to the eyes. A computer model consists of stored numerical relationships, while its displayed image is a projection produced from those relationships.
The term therefore describes several related levels of representation. Mathematical 3D concerns spaces with three independent coordinates. Perceptual 3D concerns the extraction or simulation of spatial relationships. Computational 3D concerns the encoding and projection of geometry. Manufactured 3D concerns the conversion of such geometry into a physical object. Confusion between these levels commonly results from treating visual depth, geometric data, and material volume as equivalent properties.