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Topic: Perspective projection distortion

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	Perspective (graphical) Summary
		One of the pioneers of mathematical perspective was Leone Battista Alberti.
		Projective geometry is the study of those properties of plane figures that are unchanged when a given set of points is projected onto a second plane.
		Perspective (from Latin perspicere, to see clearly) in the graphic arts, such as drawing, is an approximate representation on a flat surface (such as paper) of an image as it is perceived by the eye.
	www.bookrags.com /Perspective_(graphical) (4993 words)

	Axonometric projections - a technical overview
		The linear perspective, which was perfected throughout the 17th century in Europe, is based on Euclidean optics: the eye as a point object that catches straight light rays and that senses only the colour, the intensity and the angle of the rays, not their length.
		NEN 2536 describes an isometric projection that is symmetric with regards to the vertical axis; the angle between the x- and y-axes, and between the z- and y-axes, is 60 degrees.
		Also note (again) that the perspective is distorted and that the angles of the top view and the side view should be taken as a "rule of thumb"; I calculated these angles in the same way as the angles in the dimetric projection presented in NEN 2536, but these calculations are, actually, no longer applicable.
	www.compuphase.com /axometr.htm (3715 words)

	Cartographic Projections
		Azimuthal projections are in fact the limiting case of conic projections as the cone becomes a plane.
		The name of this projection is sufficient to describe it; it is conformal (preserves angles) except at the pole of the projection, and it is a conic with all the general characteristics of conic projections.
		A transverse aspect of this projection is known as Cassini's projection and is frequently used for areas near a specific meridian which is taken for the equator of the projection.
	www.research.ibm.com /dx/proceedings/cart/cart.htm (1671 words)

	OMC: supported projections
		Projection number five (Azimuthal Equidistant Projection) returns a circular plot of the entire world with the area of interest as the center, distances measured from there are true.
		Distortion is zero at the center of the projection, and increases radially away from this point.
		The orthographic azimuthal projection is a perspective projection from infinite distance.
	www.aquarius.geomar.de /omc/omc_project.html (628 words)

	Azimuthal Projections
		The gnomonic projection is the perspective, azimuthal projection with point of perspective at the center of the globe.
		Hence, with the gnomonic projection, the interior of a hemispherical region of the globe is projected to the
		The default clipping region for the gnomonic projection is a circle with a radius of 60 degrees at the center of projection.
	hesperia.gsfc.nasa.gov /~jma/idl_html_help/projections9.html (1384 words)

	Perspective projection distortion - Wikipedia, the free encyclopedia
		Perspective projection distortion refers here to the inevitable deviation from the appearance of reality which is introduced into drawing and photography by the use of "projectors" onto a flat surface.
		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 small-angle approximation which results from projecting onto a flat surface.
		In an artificial 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 viewer that car would be.
	en.wikipedia.org /wiki/Perspective_projection_distortion (2367 words)

	Map Projections Poster
		Some projections are suited for mapping large areas that are mainly north-south in extent, others for large areas that are mainly east-west in extent, and still others for large areas that are oblique to the Equator.
		For example, the almost grotesque distortion that is obvious at high latitudes on a small-scale Mercator map of the world disappears almost completely on a properly oriented large-scale Transverse Mercator map of a small area in the same high latitudes.
		Cylindrical— Mathematically projected on a cylinder tangent to the Equator.
	erg.usgs.gov /isb/pubs/MapProjections/projections.html (3465 words)

	What Is Perspective?
		Linear perspective is a mathematical system for projecting the three-dimensional world onto a two-dimensional surface, such as paper or canvas.
		Linear perspective soon emerged as the tool for artists to capture the world around them in a remarkably illusionistic manner (this was the same time that cartographers were mapping the surface of the earth using a similar system of mathematical projection).
		Curvilinear perspective is an alternate to linear perspective.
	www.artic.edu /aic/students/sciarttech/2d1.html (1292 words)

	Tilted Perspective (Site not responding. Last check: 2007-09-17)
		Perspective projections are neither conformal nor equal area.
		Vertical Perspective projections are azimuthal; Tilted Perspective projections are not.
		Considerable distortion near the edges of the projection and throughout Tilted Perspective projections.
	www.manifold.net /doc/700/tilted_perspective.htm (199 words)

	Distortion correction in the Zapruder film (Site not responding. Last check: 2007-09-17)
		Finding the distortion from the wall is strictly a two dimensional problem and is not affected by the perspective projection that forms the image on the film.
		This field of view determination is itself affected by distortion (and to a very slight degree, the placement of the center), so the wall in 249 and a reasonable guess at the center were used to get an estimate of the distortion.
		The effect of distortion on the field of view determination is not large, so after three iterations, the field of view constant remained the same to four decimal places.
	home.att.net /~joliraja2/DistortionDiscussion.html (1394 words)

	Perspective: Perspective: Principle II
		The projections of all lines that are parallel in space either remain parallel in the picture plane or intersect at a single vanishing point (Fig.
		Thus, the first job in perspective projection is to identify all the lines in the scene that are parallel to each other, then make sure that they are drawn so as to project to a common vanishing point.
		Depiction of the scene as projecting to the plane of the canvas, with the perpendicular lines converging at the central vanishing point, while the transversals remain parallel to it.
	webexhibits.org /sciartperspective/tylerperspective2.html (353 words)

	Perspective View - Rayfront User Manual
		The perspective view projection comes closest to what we usually consider a "natural" view, as it most closely reproduces the way that the human eyes perceive reality.
		A perspective projection is calculated by determining the point where a ray travelling towards the eye from a specific angle penetrates a virtual image plane.
		This diagram illustrates the projection of angles relative to the view direction onto the image plane.
	www.schorsch.com /rayfront/manual/viewedit_v.html (193 words)

	Map Projection Overview
		Map projections are attempts to portray the surface of the earth or a portion of the earth on a flat surface.
		Gall's stereographic cylindrical projection results from projecting the earth's surface from the equator onto a secant cylinder intersected by the globe at 45 degrees north and 45 degrees south.
		The Peters projection is a cylindrical equal-area projection that de-emphasizes area exaggerations in high latitudes by shifting the standard parallels to 45 or 47 degrees.
	www.colorado.edu /geography/gcraft/notes/mapproj/mapproj.html (1829 words)

	Perspective projection - Wikipedia, the free encyclopedia
		Perspective projection is a type of drawing, or rendering, that graphically approximates on a planar (two-dimensional) surface (e.g.
		A perspective projection drawing must be made according to an established geometric protocol.
		All perspectives on a planar surface have some degree of distortion, similar to the distortion created when portraying the earth's surface on a planar map.
	en.wikipedia.org /wiki/Perspective_projection (248 words)

	Mathematics of Perspective Drawing
		The isometric projections are that class or parallel projections for which a round sphere projects to a round circle.
		Perspective transformations have the property that parallel lines on the object are mapped to pencils of lines passing through a fixed point in the drawing plane.
		The usual construction is to draw a square around the circle, and then project the perspective view of the square by finding its edges using the vanishing points and measuring points, the center by drawing the diagonals, and then sketching the projected circle by drawing it tangent to the projected square.
	www.math.utah.edu /~treiberg/Perspect/Perspect.htm (5252 words)

	The Perspective Projection
		Perspective, and is epitomised by the mathematical technique known as the Perspective Projection.
		The first is that the perspective is for viewing from only one point in space: the eyepoint at the centre of the projection, which in this case is the origin of the coordinates.
		The second problem is that the perspective image is projected mathematically onto a flat plane, whereas the retina of the eye is approximately hemi-spherical.
	www-staff.it.uts.edu.au /~don/pubs/persp.html (782 words)

	Map Projections -- 3DSoftware.com (Site not responding. Last check: 2007-09-17)
		Cylindrical projections are used primarily for complete world maps, or for maps along narrow strips of a great circle arc, such as the Equator, a meridian, or an oblique great circle.
		The Cassini map projection is the transverse aspect of the Plate Carree projection.
		While cylindrical and conic projections are related to cylinders and cones wrapped around the globe, the azimuthal projections are formed onto a plane which is tangent to the globe.
	www.3dsoftware.com /Cartography/USGS/MapProjections (977 words)

	Wide-angle geometry (Site not responding. Last check: 2007-09-17)
		Images created by perspective lenses (as opposed to composites made from several images) are effectively cropped at 120-140 degrees by a cos^4 drop-off in intensity.
		Stereographic projection is also a closer approximation to the behavior of our video camera lenses than perspective projection.
		In addition to the distortion of area and shape, the perspective image is missing many of the objects visible in the stereographic image.
	www.cs.hmc.edu /~fleck/iowa-lab/wide.html (623 words)

	5.3.5 Gnomonic Projection (-Jf -JF)
		The Gnomonic azimuthal projection is a perspective projection from the center onto a plane tangent to the surface.
		The projection is neither equal-area nor conformal, and much distortion is introducted near the edge of the hemisphere; in fact, less than a hemisphere may be shown around a given center.
		Scale as 1:xxxxx or as radius/latitude where radius is distance on map in inches from projection center to a particular [possibly oblique] latitude (-Jf), or map width (-JF).
	www.ucalgary.ca /appinst/pubdoc/gmt344/doc/html/GMT_Docs/node55.html (182 words)

	Perspective Camera Models
		Most Structure from Motion (linear and non-linear) techniques begin by assuming a perspective projection model as shown in Figure 3 which can be traced back to Durer and Renaissance painters.
		The projection of the COP onto the image plane along the optical axis is called the principal point.
		This perspective projection is often referred to as a pinhole camera.
	www1.cs.columbia.edu /~jebara/htmlpapers/SFM/node6.html (282 words)

	SquaringTheCircleInPanoramas
		This is a cylindrical projection where the horizontal axis is linearly proportional to the horizontal angle and the vertical axis is linearly proportional to the vertical angle (same proportion for both axes).
		The horizontal axis is linearly proportional to the horizontal angle.
		Project the sphere onto multiple planes which are then unfolded into a single flat image.
	www.vision.caltech.edu /lihi/Demos/SquarePanorama.html (437 words)

	Correction of geometric distortions in pictures
		The perspective projection is the viewing transformation that has been primarily used for producing realistic images, in art, photography and in computer graphics.
		One motivation for using perspective projection in computer graphics is the idea of photorealism: since photographic images have one of the highest degrees of realism that we can achieve, perhaps realistic rendering should model the photographic process.
		It turns out that some of these transformations can be implemented as a postprocessing stage after perspective projection and can be applied to existing images and photos.
	graphics.stanford.edu /~dzorin/perception/sig95 (639 words)

	Basic Perspective (Site not responding. Last check: 2007-09-17)
		Perspective helps to create the illusion of reality by locating forms in space according to certain conventions.
		Perspective Made Easy by Ernest Norling is one of the best books for artists on basic perspective.
		Perspective Drawing Handbook by Joseph d'Amelio helped me to learn perspective and I still use it as a reference.
	www.nitaleland.com /articles/perspective.htm (286 words)

	Geometric distortion for dome (fisheye) projection
		In what follows an alternative is discussed, namely, the geometry is distorted in such a way that when viewed using an orthographic projection the result is a correct fisheye image.
		As a result some fisheye projection and lens arrangements truncate the fisheye, the truncated portion of the fisheye is usually located at the back of a planetarium dome or at the lower half of a vertical dome display such as the visionstation.
		A geometric reality of fisheye projection is that the shortest distance between two points is not a straight line in fisheye space, like it is in a perspective projection.
	astronomy.swin.edu.au /~pbourke/projection/domegeom (884 words)

	Lens Correction / Distortion
		To illustrate the type of distortion involved consider a reference grid, with a 35mm lens it would look something line the image on the left, a traditional perspective projection would look like the image on the right.
		An important requirement for these distortion functions is they need to be strictly one-to-one, that is, there is a unique vertical value for each horizontal value (and visa-versa).
		Note that in both cases (distorting the cartesian coordinates or polar coordinates) it is possible for there to be an unmappable region, that is, coordinates in the new image which when distorted lie outside the bounds of the input image.
	local.wasp.uwa.edu.au /~pbourke/projection/lenscorrection (1850 words)

	Basic Light Reflection Model
		This case can arise when a number of texels map onto a surface, but the projection of the surface in screen space is small, either because of its depth or because of its orientation with respect to the viewing direction.
		Pre distorting an environment to fit the interior surface of a sphere is difficult.
		Integration over a two-dimensional environment map projection, rather than over a three-dimensional environment (which is what anti-aliasing operations in ray tracing do), can produce gross errors, but Greene points out that 'the subjective quality of reality cues produced by environment mapping may be superior to those produced by a naive ray tracer'.
	www-viz.tamu.edu /faculty/parke/ends375f04/notes/sec7_4.html (4523 words)

	Hugin tutorial — Perspective correction
		Notice that there are two identical images side by side, each pair of control-points that you set involves selecting a related feature in each side.
		This elevation of the Capitoline Museum was stitched from four photographs and perspective corrected at the same time.
		The Simulating an architectural projection tutorial shows how to use just vertical control points to remove one degree of perspective.
	hugin.sourceforge.net /tutorials/perspective/en.shtml (1066 words)

	GameDev.net - Axonometric Projections - A Technical Overview
		Due to the coarseness of digital coordinates and the requirement that the edges of (checkerboard) tiles should match precisely, without any pixel overlaps or gaps, the skewing angles and scaling factors that game designers use are an approximation of the visually optimal proportions.
		These calculations would be valid if the perspective projections had not distored the original cube.
		Other dimetric projections are summarized in the table below.
	www.gamedev.net /reference/articles/article1269.asp (3733 words)

	The camera model (Site not responding. Last check: 2007-09-17)
		In this work the perspective camera model is used.
		The process is completely determined by choosing a perspective projection center and a retinal plane.
		The projection of a scene point is then obtained as the intersection of a line passing through this point and the center of projection
	www.cs.unc.edu /~marc/tutorial/node35.html (102 words)

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