3D projection

projectiongraphical projection3D
B_z is the focal length—the axial distance from the camera center to the image plane. A_z is the subject distance. 3D computer graphics. Camera matrix. Computer graphics. Cross section. Curvilinear perspective. Cutaway. Descriptive geometry. Engineering drawing. Exploded-view drawing. Graphics card. Homogeneous coordinates. Homography. Map projection (including Cylindrical projection). Multiview projection. Perspective (graphical). Plan (drawing). Technical drawing. Texture mapping. Transform and lighting. Viewing frustum. Virtual globe.

3D computer graphics

3D3D graphicsthree-dimensional
3D computer graphics or three-dimensional computer graphics (in contrast to 2D computer graphics), are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for the purposes of performing calculations and rendering 2D images. Such images may be stored for viewing later or displayed in real-time. 3D computer graphics rely on many of the same algorithms as 2D computer vector graphics in the wire-frame model and 2D computer raster graphics in the final rendered display.


z-bufferdepth bufferdepth
The range of depth values in camera space to be rendered is often defined between a and value of z. After a perspective transformation, the new value of z, or z', is defined by: After an orthographic projection, the new value of z, or z', is defined by: where z is the old value of z in camera space, and is sometimes called w or w'. The resulting values of z' are normalized between the values of -1 and 1, where the plane is at -1 and the plane is at 1. Values outside of this range correspond to points which are not in the viewing frustum, and shouldn't be rendered. Typically, these values are stored in the z-buffer of the hardware graphics accelerator in fixed point format.

Graphics pipeline

pixel pipelinerendering pipelinepipeline
With vp=Viewport; v=Point after projection. Pipeline (computing). Instruction pipelining. Hardware acceleration. Tomas Akenine-Möller, Eric Haines: Real-Time Rendering. AK Peters, Natick, Mass. 2002, ISBN: 1-56881-182-9. Michael Bender, Manfred Brill: Computergrafik: ein anwendungsorientiertes Lehrbuch. Hanser, München 2006, ISBN: 3-446-40434-1.

Picture plane

integrity of the picture planeperspective planephotographical plane
Morehead Jr. (1911) Perspective and Projective Geometries: A Comparison from Rice University. Perspective projection. Projection plane. Image plane.

Cube mapping

cube mapcube mapsCube-mapped
Another application which found widespread use in video games is projective texture mapping. It relies on cube maps to project images of an environment onto the surrounding scene; for example, a point light source is tied to a cube map which is a panoramic image shot from inside a lantern cage or a window frame through which the light is filtering. This enables a game developer to achieve realistic lighting without having to complicate the scene geometry or resort to expensive real-time shadow volume computations. A cube texture indexes six texture maps from 0 to 5 in order Positive X, Negative X, Positive Y, Negative Y, Positive Z, Negative Z.

Orthographic projection

An orthographic projection map is a map projection of cartography. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective (or azimuthal) projection, in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges. The orthographic projection has been known since antiquity, with its cartographic uses being well documented.

List of computer graphics and descriptive geometry topics

Reverse perspective. Scan line rendering. Scrolling. Technical drawing. Texture mapping. Trimetric projection. Vanishing point. Vector graphics. Vector graphics editor. Vertex shaders. Volume rendering. Voxel. List of geometry topics. List of graphical methods.

Rendering (computer graphics)

Some are integrated into larger modeling and animation packages, some are stand-alone, some are free open-source projects. On the inside, a renderer is a carefully engineered program, based on a selective mixture of disciplines related to: light physics, visual perception, mathematics, and software development. In the case of 3D graphics, rendering may be done slowly, as in pre-rendering, or in realtime. Pre-rendering is a computationally intensive process that is typically used for movie creation, while real-time rendering is often done for 3D video games which rely on the use of graphics cards with 3D hardware accelerators.

Computer graphics

Ray tracing is a technique for generating an image by tracing the path of light through pixels in an image plane. The technique is capable of producing a very high degree of photorealism; usually higher than that of typical scanline rendering methods, but at a greater computational cost. Texture mapping. Texture mapping is a method for adding detail, surface texture, or colour to a computer-generated graphic or 3D model. Its application to 3D graphics was pioneered by Dr Edwin Catmull in 1974. A texture map is applied (mapped) to the surface of a shape, or polygon. This process is akin to applying patterned paper to a plain white box.

Mathematics and art

mathematical artmathematics of artartistic and imaginative pursuit
These, such as the rhombicuboctahedron, were among the first to be drawn to demonstrate perspective by being overlaid on top of each other. The work discusses perspective in the works of Piero della Francesca, Melozzo da Forlì, and Marco Palmezzano. Da Vinci studied Pacioli's Summa, from which he copied tables of proportions. In Mona Lisa and The Last Supper, Da Vinci's work incorporated linear perspective with a vanishing point to provide apparent depth. The Last Supper is constructed in a tight ratio of 12:6:4:3, as is Raphael's The School of Athens, which includes Pythagoras with a tablet of ideal ratios, sacred to the Pythagoreans.

Projective geometry

projectiveprojective geometriesProjection
The first geometrical properties of a projective nature were discovered during the 3rd century by Pappus of Alexandria. Filippo Brunelleschi (1404–1472) started investigating the geometry of perspective during 1425 (see the history of perspective for a more thorough discussion of the work in the fine arts that motivated much of the development of projective geometry). Johannes Kepler (1571–1630) and Gérard Desargues (1591–1661) independently developed the concept of the "point at infinity". Desargues developed an alternative way of constructing perspective drawings by generalizing the use of vanishing points to include the case when these are infinitely far away.


light maplight mappinglightmapping
Texture mapping. Baking (computer graphics).

Clipping (computer graphics)

clippingclippedCircle and B-Splines clipping algorithms
Beyond projection of vertices & 2D clipping, near clipping is required to correctly rasterise 3D primitives; this is because vertices may have been projected behind the eye. Near clipping ensures that all the vertices used have valid 2D coordinates. Together with far-clipping it also helps prevent overflow of depth-buffer values. Some early texture mapping hardware (using forward texture mapping) in video games suffered from complications associated with near clipping and UV coordinates.

Multiview projection

elevationplanformplan view
To render each such picture, a ray of sight (also called a projection line, projection ray or line of sight) towards the object is chosen, which determines on the object various points of interest (for instance, the points that are visible when looking at the object along the ray of sight); those points of interest are mapped by an orthographic projection to points on some geometric plane (called a projection plane or image plane) that is perpendicular to the ray of sight, thereby creating a 2D representation of the 3D object.

History of computer animation

during the late 70s and early 80s1940s and 1950sbirthplace
The computer science faculty was founded by David Evans in 1965, and many of the basic techniques of 3D computer graphics were developed here in the early 70s with ARPA funding (Advanced Research Projects Agency). Research results included Gouraud, Phong, and Blinn shading, texture mapping, hidden surface algorithms, curved surface subdivision, real-time line-drawing and raster image display hardware, and early virtual reality work.


If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the Bolyai–Gerwien theorem. The area of a regular polygon is also given in terms of the radius r of its inscribed circle and its perimeter p by This radius is also termed its apothem and is often represented as a.


Distance is a numerical measurement of how far apart objects are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). In most cases, "distance from A to B" is interchangeable with "distance from B to A". In mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific set of rules, and is a way of describing what it means for elements of some space to be "close to" or "far away from" each other.

Shadow mapping

shadowsshadow mapCascaded Shadow Maps
For a point light source, the view should be a perspective projection as wide as its desired angle of effect (it will be a sort of square spotlight). For directional light (e.g., that from the Sun), an orthographic projection should be used. From this rendering, the depth buffer is extracted and saved. Because only the depth information is relevant, it is common to avoid updating the color buffers and disable all lighting and texture calculations for this rendering, in order to save drawing time. This depth map is often stored as a texture in graphics memory.

Homogeneous coordinates

homogeneous coordinateprojective coordinateshomogeneous co-ordinates
Homogeneous coordinates are ubiquitous in computer graphics because they allow common vector operations such as translation, rotation, scaling and perspective projection to be represented as a matrix by which the vector is multiplied. By the chain rule, any sequence of such operations can be multiplied out into a single matrix, allowing simple and efficient processing. By contrast, using Cartesian coordinates, translations and perspective projection cannot be expressed as matrix multiplications, though other operations can.

Virtual camera system

third-personthird-person perspectivethird-person view
The Walkthrough Project at the University of North Carolina at Chapel Hill produced a number of physical input devices for virtual camera view control including dual three-axis joysticks and a billiard-ball shaped prop known as the UNC Eyeball that featured an embedded six-degree of freedom motion tracker and a digital button. Camera matrix. Game engine. Virtual cinematography. First-person (video games).


orthogonalorthogonallylogically orthogonal
In art, the perspective (imaginary) lines pointing to the vanishing point are referred to as "orthogonal lines". The term "orthogonal line" often has a quite different meaning in the literature of modern art criticism. Many works by painters such as Piet Mondrian and Burgoyne Diller are noted for their exclusive use of "orthogonal lines" — not, however, with reference to perspective, but rather referring to lines that are straight and exclusively horizontal or vertical, forming right angles where they intersect.

Architectural drawing

elevationarchitectural drawingsarchitectural
One-point perspective where objects facing the viewer are orthogonal, and receding lines converge to a single vanishing point. Two-point perspective reduces distortion by viewing objects at an angle, with all the horizontal lines receding to one of two vanishing points, both located on the horizon. Three-point perspective introduces additional realism by making the verticals recede to a third vanishing point, which is above or below depending upon whether the view is seen from above or below. Colen Campbell's Vitruvius Brittanicus, illustrations of English buildings by Inigo Jones and Sir Christopher Wren, as well as Campbell himself and other prominent architects of the era.

Parallax occlusion mapping

Parallax occlusion mapping is used to procedurally create 3D definition in textured surfaces, using a displacement map (similar to a topography map) instead of through the generation of new geometry. This allows developers of 3D rendering applications to add 3D complexity in textures, which correctly change relative to perspective and with self occlusion in real time (self-shadowing is additionally possible), without sacrificing the processor cycles required to create the same effect with geometry calculations. Parallax occlusion mapping was first published in 2005 by Zoe Brawley and Natalya Tatarchuk in ShaderX3.