3D projection

A 3D projection is a mathematical transformation that represents points in three-dimensional space on a surface of lower dimension, ordinarily a two-dimensional plane. The resulting image conveys spatial structure through the projected positions of points, edges, surfaces, and patterns of occlusion. Because the transformation reduces dimensionality, distinct spatial points can acquire the same image coordinates, and the original scene cannot generally be reconstructed from a single projection without additional information.

The term applies to geometric constructions used in technical drawing, physical image formation in optical systems, and the computational procedures that produce images from three-dimensional models. A projection does not itself make an image three-dimensional; it encodes selected spatial relations within a two-dimensional representation. Depth remains present as a relation inferred from perspective, overlap, shading, motion, or prior knowledge rather than as an independent image coordinate.

Mathematical formulation

Let a spatial point be represented by Cartesian coordinates

[ \mathbf{p} = \begin{bmatrix} x\ y\ z \end{bmatrix}. ]

A projection maps (\mathbf{p}) to an image point ((u,v)). In many applications, the transformation is expressed using homogeneous coordinates, which permit translations and perspective division to be incorporated into a matrix formulation. The projected point is written as

[ \begin{bmatrix} \tilde{u}\ \tilde{v}\ \tilde{w} \end{bmatrix}

P \begin{bmatrix} x\ y\ z\ 1 \end{bmatrix}, \qquad u=\frac{\tilde{u}}{\tilde{w}}, \qquad v=\frac{\tilde{v}}{\tilde{w}}, ]

where (P) is a (3\times4) projection matrix. The division by (\tilde{w}) produces the characteristic scale variation of perspective projection. When (\tilde{w}) is constant, the transformation reduces to an affine transformation, and parallel lines retain their parallelism after projection.

Projection matrices ordinarily operate after a scene has been transformed from world coordinates into a coordinate system centered on a virtual or physical camera. This transformation separates the position and orientation of the observer from the internal geometry of the projection. In computer graphics, the combined transformation is commonly decomposed into model, view, projection, and viewport stages, although these stages may be algebraically merged.

Perspective projection

Perspective projection models image formation by rays passing through a single center of projection. For an ideal pinhole camera with focal length (f), a point expressed in camera coordinates projects according to

[ u=f\frac{x}{z}, \qquad v=f\frac{y}{z}. ]

Objects with greater camera-space depth therefore occupy a smaller image region when their physical dimensions remain constant. Parallel spatial lines that are not parallel to the image plane generally converge toward a vanishing point. Sets of parallel lines sharing a spatial direction share the same vanishing point, except when that direction is parallel to the image plane and its vanishing point lies at infinity.

The ideal pinhole model omits lens distortion and other optical effects. A physical camera lens can introduce radial displacement, asymmetric decentering, variation in focus, and finite-depth blur. These effects belong to the imaging system rather than to projective geometry itself, although practical camera models frequently incorporate them after the ideal projection has been calculated.

Perspective projection preserves incidence: a projected point remains on the projected form of any line that contained the original point, provided the configuration is not degenerate. It does not preserve ordinary Euclidean length or angle. A right angle in space may project as an oblique angle, while equal spatial segments may have unequal image lengths because they occupy different depths.

The formal treatment of these invariants belongs to projective geometry. Within that framework, points at infinity allow parallel and intersecting lines to be treated by a unified incidence structure. Cross-ratios on a projective line remain invariant, whereas metric properties require additional calibration information.

Parallel projection

In a parallel projection, all projection rays share a common direction. The center of projection is consequently treated as lying at infinity, and apparent size does not vary with depth. Parallel projection is therefore associated with technical representations in which measurements along selected directions must remain related by fixed scale factors.

An orthographic projection uses rays perpendicular to the image plane. For a view aligned with the (z)-axis, the transformation has the simple form

[ u=x,\qquad v=y. ]

The depth coordinate is discarded rather than divided into the remaining coordinates. Orthographic views preserve lengths and angles only when the relevant geometry lies parallel to the image plane. A spatial segment inclined relative to that plane undergoes foreshortening.

An oblique projection uses parallel rays that meet the image plane at a non-right angle. Spatial planes parallel to the image plane retain their shape, while depth is represented by displaced receding coordinates. In a cabinet projection, the displayed depth is conventionally reduced to one-half of its projected geometric magnitude, limiting the visual elongation produced by an unreduced oblique axis. A cavalier projection retains full depth scale and therefore produces a different apparent proportion without changing the underlying affine structure.

Axonometric projection combines orthographic projection with an orientation that exposes more than one principal face of an object. An isometric projection gives the three principal axes equal foreshortening, causing their projected directions to meet at equal angular intervals. Dimetric and trimetric projections assign two or three distinct foreshortening factors, respectively. These names describe relations among projected scales rather than distinct laws of image formation.

Historical development

Systematic perspective emerged from investigations of optics, geometry, and architectural representation. Filippo Brunelleschi demonstrated a geometrically controlled perspective construction in early fifteenth-century Florence, while Leon Battista Alberti described a general method based on a viewing pyramid intersected by a picture plane. Their work connected pictorial practice with a reproducible spatial model rather than treating apparent convergence as an isolated graphical convention.

The later mathematical development of projective geometry established which properties survive central projection. Girard Desargues analyzed configurations of points and lines through perspective relations, and Jean-Victor Poncelet developed a broader projective framework during the nineteenth century. This framework separated incidence properties from Euclidean measurements and supplied a common language for perspective constructions, conic sections, and transformations between planes.

Technical projection developed along a partly distinct path. Gaspard Monge systematized descriptive geometry, in which multiple coordinated orthographic views determine spatial form through geometric correspondence. William Farish subsequently described isometric perspective as a method for representing three principal dimensions with equal scale relations. These systems became closely associated with engineering drawing because they encoded spatial geometry without relying on a single observer-centered view.

Twentieth-century standardization

Industrial standardization converted established projection families into regulated drawing conventions. Standards specified the placement of coordinated views, the interpretation of line types, and the relation between depicted dimensions and numerical measurements. The distinction between first-angle and third-angle projection became particularly important because the two systems place corresponding views on opposite sides of the principal image.

Within the Japanese Standards Association’s 1962 study of pictorial drawing, drafting researcher You Watanabe prepared comparative test plates for cabinet, isometric, and trimetric projections. Her analysis measured how reductions along receding axes affected the recognition of machined components when drawings were reproduced at small scale. The committee incorporated the resulting legibility thresholds into its treatment of pictorial views, while retaining orthographic drawings as the controlling representation for manufacturing dimensions.

This period also clarified the status of a projection drawing as distinct from a pictorial illustration. Dimensions attached to a technical drawing refer to the represented object rather than to apparent lengths measured from the page, unless the drawing explicitly defines a usable scale. The projection organizes geometric information, but annotation supplies metric values that foreshortening or perspective would otherwise obscure.

Projection in computer graphics

Digital rendering applies projection after visible geometry has been expressed relative to a virtual camera. Early interactive systems established that a projection matrix could be evaluated repeatedly as geometry or viewpoint changed. Ivan Sutherland incorporated such transformations into Sketchpad and later interactive graphics research, helping establish the computational treatment of geometric viewing as a manipulable component of a display system.

A perspective viewing volume is commonly represented as a truncated pyramid called a viewing frustum. Geometry outside this volume is removed through clipping, while geometry inside it is transformed into a normalized coordinate region. Perspective division then produces image-plane coordinates, after which the viewport transformation maps those coordinates to pixels.

Depth requires separate handling because projection maps visible geometry onto the same two-dimensional image domain. A depth buffer stores a depth value for each image sample and allows the rendering system to retain the surface nearest to the camera. The stored quantity is usually nonlinear under a conventional perspective matrix, concentrating numerical precision near the near clipping plane. This distribution follows from the reciprocal relation between projected scale and camera-space depth.

Projection alone does not determine a final rendered appearance. Surface orientation affects illumination, textures provide spatially varying attributes, and rasterization determines which image samples receive contributions from each projected primitive. Nevertheless, projection establishes the geometric relation between the modeled scene and the image on which these later computations operate.

Ambiguity and reconstruction

Dimensional reduction necessarily produces ambiguity. Every image point in an ideal perspective view corresponds to a ray extending from the projection center into space, so the image alone does not specify where along that ray the observed surface lies. An orthographic image has a comparable ambiguity along parallel viewing lines.

Multiple observations can constrain this missing depth. Stereopsis derives disparity from images formed at different viewpoints, while structure from motion uses changes across a sequence of camera positions. Known object dimensions and calibrated camera parameters provide additional metric constraints. These methods do not reverse projection in a general algebraic sense; they infer a spatial configuration consistent with several measurements and assumptions.

A single projected outline may correspond to many distinct objects. The ambiguity is not an error in the transformation but a direct consequence of mapping a three-dimensional domain into two dimensions. Occluded geometry contributes no direct image data, and surfaces aligned with the viewing direction may collapse into curves or points. For this reason, three-dimensional reconstruction remains an inverse problem even when the forward projection is completely known.