query: where do the projection lines converge in a perspective sketch

pos: Vanishing point or three mutually orthogonal directions can be found. Vanishing 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.

pos: Graphical projection Perspective projection is a linear projection where three dimensional objects are projected on a "picture plane". This has the effect that distant objects appear smaller than nearer objects. It also means that lines which are parallel in nature (that is, meet at the point at infinity) appear to intersect in the projected image, for example if railways are pictured with perspective projection, they appear to converge towards a single point, called vanishing point. Photographic lenses and the human eye work in the same way, therefore perspective projection looks most realistic. Perspective projection is usually categorized into "one-point", "two-point" and "three-point

pos: Horizon science of graphical perspective. Assuming the picture plane stands vertical to ground, and "P" is the perpendicular projection of the eye point "O" on the picture plane, the horizon is defined as the horizontal line through "P". The point "P" is the vanishing point of lines perpendicular to the picture. If "S" is another point on the horizon, then it is the vanishing point for all lines parallel to "OS". But Brook Taylor (1719) indicated that the horizon plane determined by "O" and the horizon was like any other plane: The peculiar geometry of perspective where parallel lines converge in

pos: Point at infinity complex field, the set of points at infinity is a manifold. In artistic drawing and technical perspective, the projection on the picture plane of the point at infinity of a class of parallel lines is called their vanishing point. In hyperbolic geometry, points at infinity are typically named ideal points. Unlike Euclidean and elliptic geometries, each line has two points at infinity: given a line "l" and a point "P" not on "l", the right- and left-limiting parallels converge asymptotically to different points at infinity. All points at infinity together form the Cayley absolute or boundary of a hyperbolic plane.

pos: Vanishing point Vanishing 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. The vanishing point may also be referred to

pos: Perspective (graphical) with one-point perspective. These parallel lines converge at the vanishing point. One-point perspective exists when the picture plane is parallel to two axes of a rectilinear (or Cartesian) scene – a scene which is composed entirely of linear elements that intersect only at right angles. If one axis is parallel with the picture plane, then all elements are either parallel to the picture plane (either horizontally or vertically) or perpendicular to it. All elements that are parallel to the picture plane are drawn as parallel lines. All elements that are perpendicular to the picture plane converge at a single point

pos: Perspective projection distortion the lines represented on the picture plane is identical, "y." because both lines are projected onto one single picture plane and its distance to the observer is measured perpendicularly. This is the case even though it is clear that the left line "x" is actually further away from "S" than the right line "x." This distortion is enhanced because the lines "x" are in a parallel position with respect to the projection plane, and thus have no vanishing point (it is a frontal perspective). And besides that, as the visual angle covers both lines, but the line of sight is

pos: Graphical projection perspective", depending on the orientation of the projection plane towards the axes of the depicted object. Graphical projection methods rely on the duality between lines and points, whereby two straight lines determine a point while two points determine a straight line. The orthogonal projection of the eye point onto the picture plane is called the "principal vanishing point" (P.P. in the scheme on the left, from the Italian term "punto principale", coined during the renaissance). Two relevant points of a line are: The principal vanishing point is the vanishing point of all horizontal lines perpendicular to the picture plane. The

pos: Perspective (graphical) towards the opposite vanishing point. Two-point perspective exists when the painting plate is parallel to a Cartesian scene in one axis (usually the z-axis) but not to the other two axes. If the scene being viewed consists solely of a cylinder sitting on a horizontal plane, no difference exists in the image of the cylinder between a one-point and two-point perspective. Two-point perspective has one set of lines parallel to the picture plane and two sets oblique to it. Parallel lines oblique to the picture plane converge to a vanishing point, which means that this set-up will require two vanishing

pos: Ponzo illusion would have to be longer than a nearer one for both to produce retinal images of the same size. One of the explanations for the Ponzo illusion is the "perspective hypothesis", which states that the perspective feature in the figure is obviously produced by the converging lines ordinarily associated with distance, that is, the two oblique lines appear to converge toward the horizon or a vanishing point. Another is the "framing-effects hypothesis", which says that the difference in the separation or gap of the horizontal lines from the framing converging lines may determine, or at least contribute to the magnitude

pos: Optics principles. For example, the Ames room, Hering, Müller-Lyer, Orbison, Ponzo, Sander, and Wundt illusions all rely on the suggestion of the appearance of distance by using converging and diverging lines, in the same way that parallel light rays (or indeed any set of parallel lines) appear to converge at a vanishing point at infinity in two-dimensionally rendered images with artistic perspective. This suggestion is also responsible for the famous moon illusion where the moon, despite having essentially the same angular size, appears much larger near the horizon than it does at zenith. This illusion so confounded Ptolemy that he incorrectly

pos: Ryan (film) camera. Each object in the three-dimensional scene had to be independently specified, and frame composition had to be independent of the projection. Three types of projection systems were used for the production of "Ryan": perspective linear projection for lines converging at a common vanishing point, parallel linear projection for objects to maintain "parallel line relationships", and nonlinear projection for curvilinear distortion of nearby objects. The latter was defined by the viewing and projection transformations of the cameras and spatial weight functions applied to each camera. Included in the model was support for independent manipulation of two-dimensional transformations, particularly for changing

pos: Drawing perspective is a method of portraying objects on a flat surface so that the dimensions shrink with distance. Each set of parallel, straight edges of any object, whether a building or a table, follows lines that eventually converge at a vanishing point. Typically this convergence point is somewhere along the horizon, as buildings are built level with the flat surface. When multiple structures are aligned with each other, such as buildings along a street, the horizontal tops and bottoms of the structures typically converge at a vanishing point. When both the fronts and sides of a building are drawn, then

pos: Graphical projection of that circle with the horizon line consists of two "distance points". They are useful for drawing chessboard floors which, in turn, serve for locating the base of objects on the scene. In the perspective of a geometric solid on the right, after choosing the principal vanishing point —which determines the horizon line— the 45° vanishing point on the left side of the drawing completes the characterization of the (equally distant) point of view. Two lines are drawn from the orthogonal projection of each vertex, one at 45° and one at 90° to the picture plane. After intersecting the ground

pos: Perspective (graphical) isometric projection, or by nonparallel ones without a vanishing point. By the later periods of antiquity, artists, especially those in less popular traditions, were well aware that distant objects could be shown smaller than those close at hand for increased realism, but whether this convention was actually used in a work depended on many factors. Some of the paintings found in the ruins of Pompeii show a remarkable realism and perspective for their time. It has been claimed that comprehensive systems of perspective were evolved in antiquity, but most scholars do not accept this. Hardly any of the many works

neg: Perspective control has been corrected. The popularity of amateur photography has made distorted photos made with cheap cameras so familiar that many people do not immediately realise the distortion. This "distortion" is relative only to the accepted norm of constructed perspective (where vertical lines in reality do not converge in the constructed image), which in itself is distorted from a true perspective representation (where lines that are vertical in reality would begin to converge above and below the horizon as they become more distant from the viewer). Professional cameras where perspective control is important control the perspective at exposure by raising the

neg: Perspective projection distortion is apparently distorted (foreshortened): to the eye, parallel lines appear to intersect (like the distant edges of a path or road), whereas in "un−distorted" nature, they do not. One of the first uses of perspective was in Giotto’s "Jesus Before the Caïf", more than 100 years before Filippo Brunelleschi’s perspectival demonstrations galvanized the proper widespread use of convergent perspective of the Renaissance. "Artificial perspective projection" was the name given by Leonardo da Vinci to what today is called "classical perspective projection" and, as noted above, is the result of a geometric protocol. 'Artificial perspective projection' is used here rather than

neg: Transformation matrix when using perspective projections. Another type of transformation, of importance in 3D computer graphics, is the perspective projection. Whereas parallel projections are used to project points onto the image plane along parallel lines, the perspective projection projects points onto the image plane along lines that emanate from a single point, called the center of projection. This means that an object has a smaller projection when it is far away from the center of projection and a larger projection when it is closer. The simplest perspective projection uses the origin as the center of projection, and "z" = 1 as the

neg: Perspective projection distortion Perspective projection distortion "This page is about graphical perspective projection, specifically about distortion" Perspective projection distortion is the mechanism that permits a draftsman or artist to produce linear perspective. It is accomplished by a geometric protocol that exhibits the inevitable distortion of three-dimensional space when “projected," i.e., drawn, on a two-dimensional surface. The words projected/projection here refer to the use of graphics’ lines in the protocol to simulate light ray traces from a Station Point (a supposed observer´s location) to the edges and corners of an object in space, creating thereby or by their extension, an image at the lines’

neg: Perspective projection distortion object would appear rotated but still with minimal projection distortion (provided that the projection plane was also rotated). The difference between the images of the same object produced by "artificial" perspective projection and by "natural" perspective projection is called "perspective distortion." It should be noted that both "artificial" and "natural" projection (the image beheld by the eye) foreshorten real objects and that parallel lines appear to intersect, so that is not the difference between them. The distortion is produced by the difference between the concave retina and the flat projection plane. If an artificial perspective would be projected onto a

neg: Oblique projection point. Some also explain the name by the fact that it was the way a rider could see a small object on the ground from his horseback. The term cabinet projection (sometimes cabinet perspective) stems from its use in illustrations by the furniture industry. Like cavalier perspective, one face of the projected object is parallel to the viewing plane, and the third axis is projected as going off at an angle (typically 63.4°). Unlike cavalier projection, where the third axis keeps its length, with cabinet projection the length of the receding lines is cut in half. As a formula, if

neg: Perspective projection distortion a 50° visual angle. To avoid this excessive distortion, the visual angle of the observer should be always under 40°, and the closer it is to 30°, the better. Perspective projection distortion "This page is about graphical perspective projection, specifically about distortion" Perspective projection distortion is the mechanism that permits a draftsman or artist to produce linear perspective. It is accomplished by a geometric protocol that exhibits the inevitable distortion of three-dimensional space when “projected," i.e., drawn, on a two-dimensional surface. The words projected/projection here refer to the use of graphics’ lines in the protocol to simulate light ray traces

neg: Perspective projection distortion possible to the real view, the line of sight should be directed to its center. On a plan drawing, the line of sight should correspond to the bisector of the angle containing the extreme projections of the object(s) to the station point. The area on a plan or on an elevation or section, between the pair of lines that correspond to the extremes of the view and converge at the station point is called the drawing´s visual angle and its breadth depends on the distance from the observer to the object(s). The line of sight should always bisect the observer´s

neg: Portrait of an Army Doctor reduced to geometrical forms. Sometimes certain indications of figures appear in this assemblage, obtained with the help of a few lines and several suggestive points. (Gleizes, 1927) While depth perception and perspective have been subdued to a large extent, the primary structural lines appear to recede or converge toward a multitude of vanishing points located at infinity. Though, unlike those of classical perspective, these vanishing points are not placed on a given horizon (a theoretical line that represents the eye level of the observer), nor do they allow the viewer to reconstruct the relative distance of parts or features of

neg: Perspective (graphical) such that, there would have been vanishing points "had there been parallel lines", and thus enjoy the sense of depth as a perspective projection. On the other hand, parallel projection such as elevation can be approximated by viewing the object in question from very far away, because projection lines from the point of view approaches parallel when the point of view (POV) approaches infinity. This may account for the confusion over zero-point perspective, since natural scenes often are viewed from very far away, and the size of objects within the scene would be insignificant compared to their distance to the

neg: Parallel projection central or perspective projection, in which the rays pass through a fixed point called the "center" or "viewpoint", as this point is moved towards infinity. Put differently, a parallel projection corresponds to a perspective projection with an infinite focal length (the distance between the lens and the focal point in photography) or "zoom". In parallel projections, lines that are parallel in three-dimensional space remain parallel in the two-dimensional projected image. A perspective projection of an object is often considered more realistic than a parallel projection, since it more closely resembles human vision and photography. However, parallel projections are popular in

neg: Orthographic projection in cartography orthographic map projection. In a wide sense, all projections with the point of perspective at infinity (and therefore parallel projecting lines) are considered as orthographic, regardless of the surface onto which they are projected. These kinds of projections distort angles and areas close to the poles. An example of an orthographic projection onto a cylinder is the Lambert cylindrical equal-area projection. Orthographic projection in cartography The use of orthographic projection in cartography dates back to antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective (or azimuthal) projection, in which the sphere is projected onto a tangent

neg: Perspective control of correcting (or distorting) the perspective to the artist's taste. See 3D projection for details. Perspective control Perspective control is a procedure for composing or editing photographs to better conform with the commonly accepted distortions in constructed perspective. The control would: Perspective projection distortion occurs in photographs when the film plane is not parallel to lines that are required to be parallel in the photo. A common case is when a photo is taken of a tall building from ground level by tilting the camera backwards: the building appears to fall away from the camera. In the two images shown

neg: Engineering drawing makes isometric drawings so attractive is the ease with which 60° angles can be constructed with only a compass and straightedge. Isometric projection is a type of axonometric projection. The other two types of axonometric projection are: An oblique projection is a simple type of graphical projection used for producing pictorial, two-dimensional images of three-dimensional objects: In both oblique projection and orthographic projection, parallel lines of the source object produce parallel lines in the projected image. Perspective is an approximate representation on a flat surface, of an image as it is perceived by the eye. The two most characteristic features

neg: Rhumb line that they would converge at certain points of the map: lines going in every direction would converge at each of these points. See compass rose. Such maps would necessarily have been in the Mercator projection therefore not all old maps would have been capable of showing rhumb line markings. The radial lines on a compass rose are also called "rhumbs". The expression ""sailing on a rhumb"" was used in the 16th–19th centuries to indicate a particular compass heading. Early navigators in the time before the invention of the marine chronometer used rhumb line courses on long ocean passages, because the

neg: Graphical projection its own protocol: In parallel projection, the lines of sight from the object to the projection plane are parallel to each other. Thus, lines that are parallel in three-dimensional space remain parallel in the two-dimensional projected image. Parallel projection also corresponds to a perspective projection with an infinite focal length (the distance from a camera's lens and focal point), or "zoom". Images drawn in parallel projection rely upon the technique of axonometry ("to measure along axes"), as described in Pohlke's theorem. In general, the resulting image is "oblique" (the rays are not perpendicular to the image plane); but in special

neg: Stereographic projection projection point. Parallel lines, which do not intersect in the plane, are transformed to circles tangent at projection point. Intersecting lines are transformed to circles that intersect transversally at two points in the sphere, one of which is the projection point. (Similar remarks hold about the real projective plane, but the intersection relationships are different there.) The loxodromes of the sphere map to curves on the plane of the form where the parameter measures the "tightness" of the loxodrome. Thus loxodromes correspond to logarithmic spirals. These spirals intersect radial lines in the plane at equal angles, just as the loxodromes

neg: Radiant (meteor shower) seen from the perspective of the observer, who views a two-dimensional projection against the sky. The geometric effect is identical to crepuscular rays, where parallel sunbeams appear to converge. A meteor that does not point back to the known radiant for a given shower is known as a "sporadic" and is not considered part of that shower. Shower meteors may appear a short time before the radiant has risen in observer's eastern sky. The radiant in such cases is above the horizon at the meteor's altitude. During the active period of most showers, the radiant moves nearly one degree eastwards,

neg: Truncated dodecahedron all even permutations of: where "φ" = is the golden ratio. The "truncated dodecahedron" has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A and H Coxeter planes. The truncated dodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. Schlegel diagrams are similar, with a perspective projection and

neg: Isometric projection isometric projection do not appear larger or smaller as they extend closer to or away from the viewer. While advantageous for architectural drawings where measurements need to be taken directly, the result is a perceived distortion, as unlike perspective projection, it is not how human vision or photography normally work. It also can easily result in situations where depth and altitude are difficult to gauge, as is shown in the illustration to the right. This can appear to create paradoxical or impossible shapes, such as the Penrose stairs. Isometric graphics were regularly used in video games during the 1980s and

neg: Perspective projection distortion visual angle in order to minimize distortion. On a longitudinal section or elevation, the line of sight should also be determined the same way, but usually it is placed horizontally because this position permits a much faster and an easier drawing process, keeping the projection plane in a vertical position. Vanishing points are manifested in perspective projection by the convergence toward an apparent intersection at an infinite distance of spatially parallel lines that obviously never intersect. This phenomenon is also considered a reality in human vision because it is all one experiences throughout life. It consists in an apparent gradual

neg: Drawing the parallel lines forming a side converge at a second point along the horizon (which may be off the drawing paper.) This is a two-point perspective. Converging the vertical lines to a third point above or below the horizon then produces a three-point perspective. Depth can also be portrayed by several techniques in addition to the perspective approach above. Objects of similar size should appear ever smaller the further they are from the viewer. Thus the back wheel of a cart appears slightly smaller than the front wheel. Depth can be portrayed through the use of texture. As the texture

neg: Perspective projection distortion picture plane ("P"), along with the line of sight and independently of the single or multiple object´s location. The projection plane can be placed at any place, not only between the object and the observer, but also behind the object, causing the apparent scale of the object. The scale doesn´t impact in any way the perspective´s distortion. Figure M1 illustrates why this occurs. The two lines, "x" are the same length and are the same distance, "z," away from the picture plane, "P." When the two lines are projected onto the picture plane, toward the viewer, "S," the size of

neg: Perspective distortion (photography) of field. Another result of using a wide-angle lens is a greater apparent perspective distortion when the camera is not aligned perpendicularly to the subject: parallel lines converge at the same rate as with a normal lens, but converge more due to the wider total field. For example, buildings appear to be falling backwards much more severely when the camera is pointed upward from ground level than they would if photographed with a normal lens at the same distance from the subject, because more of the subject building is visible in the wide-angle shot. Because different lenses generally require a

neg: General Perspective projection General Perspective projection 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. They are also azimuthal projections, meaning that the projection surface

neg: Perspective projection distortion reduction in object´s size as the objects recede from the observer until they can no more be distinguished (among other optical effects). Parallel lines never intersect in nature, but if sufficiently extended they nearly always seem to intersect in perspective projections and invariantly in human vision. The rare one exception is in the former wherein a plane of a projected object is parallel to the projection plane and thus its edges have no vanishing points. The horizon is depicted in a perspective by a line which designates the infinite limit of a horizontal plane at the observer eyes height. In

neg: Axonometric projection space on a two-dimensional picture plane. It usually comes as a standard feature of CAD systems and other visual computing tools. According to Jan Krikke (2000), "axonometry originated in China. Its function in Chinese art was similar to linear perspective in European art. Axonometry, and the pictorial grammar that goes with it, has taken on a new significance with the advent of visual computing". As with all types of parallel projection, objects drawn with axonometric projection do not appear larger or smaller as they lie closer to or farther away from the viewer. While advantageous for architectural drawings, where measurements

neg: Real-time computer graphics The main characteristic of orthographic projection is that parallel lines remain parallel after the transformation. Perspective projection utilizes the concept that if the distance between the observer and model increases, the model appears smaller than before. Essentially, perspective projection mimics human sight. Clipping is the process of removing primitives that are outside of the view box in order to facilitate the rasterizer stage. Once those primitives are removed, the primitives that remain will be drawn into new triangles that reach the next stage. The purpose of screen mapping is to find out the coordinates of the primitives during the clipping

neg: Orthographic projection in cartography Orthographic projection in cartography The use of orthographic projection in cartography dates back to antiquity. 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.

neg: Graphical projection as going off in an angle (typically 30° or 45° or arctan(2)= 63.4°). Unlike cavalier projection, where the third axis keeps its length, with cabinet projection the length of the receding lines is cut in half. A variant of oblique projection is called "military projection". In this case the horizontal sections are isometrically drawn so that the floor plans are not distorted and the verticals are drawn at an angle. The military projection is given by rotation in the "xy"-plane and a vertical translation an amount "z". Objects drawn with parallel projection do not appear larger or smaller as they

neg: Stereographic projection a local isometry and would preserve Gaussian curvature. The sphere and the plane have different Gaussian curvatures, so this is impossible. Circles on the sphere that do "not" pass through the point of projection are projected to circles on the plane. Circles on the sphere that "do" pass through the point of projection are projected to straight lines on the plane. These lines are sometimes thought of as circles through the point at infinity, or circles of infinite radius. All lines in the plane, when transformed to circles on the sphere by the inverse of stereographic projection, meet at the

neg: Orthographic projection 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. Hipparchus used the projection in the 2nd century BC to determine the places of star-rise and star-set.

neg: 3D projection "weak" perspective projection uses the same principles of an orthographic projection, but requires the scaling factor to be specified, thus ensuring that closer objects appear bigger in the projection, and vice versa. It can be seen as a hybrid between an orthographic and a perspective projection, and described either as a perspective projection with individual point depths formula_9 replaced by an average constant depth formula_10, or simply as an orthographic projection plus a scaling. The weak-perspective model thus approximates perspective projection while using a simpler model, similar to the pure (unscaled) orthographic perspective. It is a reasonable approximation when the

neg: Nautical chart always true north, rather than magnetic north, towards which a compass points. Most charts include a compass rose depicting the variation between magnetic and true north. However, the use of the Mercator projection is not without its drawbacks. Mercator's technique was to make the lines of longitude parallel. On the real globe, the lines of longitude converge as one goes north or south away from the equator, until they meet at the pole. This distortion means that horizontal distances are exaggerated. Mercator's solution, imperfect as it might be, was to increase the distance between lines of latitude in proportion; in

neg: Leaning tower illusion towers rise in parallel but are viewed from below, their corresponding outlines converge in the retinal image due to perspective. The visual system normally "corrects" for the perspective distortion and as a result perceives the towers correctly, i.e. as rising in parallel. However in the case of the two identical images of the Pisa tower, the corresponding outlines of the towers do not converge but run in parallel, and as a result the towers are perceived as non-parallel, i.e. as diverging. The illusion reveals that the visual system is obliged to treat the two images as part of the same

neg: Curvilinear perspective Curvilinear perspective Curvilinear perspective is a graphical projection used to draw 3D objects on 2D surfaces. It was formally codified in 1968 by the artists and art historians André Barre and Albert Flocon in the book "La Perspective curviligne", which was translated into English in 1987 as "Curvilinear Perspective: From Visual Space to the Constructed Image" and published by the University of California Press. Earlier, less mathematically precise versions can be seen in the work of the miniaturist Jean Fouquet. Leonardo da Vinci in a lost notebook spoke of curved perspective lines. Examples of approximated five-point perspective can also be

neg: Perspective projection distortion viewer that car would be. However, this seeming incongruity is cancelled out if the perspective meets three conditions: A) That the line of sight bisects the angle of view, B) That the angle of view is close to 30° (this will be explained below) and C) That it is viewed from the same point as the generated perspective. Mathematically, the difference between artificial perspective projection (perspective projection onto a flat surface) and natural perspective projection (perspective projection onto a spherical surface) is a distortion resulting from the difference in the size of the projection on a concave surface like the

neg: Homogeneous coordinates fact, the most general type of system of homogeneous coordinates for points in the plane if none of the lines is the line at infinity. 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. Modern

neg: Perspective control Perspective control Perspective control is a procedure for composing or editing photographs to better conform with the commonly accepted distortions in constructed perspective. The control would: Perspective projection distortion occurs in photographs when the film plane is not parallel to lines that are required to be parallel in the photo. A common case is when a photo is taken of a tall building from ground level by tilting the camera backwards: the building appears to fall away from the camera. In the two images shown to the right, the first suffers from perspective distortion — in the second that distortion

neg: Perspective projection distortion a single scene there may exist an unlimited number of differing sets of parallel lines which, of course, are not necessarily horizontal defining an unlimited number of planes and an accompanying unlimited number of vanishing points (which correspond to their infinite extensions), which can be on, over or under the horizon. If one would have bilateral vision so as to view 180°, one would see the parallel lines appear to intersect in opposite directions simultaneously even if viewed from between the lines, but this bilateral vision is impossible, both in human vision and in perspective. We must not forget that

neg: Perspective projection distortion two distortions exist simultaneously. In the special case — and the only instance — in which perspective imagery appears affected only by the perspective projection distortion, the real observer must view the perspective imagery from precisely the supposed station point of the perspective. Real human vision and perspective projection should (unless it is otherwise desired) look the same. The difference should be imperceptible. The base to rate the quality of the perspective projection is the real vision and the difference between them is the perspective projection distortion. Normal human vision should not be considered to present any distortion unless a

neg: Masaccio who is represented to the left of the Virgin in the painting, while his wife is right of St. John the Evangelist. The fresco, considered by many to be Masaccio's masterwork, is the earliest surviving painting to use systematic linear perspective, possibly devised by Masaccio with the assistance of Brunelleschi. According to the reconstruction Masaccio started by producing a rough drawing of the composition and perspective lines on the wall. The drawing was covered with fresh plaster for making the fresco. To ensure the precise transfer of the perspective lines from the sketch to the plaster, Masaccio inserted a nail

neg: Perspective projection distortion 'classical perspective projection' in order to recognize Leonardo's priority in developing the concept. "Natural perspective projection" is the name given by Leonardo to the projective., i.e., reflected light, image beheld by the human eye and which is impossible to replicate on a plane surface. Figures 1-2 illustrate the principle of an artificial perspective projection. The artificial perspective projection image appears upon the "projection plane" ("P"). A supposed human eye is placed at the "station point" ("S"). In Figure 1 this human eye views both the object and the image of the object as if the projection plane did not exist.

neg: Oblique projection the advent of 3D games) also often use a form of oblique projection. Examples include "SimCity", "Ultima VII", "Ultima Online", "EarthBound", "Paperboy" and, more recently, "Tibia". Oblique 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, but the technique does yield somewhat convincing and useful images. Oblique projection is commonly used in technical drawing. The cavalier projection was used by French military artists in the

neg: Oblique projection Oblique 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, but the technique does yield somewhat convincing and useful images. 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

neg: Archaeo-optics of a megalithic cairn, would very effectively generate a sense of "otherness". Furthermore, people in the Neolithic did not share present-day knowledge of optical physics, and would have had their own understandings. Even from a scientific perspective, however, the act of witnessing the projection of the sun’s disc, or human figures emerging from a stone wall, is rather suggestive of an encounter with the supernatural. It has been suggested that projections of the sun may have been interpreted as the manifestation of a Neolithic solar deity. Finally, optical projections converge with other sensory qualities of these places, potentially including the

neg: General Perspective projection not for minimal distortion. A picture taken with a hand-held camera from the window of a spacecraft has a tilted vertical perspective, so the manned Gemini and Apollo space missions sparked interest in this projection. Some prominent Internet mapping tools also use the tilted perspective projection. For example, Google Earth and NASA World Wind show the globe as it appears from space. These applications permit a wide variety of interactive pan and zoom operations, including fly-through simulations, mimicking pictures or movies taken with a hand-held camera from an airplane or spacecraft. General Perspective projection The General Perspective projection is a

neg: Orthographic projection Orthographic projection Orthographic projection (sometimes orthogonal projection), is a means of representing three-dimensional objects in two dimensions. It is a form of parallel projection, in which all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are "not" orthogonal to the projection plane. The term "orthographic" is sometimes reserved specifically for depictions of objects where the principal axes or planes of the object are also

neg: Angle of view Another result of using a wide angle lens is a greater apparent perspective distortion when the camera is not aligned perpendicularly to the subject: parallel lines converge at the same rate as with a normal lens, but converge more due to the wider total field. For example, buildings appear to be falling backwards much more severely when the camera is pointed upward from ground level than they would if photographed with a normal lens at the same distance from the subject, because more of the subject building is visible in the wide-angle shot. Because different lenses generally require a different

neg: Wide-angle lens perspective distortion when the camera is not aligned perpendicularly to the subject: parallel lines converge at the same rate as with a normal lens, but converge more due to the wider total field. For example, buildings appear to be falling backwards much more severely when the camera is pointed upward from ground level than they would if photographed with a normal lens at the same distance from the subject, because more of the subject building is visible in the wide-angle shot. Because different lenses generally require a different camera–subject distance to preserve the size of a subject, changing the angle

neg: Iconorama Iconorama Iconorama was a Cold War electronic projection system for graphic presentation ("stylized display using an etched plate to produce symbols") ordered by the USAF in 1959. The mechanism used a rear projection display, showing both a map overlay from a fixed lantern slide and dynamically updated data from a mechanical plotter. The mechanism for etching the slide is somewhat similar to an Etch A Sketch. The display can draw lines and characters, but cannot erase them individually. The unit was used in the IBM 473L Command and Control System's Large Panel Display Subsystem (e.g., at the National Military Command

neg: Transverse Mercator projection directions. Grid lines of the transverse projection, other than the "x" and "y" axes, do not run north-south or east-west as defined by parallels and meridians. This is evident from the global projections shown above. Near the central meridian the differences are small but measurable. The difference between the north-south grid lines and the true meridians is the angle of convergence. Transverse Mercator projection The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the UTM. When paired with a suitable

neg: Ultra wide angle lens (as the viewing angle increases). However, with modern technology and understanding of optics, ultra wide-angle lenses can these days eliminate distortion almost completely. Longer lenses magnify the subject more, apparently compressing distance and (when focused on the foreground) blurring the background because of their shallower depth of field. Wider lenses tend to magnify distance between objects while allowing greater depth of field. Another result of using a wide-angle lens is a greater apparent perspective distortion when the camera is not aligned perpendicularly to the subject: parallel lines converge at the same rate as with a normal lens, but converge more

neg: Parallel projection Parallel projection A parallel projection is a projection of an object in three-dimensional space onto a fixed plane, known as the "projection plane" or "image plane", where the "rays", known as "lines of sight" or "projection lines", are parallel to each other. It is a basic tool in descriptive geometry. The projection is called "orthographic" if the rays are perpendicular (orthogonal) to the image plane, and "oblique" or "skew" if they are not. A parallel projection is a particular case of "projection" in mathematics and "graphical projection" in technical drawing. Parallel projections can be seen as the limit of a

neg: Octant projection Octant projection The octant projection or octants projection, is a type of projection proposed the first time, in 1508, by Leonardo da Vinci in his Codex Atlanticus. Leonardo's authorship would be demonstrated by Christopher Tyler, who stated "For those projections dated later than 1508, his drawings should be effectively considered the original precursors ...". In fact, there is a sketch of it on a page in the Codex Atlanticus manuscripts, made from the very hand of Leonardo, being Leonardo's sketch, the first known description of the "octant projection". The same page of the Codex contains sketches of eight other projections

neg: Single-lens reflex camera photographer can correct certain distortions such as "keystoning", where the image 'lines' converge (i.e., photographing a building by pointing a typical camera upward to include the top of the building). Perspective correction lenses are available in the 35 mm and medium formats to correct this distortion with film cameras, and it can also be corrected after the fact with photo software when using digital cameras. The photographer can also extend the bellows to its full length, tilt the front standard and perform photomacrography (commonly known as 'macro photography'), producing a sharp image with depth-of-field without stopping down the lens diaphram.

neg: Crepuscular rays Crepuscular rays Crepuscular rays (more commonly known as sunbeams, sun rays, splintered light, or god rays), in atmospheric optics, are rays of sunlight that appear to radiate from the point in the sky where the Sun is located. Shining through openings in clouds (particularly stratocumulus) or between other objects, these columns of sunlit particulates are separated by darker cloud-shadowed volumes. Despite seeming to converge toward the light source, the rays are in fact near-parallel shafts of sunlight. Their apparent convergence is a perspective effect. This phenomenon mimics the way parallel railway lines or long hallways, for example, seem to seek

neg: Gall stereographic projection Gall stereographic projection The Gall stereographic projection, presented by James Gall in 1855, is a cylindrical projection. It is neither equal-area nor conformal but instead tries to balance the distortion inherent in any projection. The projection is conventionally defined as: where "λ" is the longitude from the central meridian in degrees, "φ" is the latitude, and "R" is the radius of the globe used as the model of the earth for projection. It is a perspective projection if the point of projection is allowed to vary with longitude: the point of projection being on the equator on the opposite side

neg: Perspective projection distortion wrongly directed to the first line´s middle point and is not obtained bisecting the visual angle (which can be deduced by the position of the projection plane), the distortion is greatly exaggerated. An example on how to contrast this distortion, is a view where one is standing facing north towards a road which runs perfectly east-west. In an artificial frontal perspective projection, every car on the road would be drawn at the same size, even though it is clear in reality that the farther away from the center of the picture that a car is, the farther away from the

neg: Map projection cylindrical projection in equatorial regions with the Collignon projection in polar areas. The term "conic projection" is used to refer to any projection in which meridians are mapped to equally spaced lines radiating out from the apex and circles of latitude (parallels) are mapped to circular arcs centered on the apex. When making a conic map, the map maker arbitrarily picks two standard parallels. Those standard parallels may be visualized as secant lines where the cone intersects the globe—or, if the map maker chooses the same parallel twice, as the tangent line where the cone is tangent to the globe.

neg: Map projection true perspective projections; that is, they can be constructed mechanically, projecting the surface of the Earth by extending lines from a point of perspective (along an infinite line through the tangent point and the tangent point's antipode) onto the plane: Other azimuthal projections are not true perspective projections: Conformal, or orthomorphic, map projections preserve angles locally, implying that they map infinitesimal circles of constant size anywhere on the Earth to infinitesimal circles of varying sizes on the map. In contrast, mappings that are not conformal distort most such small circles into ellipses of distortion. An important consequence of conformality is

neg: General Perspective projection is a plane tangent to the sphere. This results in correct directions from the center to all other points. The "point of perspective", or vantage point, for the General Perspective Projection is at a finite distance. It depicts the earth as it appears from some relatively short distance above the surface, typically a few hundred to a few tens of thousands of kilometers. When tilted, the General Perspective projection is not azimuthal (see second figure below); directions are not true from the central point, and the projection plane is not tangent to the sphere. Tilted perspectives are common from aerial

neg: Geo warping of the radar projection that equal-range lines are circles and equal-azimuth lines are straight lines. This is necessary to display radar video consistently with other map data when using a radar projection where the projection center has to be the radar position. This section explains the actual geo warping or re-projection process when applied to radar video in real time. Assume we want to display radar video on top of a satellite image. As an example we use the CIB projection which is used to display satellite data in CIB (Controlled Image Base) format. The Figure "Geo Warping Radar to

neg: Projective space Projective space In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space "V". The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R denotes ordered pairs of real numbers, and R denotes ordered triplets of real numbers. The idea of a projective space relates to perspective, more precisely to the way an eye or a camera projects a 3D scene to a 2D image. All points that lie on a projection line (i.e., a "line

neg: Steiner conic uniquely determined by prescribing the images of three lines. That means that, for the Steiner generation of a conic section, besides two points formula_3 only the images of 3 lines have to be given. These 5 items (2 points, 3 lines) uniquely determine the conic section. "Remark:" The notation "perspective" is due to the dual statement: The projection of the points on a line formula_12 from a center formula_19 onto a line formula_20 is called a perspectivity (see below). For the following example the images of the lines formula_21 (see picture) are given: formula_22. The projective mapping formula_6 is the

neg: Mathematics and art Sketch", Escher showed that art can be created with polygons or regular shapes such as triangles, squares, and hexagons. Escher used irregular polygons when tiling the plane and often used reflections, glide reflections, and translations to obtain further patterns. Many of his works contain impossible constructions, made using geometrical objects which set up a contradiction between perspective projection and three dimensions, but are pleasant to the human sight. Escher's "Ascending and Descending" is based on the "impossible staircase" created by the medical scientist Lionel Penrose and his son the mathematician Roger Penrose. Some of Escher's many tessellation drawings were inspired

neg: The Blueprint 3 Kanye West, Rihanna, Drake, Kid Cudi, Young Jeezy, J. Cole, Alicia Keys, and more.) The cover consists of a large amount of all-white instruments and recording tools stacked in a corner, with three red lines across the image. Rather than simply using photoshop, the album's design team carefully stacked all the equipment in a corner then used a projector to create the bars. They then painted red onto the equipment where the projection of the bars was, and replaced the projector with a camera to achieve the correct perspective for the image. "Blueprint 3" would be Jay-Z's first album cover

neg: Eckert II projection Eckert II projection The Eckert II projection is an equal-area pseudocylindrical map projection. In the equatorial aspect (where the equator is shown as the horizontal axis) the network of longitude and latitude lines consists solely of straight lines, and the outer boundary has the distinctive shape of an elongated hexagon. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. Within each pair, the meridians have the same shape, and the odd-numbered projection has equally spaced parallels, whereas the even-numbered projection has parallels spaced to preserve area. The pair to

neg: Planar projection (e.g., photographic negative, photographic print, computer monitor). When the centre of projection is at a finite distance from the projection plane, a perspective projection is obtained. When the centre of projection is at infinity, all the projectors are parallel, and the corresponding subset of planar projections are referred to as parallel projections. Mathematically, planar projections are linear transformations acting on a point in three-dimensional space formula_1 to give a point formula_2 on the projection plane. These transformations consist of various compositions of the five transformations: orthographic projection, rotation, shear, translation and perspective. It is also used in maps to show

neg: General Perspective projection and low orbit photography, generally taken from at a height measured in kilometers to hundreds of kilometers, rather than the hundreds or thousands of kilometers typical of a vertical perspective. Some forms of the projection were known to the Greeks and Egyptians 2,000 years ago. It was studied by several French and British scientists in the 18th and 19th centuries. However, the projection had little practical value at that time; computationally simpler nonperspective azimuthal projections could be used instead. Space exploration led to a renewed interest in the perspective projection. Now the concern was for a pictorial view from space,

neg: Arnaud Courlet de Vregille the catalog "Le deuxième regard" wrote : ""The painting of Courlet de Vregille enters into an almost natural relation with the inspiration of Ledoux ; indeed both explore those paths which Bachelard referred to as" "materialised imagination, that spiritual place where all forms of art converge and meet." ""Following a very contemporary trend, an expression of the point of view of one art on another, which resumes here in an independent vision the wonder of new discovery"", a portrait of the Lumière brothers : "Lumière ou Projection privée" ("Light or Private projection"), is exhibited in l'Eden Théâtre de La Ciotat.

neg: 3D projection depth of the object along the line of sight is small compared to the distance from the camera, and the field of view is small. With these conditions, it can be assumed that all points on a 3D object are at the same distance formula_10 from the camera without significant errors in the projection (compared to the full perspective model). Equation assuming focal length formula_13. When the human eye views a scene, objects in the distance appear smaller than objects close by - this is known as perspective. While orthographic projection ignores this effect to allow accurate measurements, perspective projection

neg: Armadillo projection 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

neg: 2.5D (visual perception) a different object depending on the perceiver. David Marr’s work on the 2.5D Sketch has found that 2.5D has visual projection constraints. 2.5D projection constraints exist because "parts of images are always (deformed) discontinuities in luminance"; therefore, in reality we do not see all of our surroundings but construct the viewer-centered three-dimensional view of our environment. A primary aspect in regards to the human visual system is blur perception. It plays a vital role in ocular focusing in order for one to attain clarity central to retinal-imagery. Visual perception is a complex system in which blur perception plays a key

neg: Projections onto convex sets simplicity of the algorithm explains some of its popularity. If the intersection of "C" and "D" is non-empty, then the sequence generated by the algorithm will converge to some point in this intersection. Unlike Dykstra's projection algorithm, the solution need not be a projection onto the intersection "C" and "D". The method of averaged projections is quite similar. For the case of two closed convex sets "C" and "D", it proceeds by It has long been known to converge globally. Furthermore, the method is easy to generalize to more than two sets; some convergence results for this case are in.

neg: 3D rendering projection is that objects that are further away are made smaller in relation to those that are closer to the eye. Programs produce perspective by multiplying a dilation constant raised to the power of the negative of the distance from the observer. A dilation constant of one means that there is no perspective. High dilation constants can cause a "fish-eye" effect in which image distortion begins to occur. Orthographic projection is used mainly in CAD or CAM applications where scientific modeling requires precise measurements and preservation of the third dimension. 3D rendering 3D rendering is the 3D computer graphics process

neg: Eckert II projection the globe to be projected. Here, "y" assumes the sign of "φ". Eckert II projection The Eckert II projection is an equal-area pseudocylindrical map projection. In the equatorial aspect (where the equator is shown as the horizontal axis) the network of longitude and latitude lines consists solely of straight lines, and the outer boundary has the distinctive shape of an elongated hexagon. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. Within each pair, the meridians have the same shape, and the odd-numbered projection has equally spaced parallels, whereas

neg: Witt vector power series in formula_15 where formula_21 are usually taken from the set formula_54 Of course, this power series usually will not converge in formula_55 using the standard metric on the reals, but it will converge in formula_28 with the p-adic metric. Viewed as a sequence, formula_16 is just formula_58,if one forgets the ring structure. What follows is a sketch of how a ring structure can be provided for the sequence. Letting formula_59 be denoted by formula_60, one might consider the following definition for addition: However, this lacks several properties needed to produce a general formula; most notably, one does not

neg: Thiotricha rhodomicta discal. Four or five ochreous-whitish lines converge to the apical projection posteriorly, one running along its lower edge. There is also a black subapical dot preceded by a silvery dot. The hindwings are grey, lighter anteriorly. Thiotricha rhodomicta Thiotricha rhodomicta is a moth of the Gelechiidae family. It was described by Meyrick in 1918. It is found in India (Assam). The wingspan is 12–16 mm. The forewings are brownish, more or less irrorated dark fuscous, sometimes partially suffused carmine-pink, sometimes mixed ochreous-whitish in the disc. There are dark fuscous dots towards the base near the costa and dorsum, and near

neg: Armadillo projection central meridian "λ" and a point with geographical latitude "φ" and longitude "λ", plane coordinates "x" and "y" can be computed using the following formulas: In this formulation, no latitude more southerly than "φ" should be plotted for the given longitude. The "y"-axis coincides with the central meridian. 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

neg: Isometric video game graphics parallelly projected objects do not change size as they move about the game field, there is no need for the computer to scale sprites or do the complex calculations necessary to simulate visual perspective. This allowed older 8-bit and 16-bit game systems (and, more recently, handheld and mobile systems) to portray large 3D areas quickly and easily. And, while the depth confusion problems of parallel projection can sometimes be a problem, good game design can alleviate this. There are also gameplay advantages to using an isometric or pseudo-isometric perspective in video games. For instance, compared to a purely top-down game,

neg: Real image (such as the images on the right), real rays of light are always represented by full, solid lines; perceived or extrapolated rays of light are represented by dashed lines. A real image occurs where rays converge, whereas a virtual image occurs where rays only "appear" to converge. Real images can be produced by concave mirrors and converging lenses, only if the object is placed further away from the mirror/lens than the focal point and this real image is inverted. As the object approaches the focal point the image approaches infinity, and when the object passes the focal point the image

neg: Gnomonic projection to find the shortest route between start and destination. The gnomonic projection is used extensively in photography, where it is called "rectilinear projection". The gnomonic projection is used in astronomy where the tangent point is centered on the object of interest. The sphere being projected in this case is the celestial sphere, "R" = 1, and not the surface of the Earth. Gnomonic projection A gnomonic map projection displays all great circles as straight lines, resulting in any straight line segment on a gnomonic map showing a geodesic, the shortest route between the segment's two endpoints. This is achieved by

neg: Punto y Raya Festival projection systems and the notion of perspective in 3d space. In the second version the festival explored all "dots and lines of a schizoid nature". That is, who suffer from an identity crisis and wish to belong to another dimension. A delightful journey through fractal dimensions inhabited by forms and movements that try to resignify the euclidean categories in a mathematical or poetic way. This first "version" of the festival features exclusively dot·lines as ends in themselves. No figuration, no perspective, just dot·lines moving on a plane. Punto y Raya Festival Punto y Raya Festival is a festival devoted to

neg: Franconian languages somewhat variable in meaning. It can refer to a West Germanic dialect continuum spoken in the Rhineland, including Dutch at one end and all the transitional dialects between Dutch and standard German, including it, which do not fully participate in the High German consonant shift or German diphthongization of long vowels. This area follows the course of the Benrath and Speyer lines, a zone along which the two changes occurred along with other diagnostic changes. From a linguistics point of view the Rhineland is an intermediate zone where the Rhenish Fan is located, a zone where eight isoglosses converge. Not