1.
2.5D
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By contrast, games using 3D computer graphics without such restrictions are said to use true 3D. Common in video games, these projections have also been useful in geographic visualization to help understand visual-cognitive spatial representations or 3D visualization, lines perpendicular to the plane become points, lines parallel to the plane have true length, and lines inclined to the plane are foreshortened. They are popular camera perspectives among 2D video games, most commonly those released for 16-bit or earlier and handheld consoles, as well as in later strategy and role-playing video games. The advantage of these perspectives are that they combine the visibility and mobility of a game with the character recognizability of a side-scrolling game. There are three divisions of axonometric projection, isometric, dimetric, and trimetric. The most common of these types in engineering drawing is isometric projection. This projection is tilted so that all three axes create equal angles at intervals of 120 degrees, the result is that all three axes are equally foreshortened. In video games, a form of projection with a 2,1 pixel ratio is more common due to the problems of anti-aliasing. In oblique projection typically all three axes are shown unforeshortened, all lines parallel to the axes are drawn to scale, and diagonals and curved lines are distorted. One tell-tale sign of oblique projection is that the face pointed toward the camera retains its right angles with respect to the image plane, two examples of oblique projection are Ultima VII, The Black Gate and Paperboy. Examples of axonometric projection include SimCity 2000, and the role-playing games Diablo, the name refers to the fact that objects are seen as if drawn on a billboard. This technique was used in early 1990s video games when consoles did not have the hardware power to render fully 3D objects. This is also known as a backdrop and this can be used to good effect for a significant performance boost when the geometry is sufficiently distant that it can be seamlessly replaced with a 2D sprite. In games, this technique is most frequently applied to such as particles. A pioneer in the use of technique was the game Jurassic Park. It has since become mainstream, and is found in games such as Rome, Total War. Other examples include early first-person shooters like Wolfenstein 3D, Doom, Hexen and Duke Nukem 3D as well as racing games like Carmageddon, skyboxes and skydomes are methods used to easily create a background to make a game level look bigger than it really is. A skydome employs the concept but uses a sphere or hemisphere instead of a cube

2.
Anamorphosis
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Anamorphosis is a distorted projection or perspective requiring the viewer to use special devices or occupy a specific vantage point to reconstitute the image. The word anamorphosis is derived from the Greek prefix ana‑, meaning back or again, an optical anamorphism is the visualization of a mathematical operation called an affine transformation. There are two types of anamorphosis, perspective and mirror. More-complex anamorphoses can be devised using distorted lenses, mirrors, or other optical transformations, examples of perspectival anamorphosis date to the early Renaissance. Examples of mirror anamorphosis were first seen in the late Renaissance, the deformed image is painted on a plane surface surrounding the mirror. By looking into the mirror, a viewer can see the image undeformed, leonardos Eye is the earliest known definitive example of perspective anamorphosis in modern times. The prehistoric cave paintings at Lascaux may also use this technique, Hans Holbein the Younger is well known for incorporating an oblique anamorphic transformation into his painting The Ambassadors. In this artwork, a distorted shape lies diagonally across the bottom of the frame, viewing this from an acute angle transforms it into the plastic image of a human skull, a symbolic memento mori. During the seventeenth century, Baroque trompe loeil murals often used anamorphism to combine actual architectural elements with illusory painted elements, when a visitor views the art work from a specific location, the architecture blends with the decorative painting. The dome and vault of the Church of St. Ignazio in Rome, painted by Andrea Pozzo, due to neighboring monks complaining about blocked light, Pozzo was commissioned to paint the ceiling to look like the inside of a dome, instead of building a real dome. As the ceiling is flat, there is one spot where the illusion is perfect. Mirror anamorphosis emerged early in the 17th century in Italy and China and it remains uncertain whether Jesuit missionaries imported or exported the technique. Anamorphosis could be used to conceal images for privacy or personal safety, a secret portrait of Bonnie Prince Charlie is painted in a distorted manner on a tray and can only be recognized when a polished cylinder is placed in the correct position. To possess such an image would have seen as treason in the aftermath of the 1746 Battle of Culloden. In the eighteenth and nineteenth centuries, anamorphic images had come to be used more as childrens games than fine art, in the twentieth century, some artists wanted to renew the technique of anamorphosis. Marcel Duchamp was interested in anamorphosis, and some of his installations are visual paraphrases of anamorphoses, Jan Dibbets conceptual works, the so-called perspective corrections are examples of linear anamorphoses. In the late century, mirror anamorphosis was revived as childrens toys. Beginning in 1967, Dutch artist Jan Dibbets based a series of photographic work titled Perspective Corrections on the distortion of reality through perspective anamorphosis

3.
Axonometric projection
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There are three main types of axonometric projection, isometric, dimetric, and trimetric projection. Axonometric means to measure along axes, axonometric projection shows an image of an object as viewed from a skew direction in order to reveal more than one side in the same picture. With axonometric projections the scale of distant features is the same as for near features, so such pictures will look distorted and this distortion is especially evident if the object to view is mostly composed of rectangular features. Despite this limitation, axonometric projection can be useful for purposes of illustration, in German literature, oblique projection is also considered an axonometric view, per Pohlkes theorem, the fundamental theorem of axonometry. In some English literature, axonometric projection is considered a sub-class of orthographic projection, farish published his ideas in the 1822 paper On Isometrical Perspective, in which he recognized the need for accurate technical working drawings free of optical distortion. This would lead him to formulate isometry, isometry means equal measures because the same scale is used for height, width, and depth. S. The popular acceptance of axonometry came in the 1920s, when modernist architects from the Bauhaus, De Stijl architects like Theo van Doesburg used axonometry for their architectural designs, which caused a sensation when exhibited in Paris in 1923. Since the 1920s axonometry, or parallel perspective, has provided an important graphic technique for artists, architects, like linear perspective, axonometry helps depict 3D space on the 2D picture plane. It usually comes as a feature of CAD systems and other visual computing tools. According to Jan Krikke axonometry originated in China and its function in Chinese art was similar to linear perspective in European art. Axonometry, and the grammar that goes with it, has taken on a new significance with the advent of visual computing. The three types of projection are isometric projection, dimetric projection, and trimetric projection, depending on the exact angle at which the view deviates from the orthogonal. Typically in axonometric drawing, one axis of space is shown as the vertical, as the distortion caused by foreshortening is uniform the proportionality of all sides and lengths are preserved, and the axes share a common scale. This enables measurements to be read or taken directly from the drawing, another advantage is that 120° angles are more easily constructed using only a compass and straightedge. In trimetric projection, the direction of viewing is such that all of the three axes of space appear unequally foreshortened, the scale along each of the three axes and the angles among them are determined separately as dictated by the angle of viewing. Approximations are common in dimetric and trimetric drawings, as with all types of parallel projection, objects drawn with axonometric projection do not appear larger or smaller as they extend closer to or away from the viewer. It also can result in situations where depth and altitude are difficult to gauge. In this isometric drawing, the sphere is two units higher than the red one

4.
Bird's-eye view
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A birds-eye view is an elevated view of an object from above, with a perspective as though the observer were a bird, often used in the making of blueprints, floor plans and maps. It can be a photograph, but also a drawing. Before manned flight was common, the birds eye was used to distinguish views drawn from direct observation at high locations. Birds eye views as a genre have existed since classical times, the last great flourishing of them was in the mid-to-late 19th century, when birds eye view prints were popular in the United States and Europe. The terms aerial view and aerial viewpoint are also sometimes used synonymous with birds-eye view, the term aerial view can refer to any view from a great height, even at a wide angle, as for example when looking sideways from an airplane window or from a mountain top. Overhead view is fairly synonymous with birds-eye view but tends to imply a less lofty vantage point than the latter term, for example, in computer and video games, an overhead view of a character or situation often places the vantage point only a few feet above human height. Recent technological and networking developments have made satellite images more accessible, microsoft Bing Maps offers direct overhead satellite photos of the entire planet but also offers a feature named Birds eye view in some locations. The Birds Eye photos are angled at 40 degrees rather than being straight down, satellite imaging programs and photos have been described as offering a viewer the opportunity to fly over and observe the world from this specific angle. In filmmaking and video production, a birds-eye shot refers to a shot looking directly down on the subject, the perspective is very foreshortened, making the subject appear short and squat. This shot can be used to give an overall establishing shot of a scene and these shots are normally used for battle scenes or establishing where the character is. It is shot by lifting the camera up by hands or by hanging it off something strong enough to support it, when a scene needs a large area shot, it is a crane shot. A distinction is drawn between a birds-eye view and a birds-flight view, or view-plan in isometrical projection. The technique was popular among local surveyors and cartographers of the sixteenth and early seventeenth centuries

5.
Cross section (geometry)
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In geometry and science, a cross section is the intersection of a body in three-dimensional space with a plane, or the analog in higher-dimensional space. Cutting an object into slices creates many parallel cross sections, conic sections – circles, ellipses, parabolas, and hyperbolas – are formed by cross-sections of a cone at various different angles, as seen in the diagram at left. Any planar cross-section passing through the center of an ellipsoid forms an ellipse on its surface, a cross-section of a cylinder is a circle if the cross-section is parallel to the cylinders base, or an ellipse with non-zero eccentricity if it is neither parallel nor perpendicular to the base. If the cross-section is perpendicular to the base it consists of two line segments unless it is just tangent to the cylinder, in which case it is a single line segment. A cross section of a polyhedron is a polygon, if instead the cross section is taken for a fixed value of the density, the result is an iso-density contour. For the normal distribution, these contours are ellipses, a cross section can be used to visualize the partial derivative of a function with respect to one of its arguments, as shown at left. In economics, a function f specifies the output that can be produced by various quantities x and y of inputs, typically labor. The production function of a firm or a society can be plotted in three-dimensional space, also in economics, a cardinal or ordinal utility function u gives the degree of satisfaction of a consumer obtained by consuming quantities w and v of two goods. Cross sections are used in anatomy to illustrate the inner structure of an organ. A cross section of a trunk, as shown at left, reveals growth rings that can be used to find the age of the tree. Cavalieris principle states that solids with corresponding sections of equal areas have equal volumes. The cross-sectional area of an object when viewed from an angle is the total area of the orthographic projection of the object from that angle. For example, a cylinder of height h and radius r has A ′ = π r 2 when viewed along its central axis, a sphere of radius r has A ′ = π r 2 when viewed from any angle. For a convex body, each ray through the object from the viewers perspective crosses just two surfaces, descriptive geometry Exploded view drawing Graphical projection Plans

6.
Cutaway drawing
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According to Diepstraten et al. the purpose of a cutaway drawing is to allow the viewer to have a look into an otherwise solid opaque object. Instead of letting the inner object shine through the surrounding surface and this produces a visual appearance as if someone had cutout a piece of the object or sliced it into parts. Cutaway illustrations avoid ambiguities with respect to spatial ordering, provide a sharp contrast between foreground and background objects, and facilitate an understanding of spatial ordering. The goal of this drawings in studies can be to identify common patterns for particular vehicle classes. Thus, the accuracy of most of these drawings, while not 100 percent, is high enough for this purpose. The technique is used extensively in computer-aided design, see first image and it has also been incorporated into the user interface of some video games. In The Sims, for instance, users can select through a control panel whether to view the house they are building with no walls, cutaway walls, the cutaway view and the exploded view were minor graphic inventions of the Renaissance that also clarified pictorial representation. This cutaway view originates in the fifteenth century notebooks of Marino Taccola. In the 16th century cutaway views in definite form were used in Georgius Agricolas mining book De Re Metallica to illustrate underground operations. It shows the many used in mining, such as the machine for lifting men and material into and out of a mine shaft. The term Cutaway drawing was already in use in the 19th century but and these factors, according to Diepstraten et al. breakaway, a cutaway realized by a single hole in the outside of the object. Some more examples of cutaway drawings, from products and systems to architectural building, similar types of technical drawings Cross-section Exploded view drawing Perspective

7.
Exploded-view drawing
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An exploded view drawing is a diagram, picture, schematic or technical drawing of an object, that shows the relationship or order of assembly of various parts. It shows the components of an object slightly separated by distance, the exploded view drawing is used in parts catalogs, assembly and maintenance manuals and other instructional material. Usually, the projection of a view is normally shown from above. It is slightly from above and shown from the side of the drawing in diagonal. An exploded view drawing is a type of drawing, that shows the assembly of mechanical or other parts. It shows all parts of the assembly and how they fit together, in mechanical systems usually the component closest to the center are assembled first, or is the main part in which the other parts get assembled. This drawing can help to represent the disassembly of parts. Exploded diagrams are common in descriptive manuals showing parts placement, or parts contained in an assembly or sub-assembly, usually such diagrams have the part identification number and a label indicating which part fills the particular position in the diagram. Many spreadsheet applications can automatically create exploded diagrams, such as exploded pie charts, in patent drawings in an exploded views the separated parts should be embraced by a bracket, to show the relationship or order of assembly of various parts are permissible, see image. When an exploded view is shown in a figure that is on the sheet as another figure. Exploded views can also be used in drawing, for example in the presentation of landscape design. An exploded view can create an image in which the elements are flying through the air above the architectural plan, the locations can be shadowed or dotted in the siteplan of the elements. Together with the view the exploded view was among the many graphic inventions of the Renaissance. The exploded view can be traced back to the fifteenth century notebooks of Marino Taccola. One of the first clearer examples of a view was created by Leonardo in his design drawing of a reciprocating motion machine. Leonardo applied this method of presentation in several studies, including those on human anatomy. May also show the sequence of assembling or disassembling the detail parts

8.
Fisheye lens
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A fisheye lens is an ultra wide-angle lens that produces strong visual distortion intended to create a wide panoramic or hemispherical image. The term fisheye was coined in 1906 by American physicist and inventor Robert W. Wood based on how a fish would see an ultrawide hemispherical view from beneath the water. Their first practical use was in the 1920s for use in meteorology to study cloud formation giving them the name whole-sky lenses, the angle of view of a fisheye lens is usually between 100 and 180 degrees while the focal lengths depend on the film format they are designed for. Mass-produced fisheye lenses for photography first appeared in the early 1960s and are used for their unique. For the popular 35 mm film format, typical lengths of fisheye lenses are between 8 mm and 10 mm for circular images, and 15–16 mm for full-frame images. For digital cameras using smaller electronic imagers such as 6.4 mm and 8.5 mm format CCD or CMOS sensors, the focal length of miniature fisheye lenses can be as short as 1 to 2 mm. These types of lenses also have other applications such as re-projecting images filmed through a lens, or created via computer generated graphics. Fisheye lenses are used for scientific photography such as recording of aurora and meteors. They are also used as peephole door viewers to give the user a wide field of view. In a circular lens, the image circle is inscribed in the film or sensor area. Further, different fisheye lenses distort images differently, and the manner of distortion is referred to as their mapping function, a common type for consumer use is equisolid angle. Although there are digital fisheye effects available both in-camera and as computer software they cant extend the angle of view of the images to the very large one of a true fisheye lens. The first types of lenses to be developed were circular fisheye — lenses which took in a 180° hemisphere. Some circular fisheyes were available in orthographic projection models for scientific applications and these have a 180° vertical angle of view, and the horizontal and diagonal angle of view are also 180°. Most circular fisheye lenses cover a smaller circle than rectilinear lenses. The first full-frame fisheye lens to be mass-produced was a 16 mm lens made by Nikon in the early 1970s, Digital cameras with APS-C sized sensors require a 10.5 mm lens to get the same effect as a 16 mm lens on a camera with full-frame sensor. Sigma currently makes a 4. 5mm fisheye lens that captures a 180-degree field of view on a crop body, sunex also makes a 5. 6mm fisheye lens that captures a circular 185-degree field of view on a 1. 5x Nikon and 1. 6x Canon DSLR cameras. Nikon produced a 6 mm circular fisheye lens that was designed for an expedition to Antarctica

9.
Floor plan
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Dimensions are usually drawn between the walls to specify room sizes and wall lengths. Floor plans may also include details of fixtures like sinks, water heaters, furnaces, floor plans may include notes for construction to specify finishes, construction methods, or symbols for electrical items. Similar to a map the orientation of the view is downward from above, but unlike a conventional map, objects below this level are seen, objects at this level are shown cut in plan-section, and objects above this vertical position within the structure are omitted or shown dashed. Plan view or planform is defined as an orthographic projection of an object on a horizontal plane. The term may be used in general to any drawing showing the physical layout of objects. For example, it may denote the arrangement of the objects at an exhibition. Drawings are now reproduced using plotters and large format xerographic copiers and this convention maintains the same orientation of the floor and ceilings plans - looking down from above. RCPs are used by designers and architects to demonstrate lighting, visible mechanical features, a floor plan is not a top view or birds eye view. It is a drawing to scale of the layout of a floor in a building. A top view or birds eye view does not show an orthogonally projected plane cut at the typical 4 foot height above the floor level, a floor plan could show, Interior walls and hallways Restrooms Windows and doors Appliances such as stoves, refrigerators, water heater etc. In other words, a plan is a section viewed from the top, in such views, the portion of the object above the plane is omitted to reveal what lies beyond. In the case of a plan, the roof and upper portion of the walls may typically be omitted. Roof plans are orthographic projections, but they are not sections as their plane is outside of the object. A plan is a method of depicting the internal arrangement of a 3-dimensional object in two dimensions. It is often used in drawing and is traditionally crosshatched. The style of crosshatching indicates the type of material the section passes through, a 3D floor plan can be defined as a virtual model of a building floor plan. Its often used to better convey architectural plans to individuals not familiar with floor plans, despite the purpose of floor plans originally being to depict 3D layouts in a 2D manner, technological expansion has made rendering 3D models much more cost effective. 3D plans show a depth of image and are often complimented by 3D furniture in the room

10.
Graphical projection
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Graphical projection is a protocol, used in technical drawing, by which an image of a three-dimensional object is projected onto a planar surface without the aid of numerical calculation. The projection is achieved by the use of imaginary projectors, the projected, mental image becomes the technician’s vision of the desired, finished picture. By following the protocol the technician may produce the picture on a planar surface such as drawing paper. The protocols provide a uniform imaging procedure among people trained in technical graphics, the orthographic projection is derived from the principles of descriptive geometry and is a two-dimensional representation of a three-dimensional object. It is the type of choice for working drawings. Within parallel projection there is a known as Pictorials. Pictorials show an image of an object as viewed from a direction in order to reveal all three directions of space in one picture. Parallel projection pictorial instrument drawings are used to approximate graphical perspective projections. Because pictorial projections inherently have this distortion, in the instrument drawing of pictorials, great liberties may then be taken for economy of effort, parallel projection pictorials rely on the technique of axonometric projection. Axonometric projection is a type of projection used to create a pictorial drawing of an object. There are three types of axonometric projection, isometric, dimetric, and trimetric projection. In isometric pictorials, the direction of viewing is such that the three axes of space appear equally foreshortened, and there is an angle of 120° between them. As the distortion caused by foreshortening is uniform the proportionality of all sides and lengths are preserved, and this enables measurements to be read or taken directly from the drawing. Approximations are common in dimetric drawings, in trimetric pictorials, the direction of viewing is such that all of the three axes of space appear unequally foreshortened. The scale along each of the three axes and the angles among them are determined separately as dictated by the angle of viewing, approximations in Trimetric drawings are common. In oblique projections the parallel projection rays are not perpendicular to the plane as with orthographic projection. In both orthographic and oblique projection, parallel lines in space appear parallel on the projected image, because of its simplicity, oblique projection is used exclusively for pictorial purposes rather than for formal, working drawings. In an oblique pictorial drawing, the angles among the axes as well as the foreshortening factors are arbitrary

11.
Isometric projection
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Isometric projection is a method for visually representing three-dimensional objects in two dimensions in technical and engineering drawings. It is a projection in which the three coordinate axes appear equally foreshortened and the angle between any two of them is 120 degrees. The term isometric comes from the Greek for equal measure, reflecting that the scale along each axis of the projection is the same. An isometric view of an object can be obtained by choosing the viewing direction such that the angles between the projections of the x, y, and z axes are all the same, or 120°. For example, with a cube, this is done by first looking straight towards one face, next, the cube is rotated ±45° about the vertical axis, followed by a rotation of approximately ±35. 264° about the horizontal axis. Note that with the cube the perimeter of the resulting 2D drawing is a regular hexagon, all the black lines have equal length. Isometric graph paper can be placed under a piece of drawing paper to help achieve the effect without calculation. In a similar way, a view can be obtained in a 3D scene. Starting with the camera aligned parallel to the floor and aligned to the axes, it is first rotated vertically by about 35. 264° as above. Another way isometric projection can be visualized is by considering a view within a cubical room starting in a corner and looking towards the opposite. The x-axis extends diagonally down and right, the y-axis extends diagonally down and left, depth is also shown by height on the image. Lines drawn along the axes are at 120° to one another, the term isometric is often mistakenly used to refer to axonometric projections generally. From the two angles needed for a projection, the value of the second may seem counterintuitive. Let’s first imagine a cube with sides of length 2, and we can calculate the length of the line from its center to the middle of any edge as √2 using Pythagoras theorem. By rotating the cube by 45° on the x-axis, the point will become as depicted in the diagram. The second rotation aims to bring the point on the positive z-axis. There are eight different orientations to obtain a view, depending into which octant the viewer looks. 264°. As explained above, this is a rotation around the axis by β

12.
Multiview projection
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Along the y-axis, The top and bottom views, which are known as plans. Along the z-axis, The front and back views, which are known as elevations. Orthographic projections show the views of an object, each viewed in a direction parallel to one of the main coordinate axes. These primary views are called plans and elevations, sometimes they are shown as if the object has been cut across or sectioned to expose the interior, these views are called sections. Auxiliary views are taken from an angle that is not one of the primary views. A plan is a view of a 3-dimensional object seen from vertically above and it may be drawn in the position of a horizontal plane passing through, above, or below the object. The outline of a shape in this view is called its planform. The plan view from above a building is called its roof plan, a section seen in a horizontal plane through the walls and showing the floor beneath is called a floor plan. An elevation is a view of a 3-dimensional object from the position of a vertical plane beside an object, in other words, an elevation is a side-view as viewed from the front, back, left or right. An elevation is a method of depicting the external configuration. Building façades are shown as elevations in architectural drawings and technical drawings, elevations are the most common orthographic projection for conveying the appearance of a building from the exterior. Perspectives are also used for this purpose. A building elevation is typically labeled in relation to the direction it faces. E. g. the North Elevation of a building is the side that most closely faces true north on the compass, interior elevations are used to show detailing such as millwork and trim configurations. In the building elevations are a non-perspective view of the structure. These are drawn to scale so that measurements can be taken for any aspect necessary, Drawing sets include front, rear and both side elevations. A developed elevation is a variant of a regular elevation view in several adjacent non-parallel sides may be shown together. For example, the north and west views may be shown side-by-side, sharing an edge, a section, or cross-section, is a view of a 3-dimensional object from the position of a plane through the object

13.
Oblique projection
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This article discusses the imaging of 3D objects. For an abstract mathematical discussion, see Projection, oblique projection is a simple type of technical drawing of graphical projection used for producing two-dimensional images of three-dimensional objects. The objects are not in perspective, so they do not correspond to any view of an object that can be obtained in practice, oblique projection is commonly used in technical drawing. The cavalier projection was used by French military artists in the 18th century to depict fortifications, oblique projection was used almost universally by Chinese artists from the first or second centuries to the 18th century, especially when depicting rectilinear objects such as houses. Oblique projection is a type of projection, it projects an image by intersecting parallel rays from the three-dimensional source object with the drawing surface. In both oblique projection and orthographic projection, parallel lines of the source object produce parallel lines in the projected image, the projectors in oblique projection intersect the projection plane at an oblique angle to produce the projected image, as opposed to the perpendicular angle used in orthographic projection. Mathematically, the projection of the point on the xy-plane gives. The constants a and b uniquely specify a parallel projection, when a = b =0, the projection is said to be orthographic or orthogonal. The constants a and b are not necessarily less than 1, in a general oblique projection, spheres of the space are projected as ellipses on the drawing plane, and not as circles as you would expect them from an orthogonal projection. Oblique drawing is also the crudest 3D drawing method but the easiest to master, oblique is not really a 3D system but a two-dimensional view of an object with forced depth. One way to using an oblique view is to draw the side of the object you are looking at in two dimensions, i. e. Even with this depth, oblique drawings look very unconvincing to the eye. For this reason oblique is rarely used by designers and engineers. In an oblique pictorial drawing, the angles displayed among the axis, more precisely, any given set of three coplanar segments originating from the same point may be construed as forming some oblique perspective of three sides of a cube. This result is known as Pohlkes theorem, from the German mathematician Pohlke, the resulting distortions make the technique unsuitable for formal, working drawings. Nevertheless, the distortions are partially overcome by aligning one plane of the parallel to the plane of projection. Doing so creates a true image of the chosen plane. Cavalier projection is the name of such a projection, where the length along the z axis remains unscaled, cabinet projection, popular in furniture illustrations, is an example of such a technique, wherein the receding axis is scaled to half-size

14.
Picture plane
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It is ordinarily a vertical plane perpendicular to the sightline to the object of interest. In the technique of graphical perspective the picture plane has several features, Given are an eye point O, a plane of reference called the ground plane γ. The orientation of the plane is always perpendicular of the axis that comes straight out of your eyes. For example, if you are looking to a building that is in front of you and your eyesight is entirely horizontal then the plane is perpendicular to the ground. If you are looking up or down the Picture plane remains perpendicular to your sight, when this happens a third vanishing point will appear in most cases depending on what you are seeing. A well-known phrase has accompanied many discussions of painting during the period of modernism, greenberg seems to be referring to the way painting relates to the picture plane in both the modern period and the Old Master period. Morehead Jr. Perspective and Projective Geometries, A Comparison from Rice University

15.
Projection plane
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A projection plane, or plane of projection, is a type of view in which graphical projections from an object intersect. Projection planes are used often in descriptive geometry and graphical representation, a picture plane in perspective drawing is a type of projection plane. With perspective drawing, the lines of sight between an object and a picture plane return to a point and are not parallel. With parallel projection the lines of sight from the object to the plane are parallel

16.
Stereographic projection
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In geometry, the stereographic projection is a particular mapping that projects a sphere onto a plane. The projection is defined on the sphere, except at one point. Where it is defined, the mapping is smooth and bijective and it is conformal, meaning that it preserves angles. It is neither isometric nor area-preserving, that is, it preserves neither distances nor the areas of figures, intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. In practice, the projection is carried out by computer or by using a special kind of graph paper called a stereographic net, shortened to stereonet. The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians and it was originally known as the planisphere projection. Planisphaerium by Ptolemy is the oldest surviving document that describes it, one of its most important uses was the representation of celestial charts. The term planisphere is still used to refer to such charts, in the 16th and 17th century, the equatorial aspect of the stereographic projection was commonly used for maps of the Eastern and Western Hemispheres. It is believed that already the map created in 1507 by Gualterius Lud was in stereographic projection, as were later the maps of Jean Roze, Rumold Mercator, in star charts, even this equatorial aspect had been utilised already by the ancient astronomers like Ptolemy. François dAguilon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles, in 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal. He used the recently established tools of calculus, invented by his friend Isaac Newton and this section focuses on the projection of the unit sphere from the north pole onto the plane through the equator. Other formulations are treated in later sections, the unit sphere in three-dimensional space R3 is the set of points such that x2 + y2 + z2 =1. Let N = be the pole, and let M be the rest of the sphere. The plane z =0 runs through the center of the sphere, for any point P on M, there is a unique line through N and P, and this line intersects the plane z =0 in exactly one point P′. Define the stereographic projection of P to be this point P′ in the plane, in Cartesian coordinates on the sphere and on the plane, the projection and its inverse are given by the formulas =, =. In spherical coordinates on the sphere and polar coordinates on the plane, here, φ is understood to have value π when R =0. Also, there are ways to rewrite these formulas using trigonometric identities. In cylindrical coordinates on the sphere and polar coordinates on the plane, the projection is not defined at the projection point N =

17.
Vanishing point
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In graphical perspective, a vanishing point is a point in the image plane where the projections of a set of parallel lines in space intersect. Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing points. The vanishing point may also be referred to as the point, as lines having the same directional vector, say D. Mathematically, let q ≡ be a point lying on the plane, where f is the focal length, and let vq ≡ be the unit vector associated with q. When the image plane is parallel to two axes, lines parallel to the axis which is cut by this image plane will meet at infinity i. e. at the vanishing point. Lines parallel to the two axes will not form vanishing points as they are parallel to the image plane. Similarly, when the plane intersects two world-coordinate axes, lines parallel to those planes will meet at infinity and form two vanishing points. In three-point perspective the image plane intersects the x, y, the vanishing point theorem is the principal theorem in the science of perspective. It says that the image in a picture plane π of a line L in space, not parallel to the picture, is determined by its intersection with π, some authors have used the phrase, the image of a line includes its vanishing point. Guidobaldo del Monte gave several verifications, and Humphry Ditton called the result the main and Great Proposition. She notes, in terms of geometry, the vanishing point is the image of the point at infinity associated with L. As a vanishing point originates in a line, so a vanishing line originates in a plane α that is not parallel to the picture π. Given the eye point O, and β the plane parallel to α and lying on O, to put it simply, the vanishing line is obtained by the intersection of the image plane with a plane parallel to the ground plane, passing through the camera center. For different sets of line, their respective vanishing points will lie on this line. The horizon line is a line that represents the eye level of the observer. If the object is below the line, its vanishing lines angle up to the horizon line. If the object is above, they slope down, all vanishing lines end at the horizon line. Proof, Consider the ground plane π, as y = c which is, for the sake of simplicity, also, consider a line L that lies in the plane π, which is defined by the equation ax + bz = d