3D projection

projectiongraphical projection3D
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.

Cartesian coordinate system

Cartesian coordinatesCartesianaxes
If a diagram (3D projection or 2D perspective drawing) shows the x- and y-axis horizontally and vertically, respectively, then the z-axis should be shown pointing "out of the page" towards the viewer or camera. In such a 2D diagram of a 3D coordinate system, the z-axis would appear as a line or ray pointing down and to the left or down and to the right, depending on the presumed viewer or camera perspective. In any diagram or display, the orientation of the three axes, as a whole, is arbitrary. However, the orientation of the axes relative to each other should always comply with the right-hand rule, unless specifically stated otherwise.

Mathematics and art

mathematical artmathematics of artartistic and imaginative pursuit
As early as the 15th century, curvilinear perspective found its way into paintings by artists interested in image distortions. Jan van Eyck's 1434 Arnolfini Portrait contains a convex mirror with reflections of the people in the scene, while Parmigianino's Self-portrait in a Convex Mirror, c. 1523–1524, shows the artist's largely undistorted face at the centre, with a strongly curved background and artist's hand around the edge. Three-dimensional space can be represented convincingly in art, as in technical drawing, by means other than perspective.

Vanishing point

abstract pointone-point perspectivevanishes to a single point
A vanishing point is a point on the image plane of a perspective drawing where the two-dimensional perspective projections (or drawings) of mutually parallel lines in three-dimensional space appear to converge. When the set of parallel lines is perpendicular to a picture plane, the construction is known as one-point perspective, and their vanishing point corresponds to the oculus, or "eye point", from which the image should be viewed for correct perspective geometry. Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing points.

Projective geometry

projectiveprojective geometriesProjection
Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is an example of a concept not invariant with respect to projective transformations, as is seen in perspective drawing. One source for projective geometry was indeed the theory of perspective.

General Perspective projection

perspective (or azimuthal) projectionVertical perspective
The General Perspective projection is a map projection. When the Earth is photographed from space, the camera records the view as a perspective projection. When the camera is aimed toward the center of the Earth, the resulting projection is called Vertical Perspective. When aimed in other directions, the resulting projection is called a Tilted Perspective. The Vertical Perspective is related to the stereographic projection, gnomonic projection, and orthographic projection. These are all true perspective projections, meaning that they result from viewing the globe from some vantage point.

M. C. Escher

EscherM.C. EscherMaurits Cornelis Escher
Escher's interest in curvilinear perspective was encouraged by his friend and "kindred spirit", the art historian and artist Albert Flocon, in another example of constructive mutual influence. Flocon identified Escher as a "thinking artist" alongside Piero della Francesca, Leonardo da Vinci, Albrecht Dürer, Wenzel Jamnitzer, Abraham Bosse, Girard Desargues, and Père Nicon. Flocon was delighted by Escher's Grafiek en tekeningen ("Graphics in Drawing"), which he read in 1959. This stimulated Flocon and André Barre to correspond with Escher and to write the book La Perspective curviligne ("Curvilinear perspective").

Geographic information system

GISgeographic information systemsgeographical information system
Web mapping has also uncovered the potential of crowdsourcing geodata in projects like OpenStreetMap, which is a collaborative project to create a free editable map of the world. These mashup projects have been proven to provide a high level of value and benefit to end users outside that possible through traditional geographic information. The condition of the Earth's surface, atmosphere, and subsurface can be examined by feeding satellite data into a GIS. GIS technology gives researchers the ability to examine the variations in Earth processes over days, months, and years.

Leonardo da Vinci

Leonardoda VinciLéonard de Vinci
He also had a scheme for diverting the flow of the Arno river, a project on which Niccolò Machiavelli also worked. Leonardo's journals include a vast number of inventions, both practical and impractical. They include musical instruments, a mechanical knight, hydraulic pumps, reversible crank mechanisms, finned mortar shells, and a steam cannon. In 1502, Leonardo produced a drawing of a single span 720 ft bridge as part of a civil engineering project for Ottoman Sultan Beyazid II of Constantinople. The bridge was intended to span an inlet at the mouth of the Bosporus known as the Golden Horn. Beyazid did not pursue the project because he believed that such a construction was impossible.


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.

Jan van Eyck

Van EyckJanVan Eyck brothers
Jan van Eyck (before c. 1390 – 9 July 1441) was an Early Netherlandish painter active in Bruges. He is one of the founders of Early Netherlandish painting and one of the most significant representatives of Early Northern Renaissance art. The few surviving records of his early life indicate that he was born around 1380–1390, most likely in Maaseik. He took employment in the Hague around 1422, when he was already a master painter with workshop assistants, and employed as painter and valet de chambre with John III the Pitiless, ruler of Holland and Hainaut.

Fisheye lens

fisheyefish-eye lensfisheye camera
Azimuthal equidistant projection. Dashcam. Little planet effect. Stereographic projection. de:Fischaugenobjektiv Fisheye lens with more information in German. Various fisheye projections. Alternative archived URL.

Perspective projection distortion

distortion of perspectiveperspective distortion
Perspective correction. Perspective distortion (photography). Projective geometry.


For instance, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry. Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons.

Plan (drawing)

Multiview projection, including:. Plan view or floor plan view. Elevation, usually a side view of an exterior. Section, a view of the interior at a particular cutting plane. Axonometric projection, including:. Isometric projection. Dimetric projection. Trimetric projection. Oblique projection, and. Perspective projection, including:. One-point perspective. Two-point perspective. Three-point perspective. General Information : The first sheets in a set may include notes, assembly descriptions, a rendering of the project, or simply the project title. Site : Site plans, including a key plan, appear before other plans and on smaller projects may be on the first sheet.

Graphics pipeline

pixel pipelinerendering pipelinepipeline
In a perspective illustration, a central projection is used. To limit the number of displayed objects, two additional clipping planes are used; The visual volume is therefore a truncated pyramid (frustum). The parallel or orthogonal projection is used, for example, for technical representations because it has the advantage that all parallels in the object space are also parallel in the image space, and the surfaces and volumes are the same size regardless of the distance from the viewer.

Projection (mathematics)

projectioncentral projectionprojection map
In cartography, a map projection is a map of a part of the surface of the Earth onto a plane, which, in some cases, but not always, is the restriction of a projection in the above meaning. The 3D projections are also at the basis of the theory of perspective. The need for unifying the two kinds of projections and of defining the image by a central projection of any point different of the center of projection are at the origin of projective geometry. However, a projective transformation is a bijection of a projective space, a property not shared with the projections of this article.


Isometric projection. Oblique projection. Perspective projection. Fischer projection. Haworth projection. Natta projection. Newman projection. Projection (mathematics), any of several different types of geometrical mappings, including. Projection (linear algebra). Projection (set theory). Projection (measure theory). 3D projection. Vector projection. Projection (relational algebra). Projective geometry, a non-Euclidean geometry that involves. Projective spaces. Projective lines. Projective planes. Projective transformations. Projective linear groups. Projective coordinates. Projective module, a generalization of a free module. Projective object, a further generalization, in category theory.

Cervin Robinson

Army where he gained an abiding interest in map projections and perspective. Impressed early in his life with physics and photography, he continued to photograph in earnest while stationed with the Army in Germany. Upon return to the U.S., he became the assistant for Walker Evans (1953–1957), and traveled through much of the American heartland. In 1958, Robinson began contract work for the Historic American Buildings Survey (HABS) photographing in the northeast sector from Maine to Pennsylvania and into the Middle West. At the same time, he acted as American representative for the London-based Architectural Review for which he photographed major new American buildings.

History of cartography

Golden Age of Dutch/Netherlandish cartographycartographycartographer
The Swiss mathematician Johann Lambert invented several hemispheric map projections. In 1772 he created the Lambert conformal conic and Lambert azimuthal equal-area projections. The Albers equal-area conic projection features no distortion along standard parallels. It was invented by Heinrich Albers in 1805. In 1715 Herman Moll published the Beaver Map, one of the most famous early maps of North America, which he copied from a 1698 work by Nicolas de Fer. In 1763–1767 Captain James Cook mapped Newfoundland. In 1777 Colonel Joseph Frederick Wallet DesBarres created a monumental four volume atlas of North America, Atlantic Neptune.

Armadillo projection

The armadillo projection is a map projection used for world maps. It is neither conformal nor equal-area but instead affords a view evoking a perspective projection while showing most of the globe instead of the half or less that a perspective would. The projection was presented in 1943 by Erwin Raisz (1893–1968) as part of a series of "orthoapsidal" projections, which are perspectives of the globe projected onto various surfaces. This one in the series has the globe projected onto half a torus. Raisz singled it out and named it the "armadillo" projection. The toroidal shape and the angle it is viewed from tend to emphasize continental areas by eliminating or foreshortening swaths of ocean.

List of graphical methods

graphical methodmethod
Isometric projection. Orthographic projection. Perspective (graphical). Technical drawing. Graphical projection. Mohr's circle. Pantograph. Circuit diagram. Smith chart. Sankey diagram. Binary decision diagram. Control flow graph. Functional flow block diagram. Information flow diagram. IDEF. N2 chart. State diagram. System context diagram. Map projection. Orthographic projection (cartography). Robinson projection. Stereographic projection. Dymaxion map. Topographic map. Craig retroazimuthal projection. Hammer retroazimuthal projection. Cladogram. Systems Biology Graphical Notation. Free body diagram. Greninger chart. Phase diagram. Wavenumber-frequency diagram. Bode plot. Nyquist plot.

Johann Heinrich Lambert

LambertJohann LambertLambert, Johann Heinrich
In A System of Logic Ratiocinative and Inductive, John Stuart Mill expresses his admiration for Johann Heinrich Lambert. * List of things named after Johann Lambert * Johann Heinrich Lambert (1728-1777): Collected Works - Sämtliche Werke Online 1) Lambert conformal conic. 2) Transverse Mercator. 3) Lambert azimuthal equal area. 4) Lagrange projection. 5) Lambert cylindrical equal area. 6) Transverse cylindrical equal area. 7) Lambert conical equal area. Asimov's Biographical Encyclopedia of Science and Technology, Isaac Asimov, Doubleday & Co., Inc., 1972, ISBN: 0-385-17771-2. A. Papadopoulos and G.

Octacube (sculpture)

OctacubeOctacube'' (sculpture)
In the image on the right a 24-cell is shown projected into space as a 3-D object (and then the image is a 2-D rendering of it, with perspective to aid the eye). Some of the distortions: To map the 24-cell, Ocneanu uses a related projection which he calls windowed radial stereographic projection. As with the stereographic projection, there are curved lines shown in 3-D space. Instead of using semitransparent surfaces, "windows" are cut into the faces of the cells so that interior cells can be seen. Also, only 23 vertices are physically present.

Cube mapping

cube mapcube mapsCube-mapped
The environment is projected onto the sides of a cube and stored as six square textures, or unfolded into six regions of a single texture. The cube map is generated by first rendering the scene six times from a viewpoint, with the views defined by a 90 degree view frustum representing each cube face. In the majority of cases, cube mapping is preferred over the older method of sphere mapping because it eliminates many of the problems that are inherent in sphere mapping such as image distortion, viewpoint dependency, and computational inefficiency.