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.

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.

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.

Plan (drawing)

drawingsplansplan
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.

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.

Projection

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

Armadillo
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.

Orthographic projection

orthographicprojectionprojections
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 Rees's Cyclopædia articles

List of Rees's ''Cyclopaedia'' articlesList of Rees's ''Cyclopædia'' articles
The Rees Project, was instigated by Professor June Zimmerman Fullmer, who independently indexed the Cyclopædia. After tapping the invisible college of scholars who knew of Rees, she convened a summer 1986 meeting in London, following which she wrote a proposal to the American Foundation for the Humanities for funding to the project, setting out the object of producing a printed concordance to the contents of the Cyclopædia. This was intended to make Rees much more widely accessible to the modern reader. Funding was not forthcoming, and the matter lapsed. The Cyclopædia lacks a classified index volume.

Filippo Brunelleschi

Brunelleschibrunelleschian
Brunelleschi wanted his new perspective "realism" to be tested not by comparing the painted image to the actual Baptistery but to its reflection in a mirror according to the Euclidean laws of geometric optics. This feat showed artists vividly how they might paint their images, not merely as flat two-dimensional shapes, but looking more like three-dimensional structures, just as mirrors reflect them. Both panels have since been lost. Around this time linear perspective, as a novel artistic tool, spread not only in Italy but throughout Western Europe. It quickly became, and remains, standard studio practice.

Piero della Francesca

Pierodella Francesca, PieroFrescoes
Piero's deep interest in the theoretical study of perspective and his contemplative approach to his paintings are apparent in all his work. In his youth, Piero was trained in mathematics, which most likely was for mercantilism. Three treatises written by Piero have survived to the present day: Trattato d'Abaco (Abacus Treatise), Libellus de Quinque Corporibus Regularibus (Short Book on the Five Regular Solids) and De Prospectiva pingendi (On Perspective in painting). The subjects covered in these writings include arithmetic, algebra, geometry and innovative work in both solid geometry and perspective. Much of Piero's work was later absorbed into the writing of others, notably Luca Pacioli.

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.