He had several stereographic images of him produced, among them this 20008 example.
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The article also misidentified one of the animals seen in a stereographic image of Mrs. Maxwell.
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Chacalall comes up with fantastical creatures and animates them with a simple stereographic aesthetic that comes off like a lenticular print—which he also makes.
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You will still see notifications, for example, as well as alarms and incoming calls — all of which will display on cards that are now rendered in stereographic 3D.
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For another, while May is new to the VR game, he clearly understands the power of old media: the Owl VR is made by his London Stereoscopic Company, which sells modern versions of the stereographic viewers that were popular during the Victorian era.
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Darren A. Cole, with the web and social media branch of the National Archives Office of Innovation, told Hyperallergic that NARA initially saw the impact of GIF content through their Today's Document blog on Tumblr, where a curious patent for a one-wheeled vehicle or stereographic portrait of Walt Whitman got a new spark through animation.
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There are various forms of transverse or oblique stereographic projections of ellipsoids. One method uses double projection via a conformal sphere, while other methods do not. Examples of transverse or oblique stereographic projections include the Miller Oblated Stereographic and the Roussilhe oblique stereographic projection.Snyder, John P. (1993).
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Stereographic projection of the world north of 30°S. 15° graticule. The stereographic projection with Tissot's indicatrix of deformation. The stereographic projection, also known as the planisphere projection or the azimuthal conformal projection, is a conformal map projection whose use dates back to antiquity.
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The parallels on the Gall stereographic projection are distributed with the same spacing as those on the central meridian of the transverse stereographic projection. The GS50 projection is formed by mapping the oblique stereographic projection to the complex plane and then transforming points on it via a tenth-order polynomial.
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Fix any point on and a hyperplane in not containing . Then the stereographic projection of a point in is the unique point of intersection of with . As before, the stereographic projection is conformal and invertible outside of a "small" set. The stereographic projection presents the quadric hypersurface as a rational hypersurface.
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During medieval times, the stereographic coordinate system was used for navigation purposes. The stereographic coordinate system was superseded by the latitude-longitude system. Although no longer used in navigation, the stereographic coordinate system is still used in modern times to describe crystallographic orientations in the fields of crystallography, mineralogy and materials science.
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Like the orthographic projection and gnomonic projection, the stereographic projection is an azimuthal projection, and when on a sphere, also a perspective projection. On an ellipsoid, the perspective definition of the stereographic projection is not conformal, and adjustments must be made to preserve its azimuthal and conformal properties. The universal polar stereographic coordinate system uses one such ellipsoidal implementation.
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As an azimuthal projection, the stereographic projection faithfully represents the relative directions of all great circles passing through its center point. As a conformal projection, it faithfully represents angles everywhere. In addition, in its spherical form, the stereographic projection is the only map projection that renders all small circles as circles. 3D illustration of the geometric construction of the stereographic projection.
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A stereographic projection is conformal and perspective but not equal area or equidistant.
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The Roussilhe oblique stereographic projection is a mapping projection developed by Henri Roussilhe in 1922. The projection uses a truncated series to approximate an oblique stereographic projection for the ellipsoid. The projection received some attention in the former Soviet Union.Snyder, John P. (1993).
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Bennett, 83. Yokoyama was the first Japanese photographer to seriously pursue stereographic photography. An early photograph of his studio equipment shows seven cameras, of which two are stereographic. By 1869 Yokoyama, accompanied by friends and students, was travelling throughout Japan to make stereoviews.
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300px Stereographic projection with its 128 blue triangular faces and its 192 green quad faces.
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The Keystone View Company was a major distributor of stereographic images, and was located in Meadville, Pennsylvania. From 1892 through 1963 Keystone produced and distributed both educational and comic/sentimental stereoviews, and stereoscopes. By 1905 it was the world's largest stereographic company. In 1963 Department A (stereoviews sold to individual families) and the Education Departments were closed down, but Keystone continued to manufacture eye- training stereographic products as a subsidiary of Mast Development Company.
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The DVD version includes a red- blue stereographic presentation intended partially to mimic the arcade original.
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However, stereographic fisheye lenses are typically more expensive to manufacture. Image remapping software, such as Panotools, allows the automatic remapping of photos from an equal-area fisheye to a stereographic projection. The stereographic projection has been used to map spherical panoramas, starting with Horace Bénédict de Saussure's in 1779. This results in effects known as a little planet (when the center of projection is the nadir) and a tube (when the center of projection is the zenith).
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Stereographic projection of a spherical cone's generating lines (red), parallels (green) and hypermeridians (blue). Due to conformal property of Stereographic Projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles or straight lines. The generatrices and parallels generates a 3D dual cone.
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Rubens for "Opticorum libri sex philosophis juxta ac mathematicis utiles", by François d'Aguilon. It demonstrates the principle of a general perspective projection, of which the stereographic projection is a special case. The stereographic projection was known to Hipparchus, Ptolemy and probably earlier to the Egyptians. It was originally known as the planisphere projection.
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The stereographic projection was exclusively used for star charts until 1507, when Walther Ludd of St. Dié, Lorraine created the first known instance of a stereographic projection of the Earth's surface. Its popularity in cartography increased after Rumold Mercator used its equatorial aspect for his 1595 atlas.Snyder, John P. 1987. "Map Projections---A Working Manual".
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Stereographic projection of the unit circle onto the x-axis. Given a point P on the unit circle, draw a line from P to the point (the north pole). The point P′ where the line intersects the x-axis is the stereographic projection of P. Inversely, starting with a point P′ on the x-axis, and drawing a line from P′ to N, the inverse stereographic projection is the point P where the line intersects the unit circle. There is a correspondence between points on the unit circle with rational coordinates and primitive Pythagorean triples.
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By 1910, Underwood & Underwood had entered the field of news photography. Due to this expansion, stereograph production was reduced until the early years of World War I. Altogether Underwood & Underwood produced between 30,000 and 40,000 stereographic titles. In 1920 stereograph production was discontinued and Underwood & Underwood sold its stereographic stock and rights to the Keystone View Company.
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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. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography. In practice, the projection is carried out by computer or by hand using a special kind of graph paper called a stereographic net, shortened to stereonet, or Wulff net.
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Also, every plane through the origin intersects the unit sphere in a great circle, called the trace of the plane. This circle maps to a circle under stereographic projection. So the projection lets us visualize planes as circular arcs in the disk. Prior to the availability of computers, stereographic projections with great circles often involved drawing large- radius arcs that required use of a beam compass.
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The fibers of the Hopf fibration stereographically project to a family of Villarceau circles in R3. The Hopf fibration has many implications, some purely attractive, others deeper. For example, stereographic projection S3 → R3 induces a remarkable structure in R3, which in turn illuminates the topology of the bundle . Stereographic projection preserves circles and maps the Hopf fibers to geometrically perfect circles in R3 which fill space.
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3D illustration of a stereographic projection from the north pole onto a plane below the sphere In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet.
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A stereographic projection of the Moon, showing regions polewards of 60° North. Craters which are circles on the sphere appear circular in this projection, regardless of whether they are close to the pole or the edge of the map. The stereographic is the only projection that maps all circles on a sphere to circles on a plane. This property is valuable in planetary mapping where craters are typical features.
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Stereographic projection or fisheye projection can be used to form a little planet panorama by pointing the virtual camera straight down and setting the field of view large enough to show the whole ground and some of the areas above it; pointing the virtual camera upwards creates a tunnel effect. Conformality of the stereographic projection may produce more visually pleasing result than equal area fisheye projection as discussed in the stereo- graphic projection's article.
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German et al. (2007). The popularity of using stereographic projections to map panoramas over other azimuthal projections is attributed to the shape preservation that results from the conformality of the projection.
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A stereoscope is a device for viewing stereographic cards, which are cards that contain two separate images that are printed side by side to create the illusion of a three- dimensional image.
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The complex plane and the Riemann sphere above it Although any stereographic projection misses one point on the sphere (the projection point), the entire sphere can be mapped using two projections from distinct projection points. In other words, the sphere can be covered by two stereographic parametrizations (the inverses of the projections) from the plane. The parametrizations can be chosen to induce the same orientation on the sphere. Together, they describe the sphere as an oriented surface (or two- dimensional manifold).
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While stereoscopic images have typically been used for amusement, including stereographic cards, 3D films, 3D television, stereoscopic video games, printings using anaglyph and pictures, posters and books of autostereograms, there are also other uses of this technology.
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The earliest evidence of use of the stereographic projection in a machine is in , which describes an anaphoric clock (it is presumed, a or water clock) in Alexandria. The clock had a rotating field of stars behind a wire frame indicating the hours of the day. The wire framework (the spider) and the star locations were constructed using the stereographic projection. Similar constructions dated from the 1st to 3rd centuries have been found in Salzburg and northeastern France, so such mechanisms were, it is presumed, fairly widespread among Romans.
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World map made by Rumold Mercator in 1587, using two equatorial aspects of the stereographic projection. The stereographic projection was likely known in its polar aspect to the ancient Egyptians, though its invention is often credited to Hipparchus, who was the first Greek to use it. Its oblique aspect was used by Greek Mathematician Theon of Alexandria in the fourth century, and its equatorial aspect was used by Arab astronomer Al- Zarkali in the eleventh century. The earliest written description of it is Ptolemy's Planisphaerium, which calls it the "planisphere projection".
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Stereographic projections map circles to circles and preserves the circular boundary of the strip. The result is a smooth embedding of the Möbius strip into R3 with a circular edge and no self- intersections. 580px The Sudanese Möbius band in the three-sphere S3 is geometrically a fibre bundle over a great circle, whose fibres are great semicircles. The most symmetrical image of a stereographic projection of this band into R3 is obtained by using a projection point that lies on that great circle that runs through the midpoint of each of the semicircles.
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Rest the south pole of a unit 2-sphere on the -plane in three-space. We map a point of the sphere (minus the north pole ) to the plane by sending to the intersection of the line with the plane. Stereographic projection of a 3-sphere (again removing the north pole) maps to three-space in the same manner. (Notice that, since stereographic projection is conformal, round spheres are sent to round spheres or to planes.) A somewhat different way to think of the one-point compactification is via the exponential map.
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Distortion is very low as well. It is not a standard projection in the sense that it uses complex polynomials (of the tenth order) rather than a trigonometric formulation, though it was developed from an oblique stereographic projection.
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Stereographic projection of the 3-sphere's parallels (red), meridians (blue) and hypermeridians (green). Lunes exist between pairs of blue meridian arcs. Lunes can be defined on higher dimensional spheres as well. In 4-dimensions a 3-sphere is a generalized sphere.
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Flattening the Earth: Two Thousand Years of Map Projections p. 169. Chicago and London: The University of Chicago Press. . The development of the Bulgarian oblique stereographic projection was done for Romania by the Bulgarian geodesist, Hristow, in the late 1930s.
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Stereographic projection map showing the density distribution of dune fields in the Planum Boreum region. The grey regions are lower density fields. The four densest dune fields are shown in black. The prime meridian is at the bottom of the map.
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For the sphere, the problem is to construct all the circles (the boundaries of spherical caps) that are tangent to three given circles on the sphere. This spherical problem can be rendered into a corresponding planar problem using stereographic projection. Once the solutions to the planar problem have been constructed, the corresponding solutions to the spherical problem can be determined by inverting the stereographic projection. Even more generally, one can consider the problem of four tangent curves that result from the intersections of an arbitrary quadratic surface and four planes, a problem first considered by Charles Dupin.
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The scale factor at the origin (the poles) is adjusted to minimize the overall distortion of scale within the mapped region. As with the Mercator projection, the region near the tangent (or secant) point on a Stereographic map remains very close to true scale for an angular distance of a few degrees. In the ellipsoidal model, a stereographic projection tangent to the pole has a scale factor of less than 1.003 at 84° latitude and 1.008 at 80° latitude. The adjustment of the scale factor in the UPS projection reduces the average scale distortion over the entire zone.
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Stereographic projection of the spherical panorama of the Last Supper sculpture by Michele Vedani in Esino Lario, Lombardy, Italy during Wikimania 2016 "Vue circulaire des montagnes qu'on découvre du sommet du Glacier de Buet", Horace-Benedict de Saussure, Voyage dans les Alpes, précédés d'un essai sur l'histoire naturelle des environs de Geneve. Neuchatel, 1779–96, pl. 8. Some fisheye lenses use a stereographic projection to capture a wide-angle view.Samyang 8 mm 3.5 Fisheye CS Compared to more traditional fisheye lenses which use an equal-area projection, areas close to the edge retain their shape, and straight lines are less curved.
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On a stereographic projection map, a loxodrome is an equiangular spiral whose center is the north or south pole. All loxodromes spiral from one pole to the other. Near the poles, they are close to being logarithmic spirals (which they are exactly on a stereographic projection, see below), so they wind around each pole an infinite number of times but reach the pole in a finite distance. The pole-to-pole length of a loxodrome (assuming a perfect sphere) is the length of the meridian divided by the cosine of the bearing away from true north.
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Stereographic projection is also applied to the visualization of polytopes. In a Schlegel diagram, an -dimensional polytope in is projected onto an -dimensional sphere, which is then stereographically projected onto . The reduction from to can make the polytope easier to visualize and understand.
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In cartography, several named map projections, including the Mercator projection and the stereographic projection are conformal. These enjoy the property that the distortion of shapes can be made as small as desired by making the diameter of the mapped region small enough.
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The details don't really matter. Any stereographic projection of a sphere onto a plane will produce one "point at infinity", and it will map the lines of latitude and longitude on the sphere into circles and straight lines, respectively, in the plane.
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He is the eponym of the Dandelin spheres, of Dandelin's theorem in geometry (for an account of that theorem, see Dandelin spheres), and of the Dandelin-Gräffe numerical method of solution of algebraic equations. He also published on the stereographic projection, algebra, and probability theory.
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Stereographic projection map showing the density distribution of dune fields in the Planum Boreum region. The grey regions are lower density fields. The four densest dune fields including Abalos Undae are shown in black. The prime meridian is at the bottom of the map.
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W. Hager, J.F. Behensky, and B.W. Drew, 1989. Defense Mapping Agency Technical Report TM 8358.2. The universal grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) and the Ordnance Survey of Great Britain.A guide to coordinate systems in Great Britain, Ordnance Survey of Great Britain.
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The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection - details are given below). In mathematics, the Riemann sphere, named after Bernhard Riemann,B. Riemann: Theorie der Abel'sche Funktionen, J. Math. (Crelle) 1857; Werke 88-144.
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The Evo 3D uses two rear-facing 5-megapixel cameras, capable of capturing videos in 720p resolution in 3D or in 1080p while recording standard 2D. It can also take photos in stereographic 3D at 5 MPx2 resolution. It features a single 1.3-megapixel front-facing camera.
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5 (4th Quarter 1895), pp. 865-866. The Chicago-based organization was dedicated to advancing the ideas of direct legislation and free coinage of silver through the supplying of inserts and stereographic plates on the topic to newspapers around the country."The Signs of the Times," Money, vol. 1, no.
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Circa 2007, Young began creating digitally edited Stereographic images and collages to be viewed through a lorgnette. This resulted in bound collection of works titled "The Optimix Suite", which now appears in the collections of several universities. A later collection became publicly available in the form of a Google Cardboard application.
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After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean plane. In the same way, removing a single point from the 3-sphere yields three-dimensional space. An extremely useful way to see this is via stereographic projection. We first describe the lower-dimensional version.
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The stereographic projection from the sphere to the plane preserves critical points of geodesic curvature. Thus simple closed spherical curves have four vertices. Furthermore, on the sphere vertices of a curve correspond to points where its torsion vanishes. So for space curves a vertex is defined as a point of vanishing torsion.
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IF'09: Stereo focused on research, production, and experiments with perception in relation to the knowledge of two-channel sound, and stereographic 3D.About IF'09: Stereo Interactive Futures, Retrieved Feb 9, 2014 The sub-themes of Stereographics, Co-Locative, and Sensory Illusions were explored through exhibitions, presentations, panel discussions, film screenings and workshops.
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Using a stereographic projection, the classical Möbius plane may be seen to be isomorphic to the geometry of plane sections (circles) on a sphere in Euclidean 3-space. Analogously to the (axiomatic) projective plane, an (axiomatic) Möbius plane defines an incidence structure. Möbius planes may similarly be constructed over fields other than the real numbers.
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Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect <0,0,0,1> have infinite radius (= straight line). A 3-sphere is a higher-dimensional analogue of a sphere.
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Nautical charts and textual descriptions known as sailing directions have been in use in one form or another since the sixth century BC.Bowditch, 2003:2. Nautical charts using stereographic and orthographic projections date back to the second century BC. In 1900, the Antikythera mechanism was recovered from Antikythera wreck. This mechanism was built around 1st century BC.
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In electron diffraction, Kikuchi line pairs appear as bands decorating the intersection between lattice plane traces and the Ewald sphere thus providing experimental access to a crystal's stereographic projection. Model Kikuchi maps in reciprocal space,M. von Heimendahl, W. Bell and G. Thomas (1964) Applications of Kikuchi line analyses in electron microscopy, J. Appl. Phys. 35:12, 3614-3616.
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How the 24-hour analog dial might be interpreted. Diagram showing how the zodiac is projected on to the ecliptic dial – the symbols are often drawn inside the dial. Stereographic projection from the North Pole. Most astronomical clocks have a 24-hour analog dial around the outside edge, numbered from I to XII then from I to XII again.
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The Samyang 8mm f/3.5 fisheye is a fisheye photographic lens using the stereographic projection and is designed for crop factor APS-C DSLRs. It is made in South Korea by Samyang Optics and marketed under several brand names, including Rokinon. The lens uses manual focus only. For most versions of the lens, the aperture must be set manually.
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Thomas Richard Williams (5 May 1824 – 5 April 1871) was a British professional photographer and one of the pioneers of stereoscopy. Williams's first business was in London around 1850. He is known for his celebrated stereographic daguerreotypes of the Crystal Palace. He also did portrait photography, now in the Getty Museum's archives, which he regarded as his greatest success.
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Mercury(II) chloride was used as a photographic intensifier to produce positive pictures in the collodion process of the 1800s. When applied to a negative, the mercury(II) chloride whitens and thickens the image, thereby increasing the opacity of the shadows and creating the illusion of a positive image.Towler, J. (1864). Stereographic negatives and landscape photography.
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It follows that is a complete metric of constant curvature 0 on the complement of , which is therefore isometric to the plane. Composing with stereographic projection, it follows that there is a smooth function such that has Gaussian curvature +1 on the complement of . The function automatically extends to a smooth function on the whole of .
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Topologically, an -sphere can be constructed as a one-point compactification of -dimensional Euclidean space. Briefly, the -sphere can be described as , which is -dimensional Euclidean space plus a single point representing infinity in all directions. In particular, if a single point is removed from an -sphere, it becomes homeomorphic to . This forms the basis for stereographic projection.
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Coleman applied stereoscopy to the existing principle of toy phantasmascopes using rotating discs. A series of still stereographic images with chronologically successive stages of action were mounted on blades of a spinning paddle and viewed through slits. The slits passed under a stereoscopic viewer. The pictures were visible within a cabinet, and were not projected onto a screen.
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Osborn Maitland Miller (1897–1979) was a Scottish-American cartographer, surveyor and aerial photographer. A member of several expeditions himself, he also acted as adviser to other explorers. He developed several map projections, including the Bipolar Oblique Conic Conformal, the Miller Oblated Stereographic, and most notably the Miller Cylindrical in 1942. The Maitland Glacier in Antarctica was named after Miller in 1952.
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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 is a plane tangent to the sphere. This results in correct directions from the center to all other points.
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Kenderdine has produced more than 70 exhibitions and installations for museums worldwide and has 35 peer-reviewed publications, including two books. She has created several interactive installations at UNESCO World Heritage Sites, including Angkor Wat;Kenderdine, S. 2004. “Stereographic Panoramas of Angkor, Cambodia.” In VSMM2004: Proceedings of the Tenth International Conference on Virtual Systems and Multimedia, edited by H. Thwaites, 612-621.
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Knowlton, K. C., "Computer Generated Movies," Science, Vol. 150, (November 1965), pp. 116–1120. Instead of raw programming, Beflix worked using simple "graphic primitives", like draw a line, copy a region, fill an area, zoom an area, and the like. In 1965, Michael Noll created computer-generated stereographic 3D movies, including a ballet of stick figures moving on a stage.
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Snyder (1993). 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.
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It is believed that already the map created in 1507 by Gualterius LudAccording to (Snyder 1993), although he acknowledges he did not personally see it was in stereographic projection, as were later the maps of Jean Roze (1542), Rumold Mercator (1595), and many others.Snyder (1989). In star charts, even this equatorial aspect had been utilised already by the ancient astronomers like Ptolemy.
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Stereographic projection map showing the density distribution of dune fields in the Planum Boreum region. The grey regions are lower density fields. The four densest dune fields are shown in black. The prime meridian is at the bottom of the map. Siton Undae is shown on the southernmost black patch left, between longitude 291.38°E to 301.4°E (43.98°W – 57.08°W).
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An ex-pickpocket, Trembler works for the Garlands in The Ruby in the Smoke and is characteristically nervous, hence his nickname. He takes to Adelaide very quickly and loves her like a daughter. The stereographic picture of Trembler with Adelaide on his knee is a clue to the true identity of the 'Cockney Queen' in "The Tin Princess". Trembler and Mrs.
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In the polar regions, a different convention is used.DMA Technical Manual 8358.1, Appendix B. Datums, Ellipsoids, Grids, and Grid Reference Systems. South of 80°S, UPS South (Universal Polar Stereographic) is used instead of a UTM projection. The west half-circle forms a grid zone with designation A; the east half-circle forms one with designation B; see figure 3.
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D'Aguilon extensively studied stereographic projection, which he wanted to use a means to aid architects, cosmographers, navigators and artists. For centuries, artists and architects had sought formal laws of projection to place objects on a screen. Aguilon's Opticorum libri sex successfully treated projections and the errors in perception. D'Aguillon adopted Alhazen's theory that only light rays orthogonal to the cornea and lens surface are clearly registered.
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The mathematician Claudius Ptolemy 'the Alexandrian' as imagined by a 16th- century artist The Planisphaerium is a work by Ptolemy. The title can be translated as "celestial plane" or "star chart". In this work Ptolemy explored the mathematics of mapping figures inscribed in the celestial sphere onto a plane by what is now known as stereographic projection. This method of projection preserves the properties of circles.
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Computers now make this task much easier. Further associated with each plane is a unique line, called the plane's pole, that passes through the origin and is perpendicular to the plane. This line can be plotted as a point on the disk just as any line through the origin can. So the stereographic projection also lets us visualize planes as points in the disk.
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Portrait photography was Buehman's primary source of income. Cartes de visite were popular and his clientele would return to his studio for updated portraits as styles changed to prefer things such as full length or three-quarters poses or cabinet cards instead of smaller hand-held images. When stereographic cards became popular in the American West, Buehman was among the first photographers to produce them.
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Many astronomical clocks use an astrolabe-style display, such as the famous clock at Prague, adopting a stereographic projection (see below) of the ecliptic plane. In recent times, astrolabe watches have become popular. For example, Swiss watchmaker Dr. Ludwig Oechslin designed and built an astrolabe wristwatch in conjunction with Ulysse Nardin in 1985. Dutch watchmaker Christaan van der Klauuw also manufactures astrolabe watches today.
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77–79), duality, and the circumcycle and incycle of a triangle (p. 104). Yaglom continues with his Galilean study to the inversive Galilean plane by including a special line at infinity and showing the topology with a stereographic projection. The Conclusion of the book delves into the Minkowskian geometry of hyperbolas in the plane, including the nine-point hyperbola. Yaglom also covers the inversive Minkowski plane.
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He produced at least three series of views that were published at the time, but that are now very hard to find. According to photography historian Rob Oechsle, Yokoyama's are the only notable Japanese- made stereographic series from the early Meiji period; they were taken from 1869 through the 1870s.Oechsle, 221. In 1870, Shimooka Renjō invited Yokoyama to join him in photographing Mount Nikkō-Shirane.
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Central projection of circles on a sphere: The center O of projection is inside the sphere, the image plane is red. As images of the circles one gets a circle (magenta), ellipses, hyperbolas and lines. The special case of a parabola does not appear in this example. (If center O were on the sphere, all images of the circles would be circles or lines; see stereographic projection).
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The Gall–Peters projection of the world map Carrubbers Close Mission Moray Free Church, Holyrood Road, Edinburgh James Gall's grave, Grange Cemetery James Gall (27 September 1808 – 7 February 1895) was a Scottish clergyman who founded the Carrubbers Close Mission. He was also a cartographer, publisher, sculptor, astronomer and author. In cartography he gives his name to three different map projections: Gall stereographic; Gall isographic; and Gall orthographic (Gall–Peters projection).
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Dr. Melen served as Vice President of R&D; for Canon Research Center of America from its inception in 1990 until 2001. During this time he developed image processing technology for document imaging, stereographic photography, and radiographic imaging. In 2001 Melen joined Toyota InfoTechnology Center, U.S.A. as Senior Advisor. At Toyota he has focused on developing technology for vehicular information systems in support of vehicle safety and efficiency.
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The upper disc contains a "horizon", that defines the visible part of the sky at any given moment, which is naturally half of the total starry sky. That horizon line is most of the time also distorted, for the same reason the constellations are distorted. The horizon line on a stereographic projection is a perfect circle. The horizon line on other projections is a kind of "collapsed" oval.
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Later, two entrepreneurs combined photographs from the actual lynching with others staged with actors and sold the 16-image production as a stereographic set. One of the original sets sits in the United States Library of Congress. On , Dan Davis, an African American man suspected of attacking a sixteen-year-old white girl named Carrie Johnson, was burned at the stake in the Smith County Courthouse Square.The New York Times.
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Orthographic projection (equatorial aspect) of eastern hemisphere 30W-150E The orthographic projection with Tissot's indicatrix of deformation. 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.
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Stereographic projection of the hypersphere's parallels (red), meridians (blue) and hypermeridians (green). Because this projection is conformal, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect have infinite radius (= straight line). In this picture, the whole 3D space maps the surface of the hypersphere, whereas in the previous picture the 3D space contained the shadow of the bulk hypersphere.
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Conformal maps containing large regions vary scales by locations, so it is difficult to compare lengths or areas. However, some techniques require that a length of 1 degree on a meridian = 111 km = 60 nautical miles. In non-conformal maps, such techniques are not available because the same lengths at a point vary the lengths on the map. In Mercator or stereographic projections, scales vary by latitude, so bar scales by latitudes are often appended.
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The circle is birationally equivalent to the line. One birational map between them is stereographic projection, pictured here. In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles.
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The Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) coordinate systems both use a metric-based Cartesian grid laid out on a conformally projected surface to locate positions on the surface of the Earth. The UTM system is not a single map projection but a series of sixty, each covering 6-degree bands of longitude. The UPS system is used for the polar regions, which are not covered by the UTM system.
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De plana spera, geometric drawing This treatise of five propositions deals with various aspects of stereographic projection (used in planispheric astrolabes). The first and historically the most important proposition proves for all cases that circles on the surface of a sphere when projected stereographically on a plane remain circles (or a circle of infinite radius, i.e., a straight line). While this property was known long before Jordanus, it had never been proved.
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Researchers in structural geology use the Lambert azimuthal projection to plot crystallographic axes and faces, lineation and foliation in rocks, slickensides in faults, and other linear and planar features. In this context the projection is called the equal-area hemispherical projection. There is also an equal-angle hemispherical projection defined by stereographic projection. The discussion here has emphasized the lower hemisphere z ≤ 0, but some disciplines prefer the upper hemisphere z ≥ 0.
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The projection can be computed as an oblique aspect of the Peirce quincuncial projection by rotating the axis 45 degrees. It can also be computed by rotating the coordinates −45 degrees before computing the stereographic projection; this projection is then remapped into a square whose coordinates are then rotated 45 degrees. The projection is conformal except for the four corners of each hemisphere’s square. Like other conformal polygonal projections, the Guyou is a Schwarz–Christoffel mapping.
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In addition, they perform analog and numerical experiments of rock deformation in large and small settings. The analysis of structures is often accomplished by plotting the orientations of various features onto stereonets. A stereonet is a stereographic projection of a sphere onto a plane, in which planes are projected as lines and lines are projected as points. These can be used to find the locations of fold axes, relationships between faults, and relationships between other geologic structures.
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In the post-war years, these concepts were extended into the Universal Transverse Mercator/Universal Polar Stereographic (UTM/UPS) coordinate system, which is a global (or universal) system of grid-based maps. The transverse Mercator projection is a variant of the Mercator projection, which was originally developed by the Flemish geographer and cartographer Gerardus Mercator, in 1570. This projection is conformal, which means it preserves angles and therefore shapes across small regions. However, it distorts distance and area.
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Brown, Lloyd Arnold : The story of maps, p.59. François d'Aguilon gave the stereographic projection its current name in his 1613 work Opticorum libri sex philosophis juxta ac mathematicis utiles (Six Books of Optics, useful for philosophers and mathematicians alike).According to (Elkins, 1988) who references Eckert, "Die Kartenwissenschaft", Berlin 1921, pp 121-123 In 1695, Edmond Halley, motivated by his interest in star charts, published the first mathematical proof that this map is conformal.Timothy Feeman. 2002.
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The two models are related through a projection on or from the hemisphere model. The Klein model is an orthographic projection to the hemisphere model while the Poincaré disk model is a stereographic projection. When projecting the same lines in both models on one disk both lines go through the same two ideal points. (the ideal points remain on the same spot) also the pole of the chord is the centre of the circle that contains the arc.
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A stereographic projection of a Clifford torus performing a simple rotation Topologically a rectangle is the fundamental polygon of a torus, with opposite edges sewn together. In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S and S (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3.
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There is a material improvement of full color images, with the cyan filter, especially for accurate skin tones. Video games, theatrical films, and DVDs can be shown in the anaglyph 3D process. Practical images, for science or design, where depth perception is useful, include the presentation of full scale and microscopic stereographic images. Examples from NASA include Mars Rover imaging, and the solar investigation, called STEREO, which uses two orbital vehicles to obtain the 3D images of the sun.
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Hill/Beck "Sky Lens" (1923, GB 225,398) In 1924, Robin Hill first described a lens with 180° coverage that had been used for a cloud survey in September 1923 The lens, designed by Hill and R. & J. Beck, Ltd., was patented in December 1923. The Hill Sky Lens is now credited as the first fisheye lens. Hill also described three different mapping functions of a lens designed to capture an entire hemisphere (stereographic, equidistant, and orthographic).
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The H.C. White Company Mill Complex, is a historic industrial complex at 140 Water Street in North Bennington, Vermont. The White Company was founded in 1879, producing stereographic viewers and stereograph cards, as well as the Kiddie-Kar, a three-wheeled wooden scooter for children. These premises were occupied by the company from then until its closure in 1935. The complex, with buildings dating from 1887 to 1919, was listed on the National Register of Historic Places in 2017.
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Inversion in a sphere is a powerful transformation. One simple example is in map projection. The usual projection of the North or South Pole (stereographic projection) is an inversion from the Earth to a plane. If instead of making a pole the centre, we chose a city, then Inversion could produce a map where all the shortest routes (great circles) for flying from that city would appear as straight lines, which would simplify the flight path, for passengers at least.
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As in crystallography, planes are typically plotted by their poles. Unlike crystallography, the southern hemisphere is used instead of the northern one (because the geological features in question lie below the Earth's surface). In this context the stereographic projection is often referred to as the equal-angle lower-hemisphere projection. The equal-area lower-hemisphere projection defined by the Lambert azimuthal equal- area projection is also used, especially when the plot is to be subjected to subsequent statistical analysis such as density contouring.
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The Klein disk model (also known as the Beltrami–Klein model) and the Poincaré disk model are both models that project the whole hyperbolic plane in a disk. The two models are related through a projection on or from the hemisphere model. The Klein disk model is an orthographic projection to the hemisphere model while the Poincaré disk model is a stereographic projection. An advantage of the Klein disk model is that lines in this model are Euclidean straight chords.
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Astronomical dial Inside the large black outer circle lies another movable circle marked with the signs of the zodiac which indicates the location of the Sun on the ecliptic. The signs are shown in anticlockwise order. In the photograph accompanying this section, the Sun is currently moving anticlockwise from Cancer into Leo. The displacement of the zodiac circle results from the use of a stereographic projection of the ecliptic plane using the North pole as the basis of the projection.
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On the occasion of the Holy Year 1500, when many pilgrims were expected to go to Rome, he designed his famous "Rom-Weg" map (= the Way to Rome), a 41 x 29 cm wood engraving in stereographic projection to a scale of about 1:5,600,000. This is the earliest printed road map of central Europe. It is, as all of Etzlaub's maps, "south up". Distances between cities can be computed by dotted lines, where a one-dot-step means one German Mile (7400m).
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Orthographic projection (equatorial aspect) of eastern hemisphere 30°W-150°E An orthographic projection map is a map projection of cartography. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective (or azimuthal) projection, in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle.
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Sphaerae atque astrorum coelestium ratio, natura, et motus, 1536 Jordanus de Nemore (fl. 13th century), also known as Jordanus Nemorarius and Giordano of Nemi, was a thirteenth-century European mathematician and scientist. The literal translation of Jordanus de Nemore (Giordano of Nemi) would indicate that he was an Italian.Bertrand Gille, Les ingénieurs de la Renaissance. He wrote treatises on at least 6 different important mathematical subjects: the science of weights; “algorismi” treatises on practical arithmetic; pure arithmetic; algebra; geometry; and stereographic projection.
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A single 30-tetrahedron ring Boerdijk–Coxeter helix within the 600-cell, seen in stereographic projection Regular tetrahedra can be stacked face-to-face in a chiral aperiodic chain called the Boerdijk–Coxeter helix. In four dimensions, all the convex regular 4-polytopes with tetrahedral cells (the 5-cell, 16-cell and 600-cell) can be constructed as tilings of the 3-sphere by these chains, which become periodic in the three-dimensional space of the 4-polytope's boundary surface.
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In Aguilon's book there are elements of perspectivities as well as the stereographic projections of Ptolemy and Hipparchus. Unaware that Johannes Kepler had already published optical theories years before him, Aguilon decided to share his insights on geometric optics. At the age of 20, the Dutch poet Constantijn Huygens read Aguilon's and was enthralled by it. He later said that it was the best book he had ever read in geometrical optics, and he thought that Aguilon should be compared to Plato, Eudoxus and Archimedes.
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The one- point compactification of the plane is homeomorphic to a sphere (see stereographic projection); the open disk is homeomorphic to a sphere with the "north pole" missing; adding that point completes the (compact) sphere. The result of this compactification is a manifold referred to as the Riemann sphere or the complex projective line. The projection from the Euclidean plane to a sphere without a point is a diffeomorphism and even a conformal map. The plane itself is homeomorphic (and diffeomorphic) to an open disk.
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A contour chart of scale factors of GS50 projection Maps reflecting directions, such as a nautical chart or an aeronautical chart, are projected by conformal projections. Maps treating values whose gradients are important, such as a weather map with atmospheric pressure, are also projected by conformal projections. Small scale maps have large scale variations in a conformal projection, so recent world maps use other projections. Historically, many world maps are drawn by conformal projections, such as Mercator maps or hemisphere maps by stereographic projection.
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The 120-cell consists of 120 dodecahedral cells. For visualization purposes, it is convenient that the dodecahedron has opposing parallel faces (a trait it shares with the cells of the tesseract and the 24-cell). One can stack dodecahedrons face to face in a straight line bent in the 4th direction into a great circle with a circumference of 10 cells. Starting from this initial ten cell construct there are two common visualizations one can use: a layered stereographic projection, and a structure of intertwining rings.
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In order to cover the unit sphere, one needs the two stereographic projections: the first will cover the whole sphere except the point and the second except the point . Hence, one needs two complex planes, one for each projection, which can be intuitively seen as glued back- to-back at . Note that the two complex planes are identified differently with the plane . An orientation-reversal is necessary to maintain consistent orientation on the sphere, and in particular complex conjugation causes the transition maps to be holomorphic.
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Here the polynomial z2 − 1 vanishes when z = ±1, so g evidently has two branch points. We can "cut" the plane along the real axis, from −1 to 1, and obtain a sheet on which g(z) is a single-valued function. Alternatively, the cut can run from z = 1 along the positive real axis through the point at infinity, then continue "up" the negative real axis to the other branch point, z = −1\. This situation is most easily visualized by using the stereographic projection described above.
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By varying the relative amplitude of the signal sent to each speaker an artificial direction (relative to the listener) can be suggested. The control which is used to vary this relative amplitude of the signal is known as a "pan-pot" (panoramic potentiometer). By combining multiple "pan-potted" mono signals together, a complete, yet entirely artificial, sound field can be created. In technical usage, true stereo means sound recording and sound reproduction that uses stereographic projection to encode the relative positions of objects and events recorded.
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It is unlikely Nicolosi knew of al-Biruni's work, and Nicolosi's name is the one usually associated with the projection. Nicolosi published a set of maps on the projection, one of the world in two hemispheres, and one each for the five known continents. Maps using the same projection appeared occasionally over the ensuring centuries, becoming relatively common in the 19th century as the stereographic projection fell out of common use for this purpose. Use of the Nicolosi projection continued into the early 20th century.
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P.G. Gormley (1947) "Stereographic projection and the linear fractional group of transformations of quaternions", Proceedings of the Royal Irish Academy, Section A 51:67–85 In 1968 Isaak Yaglom's Complex Numbers in Geometry appeared in English, translated from Russian. There he uses P(D) to describe line geometry in the Euclidean plane and P(M) to describe it for Lobachevski's plane. Yaglom's text A Simple Non-Euclidean Geometry appeared in English in 1979. There in pages 174 to 200 he develops Minkowskian geometry and describes P(M) as the "inversive Minkowski plane".
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3D Smith chart representation A generalized 3D Smith chart based on the extended complex plane (Riemann sphere) and inversive geometry was proposed in 2011. The chart unifies the passive and active circuit design on little and big circles on the surface of a unit sphere using the stereographic conformal map of the reflection coefficient's generalized plane. Considering the point at infinity, the space of the new chart includes all possible loads. The north pole is the perfect matching point, while the south pole is the perfect mismatch point.
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The size of the separators it produces can be further improved, in practice, by using a nonuniform distribution for the random cutting planes.. The stereographic projection in the Miller et al. argument can be avoided by considering the smallest circle containing a constant fraction of the centers of the disks and then expanding it by a constant picked uniformly in the range [1,2]. It is easy to argue, as in Miller et al., that the disks intersecting the expanded circle form a valid separator, and that, in expectation, the separator is of the right size.
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In 1872, Bell joined George Wheeler's survey expedition, which was tasked with surveying American lands west of the 100th meridian, as a replacement for photographer Timothy H. O'Sullivan. As part of the expedition, he captured numerous large format and stereographic landscapes of relatively unexplored areas of the Colorado River basin in Utah and Arizona. While on the expedition, he experimented with the dry plate process, for which he would eventually become an expert. After the expedition, Bell returned to his studio in Philadelphia, and exhibited his work at the city's 1876 Centennial Exposition.
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A consequence of the hairy ball theorem is that any continuous function that maps an even-dimensional sphere into itself has either a fixed point or a point that maps onto its own antipodal point. This can be seen by transforming the function into a tangential vector field as follows. Let s be the function mapping the sphere to itself, and let v be the tangential vector function to be constructed. For each point p, construct the stereographic projection of s(p) with p as the point of tangency.
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The Samyang 8mm F3.5 UMC Fish-Eye CS II is a fisheye photographic lens using the stereographic projectionephotozine and is designed for crop factor APS-C DSLRs. It is made in South Korea by Samyang Optics and marketed under several brand names besides Samyang, including Bower, Falcon, Polar, Pro-Optic, Rokinon, Vivitar and Walimex Pro (Walser GmbH & Co. KG). There are versions for the Canon EF, Fujifilm X, Nikon F, MFT, Pentax K, Samsung NX, Sony E, Sony α/Minolta A mounts. The lens uses manual focus only.
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The identity component of the Lorentz group is isomorphic to the Möbius group . This group can be thought of as conformal mappings of either the complex plane or, via stereographic projection, the Riemann sphere. In this way, the Lorentz group itself can be thought of as acting conformally on the complex plane or on the Riemann sphere. In the plane, a Möbius transformation characterized by the complex numbers acts on the plane according to and can be represented by complex matrices since multiplication by a nonzero complex scalar does not change .
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His new quadrant was based upon the idea that the stereographic projection that defines a planispheric astrolabe can still work if the astrolabe parts are folded into a single quadrant. The result was a device that was far cheaper, easier to use and more portable than a standard astrolabe. Tibbon's work had a far reach and influenced Copernicus, Christopher Clavius and Erasmus Reinhold; and his manuscript was referenced in Dante's Divine Comedy. As the quadrant became smaller and thus more portable, its value for navigation was soon realized.
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Now, Younghusband encouraged Mason to follow in his footsteps. Through the 1920s, the interest of the British authorities had grown in the unmapped and uninhabited territories of the Shaksgam Valley because it provided access to the Aghil Pass linking China to Ladakh, India. Mason began a survey using a photo-theodolite and stereographic techniques, laboriously collecting great quantities of data. His results, plotted in Switzerland using what, at the time, was the world's most advanced Stereoplotter, were acclaimed as brilliantly successful and won him the award of the 1927 Royal Geographical Society's Founder's Gold Medal.
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The Red Room; Pedrocchi Café (signs on glass partitions), obtained 10/21/2015 The walls opposite the windows you can find the paintings of the two hemispheres of the globe in stereographic projection, with the north at the bottom and using French nomenclature. It was first called the black room as the furniture designed by Jappelli was painted black. Green Room: The Green Room is similar in size and decoration to the parallel white room, except for the color of the tapestry that is green, and has a mirror over the fireplace.
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The isomorphisms are all conformal bijections of the complex plane, but the only possibilities are maps that correspond to the composition of a multiplication by a complex number and a translation. In addition, the Euclidean geometry (which has zero curvature everywhere) is not the only geometry that the plane may have. The plane may be given a spherical geometry by using the stereographic projection. This can be thought of as placing a sphere on the plane (just like a ball on the floor), removing the top point, and projecting the sphere onto the plane from this point).
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Herman of Carinthia, translator of Planisphaerium, with an astrolabe Originally written in Ancient Greek, Planisphaerium was one of many scientific works which survived from antiquity in Arabic translation. One reason why Planisphaerium attracted interest was that stereographic projection was the mathematical basis of the plane astrolabe, an instrument which was widely used in the medieval Islamic world. In the 12th century the work was translated from Arabic into Latin by Herman of Carinthia, who also translated commentaries by Maslamah Ibn Ahmad al-Majriti. The oldest known translation is in Arabic done by an unknown scholar as part of the Translation Movement in Baghdad.
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Use of lower hemisphere stereographic projection to plot planar and linear data in structural geology, using the example of a fault plane with a slickenside lineation Researchers in structural geology are concerned with the orientations of planes and lines for a number of reasons. The foliation of a rock is a planar feature that often contains a linear feature called lineation. Similarly, a fault plane is a planar feature that may contain linear features such as slickensides. These orientations of lines and planes at various scales can be plotted using the methods of the Visualization of lines and planes section above.
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In July 1862, he made his first trip to Niagara Falls, New York, where he found a job working for Platt D. Babbitt. By the late 1860s, he had studios in both London and Niagara Falls, with the Niagara studio called Barker's Stereoscopic View Manufactory and Photograph Rooms, and had become known nationwide for his large-format (up to ) and stereographic prints of the falls. In 1866, he won a gold medal for landscape photography at the convention for the Photographers Association of America, held in Saint Louis. Barker's Niagara studio was destroyed by fire on February 7, 1870, but his negatives survived.
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She undertook intensive tutoring in mathematics and enrolled at MIT in 1901 as a special student, one of four women enrolled on the architecture course and one of two studying landscape architecture. The four women on the course were the only female members of a 500-strong student body. Stereographic card showing a mechanical drafting studio at MIT, where Coffin studied from 1901–4 Coffin took the full range of architectural courses including studying engineering, physics, maths, mechanical drafting and freehand drawing in addition to architectural and landscape design. She also studied botany and horticulture under Charles Sprague Sargent at the Arnold Arboretum.
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The VistaScreen Co Ltd was a stereographic photography outfit launched in the late 1950s by Jack & Jeff Spring, who, at the time, owned a paper merchanting company called Capital Paper Company, and Stanley Long, a former RAF photographer. Long shot the vast majority of the stereo images, mostly using a 1920s Franke & Heidecke Heidoscop stereo camera with a 6x13 cm plate back. The VistaScreen viewers were manufactured in ivory-colored plastic and were designed to fold flat in order to be able to be compactly stored. The viewers were priced at 1/6d (around 7.5p in today's terms).
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In Riemannian geometry, two Riemannian metrics g and h on a smooth manifold M are called conformally equivalent if g = u h for some positive function u on M. The function u is called the conformal factor. A diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map. One can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics.
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One stronger form of the circle packing theorem, on representing planar graphs by systems of tangent circles, states that every polyhedral graph can be represented by a polyhedron with a midsphere. The horizon circles of a canonical polyhedron can be transformed, by stereographic projection, into a collection of circles in the Euclidean plane that do not cross each other and are tangent to each other exactly when the vertices they correspond to are adjacent.; . Schramm states that the existence of an equivalent polyhedron with a midsphere was claimed by , but that Koebe only proved this result for polyhedra with triangular faces.
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In 1904, he published a second book, Sod Houses; or, The Development of the Great American Plains, at the urging of a lawyer who hoped to use Butcher's photographs and accounts to sell land in Nebraska. In 1909, he visited Yellowstone National Park and produced a set of 100 stereographic postcards. Butcher abandoned the history of Buffalo and Dawson counties after spending more than a thousand dollars on the project. Discontented with his profession as photographer, which had failed to make him a fortune or even to put him on a sound financial footing, he turned his efforts elsewhere.
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Where traditionally, this has been a largely black & white format, recent digital camera and processing advances have brought very acceptable color images to the internet and DVD field. With the online availability of low cost paper glasses with improved red-cyan filters, and plastic framed glasses of increasing quality, the field of 3D imaging is growing quickly. Scientific images where depth perception is useful include, for instance, the presentation of complex multi-dimensional data sets and stereographic images of the surface of Mars. With the recent release of 3D DVDs, they are more commonly being used for entertainment.
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The HTC Evo 3D is a 3D-enabled Android smartphone developed by HTC, released exclusively in the United States through Sprint, and was re-released as a pre- paid smartphone by Virgin Mobile in May 2012 as the HTC Evo V 4G. A variation of Sprint's flagship HTC Evo 4G, the device is distinguished by its pair of 5 MP rear cameras, which can be used to take photos or video in stereographic 3D, which can be viewed on its autostereoscopic display without the need for 3D glasses. Several GSM variants are also available in Canada, Europe and Asia.
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The circles of Apollonius are any of several sets of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for example, counterparts on the surface of a sphere can be defined through stereographic projection. The main uses of this term are fivefold: # Apollonius showed that a circle can be defined as the set of points in a plane that have a specified ratio of distances to two fixed points, known as foci. This Apollonian circle is the basis of the Apollonius pursuit problem.
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The lines of latitude are all parallel to the equator, so they will become perfect circles centered on the origin . And the lines of longitude will become straight lines passing through the origin (and also through the "point at infinity", since they pass through both the north and south poles on the sphere). This is not the only possible yet plausible stereographic situation of the projection of a sphere onto a plane consisting of two or more values. For instance, the north pole of the sphere might be placed on top of the origin in a plane that is tangent to the circle.
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Polar-stereographic projection showing 12 hours of measurements from three AMSU-B instruments The satellites that most concern us are those with a low-Earth, polar orbit since geostationary satellites view the same point throughout their lifetime. The diagram shows measurements from AMSU-B instruments mounted on three satellites over a period of 12 hours. This illustrates both the orbit path and the scan pattern which runs crosswise. Since the orbit of a satellite is deterministic, barring orbit maneuvers, we can predict the location of the satellite at a given time and, by extension, the location of the measurement pixels.
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He wrote treatises on mechanics ("the science of weights"), on basic and advanced arithmetic, on algebra, on geometry, and on the mathematics of stereographic projection. Villard de Honnecourt (fl. 13th century), a French engineer and architect who made sketches of mechanical devices such as automatons and perhaps drew a picture of an early escapement mechanism for clockworks. Roger BaconRoger Bacon (1214–94), Doctor Admirabilis, joined the Franciscan Order around 1240 where, influenced by Grosseteste, Alhacen and others, he dedicated himself to studies where he implemented the observation of nature and experimentation as the foundation of natural knowledge.
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Rotating model of the diamond cubic crystal structure 3D ball-and-stick model of a diamond lattice Pole figure in stereographic projection of the diamond lattice showing the 3-fold symmetry along the [111] direction. The diamond cubic crystal structure is a repeating pattern of 8 atoms that certain materials may adopt as they solidify. While the first known example was diamond, other elements in group 14 also adopt this structure, including α-tin, the semiconductors silicon and germanium, and silicon/germanium alloys in any proportion. Although often called the diamond lattice, this structure is not a lattice in the technical sense of this word used in mathematics.
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When M.A.X. starts telling jokes, the Professor tells him to explain the difference between 2-D and 3-D, which he does, explaining that 3-D has depth to it unlike 2-D. After pictures from the stereographic archives are shown, the Professor tries his demonstration again, but this time the machines turn Elvira into cardboard. As he tries to fix this problem, he tells the audience about 3-D movies, and several clips are shown, particularly clips where objects or people are thrown at the camera. Afterward the Professor opens up the Diorama of 3-D videos, such as Dino Island, T-2 3-D and the Abandoned Mine.
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Hipparchus was the first to show that the stereographic projection is conformal, and that it transforms circles on the sphere that do not pass through the center of projection to circles on the plane. This was the basis for the astrolabe. Besides geometry, Hipparchus also used arithmetic techniques developed by the Chaldeans. He was one of the first Greek mathematicians to do this, and in this way expanded the techniques available to astronomers and geographers. There are several indications that Hipparchus knew spherical trigonometry, but the first surviving text discussing it is by Menelaus of Alexandria in the 1st century, who on that basis is now commonly credited with its discovery.
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An astrolabe consists of a disk, called the mater (mother), which is deep enough to hold one or more flat plates called tympans, or climates. A tympan is made for a specific latitude and is engraved with a stereographic projection of circles denoting azimuth and altitude and representing the portion of the celestial sphere above the local horizon. The rim of the mater is typically graduated into hours of time, degrees of arc, or both. Above the mater and tympan, the rete, a framework bearing a projection of the ecliptic plane and several pointers indicating the positions of the brightest stars, is free to rotate.
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The computational conversion of the ion sequence data, as obtained from a position sensitive detector, to a three-dimensional visualisation of atomic types, is termed "reconstruction". Reconstruction algorithms are typically geometrically based, and have several literature formulations. Most models for reconstruction assume that the tip is a spherical object, and use empirical corrections to stereographic projection to convert detector positions back to a 2D surface embedded in 3D space, R3. By sweeping this surface through R3 as a function of the ion sequence input data, such as via ion-ordering, a volume is generated onto which positions the 2D detector positions can be computed and placed three-dimensional space.
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The torus plays a central role in the Hopf fibration of the 3-sphere, S3, over the ordinary sphere, S2, which has circles, S1, as fibers. When the 3-sphere is mapped to Euclidean 3-space by stereographic projection, the inverse image of a circle of latitude on S2 under the fiber map is a torus, and the fibers themselves are Villarceau circles. Banchoff (1990) has explored such a torus with computer graphics imagery. One of the unusual facts about the circles is that each links through all the others, not just in its own torus but in the collection filling all of space; Berger (1987) has a discussion and drawing.
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Stereographic photograph (1903), captioned "Famous 'man- eater' at Calcutta—devoured 200 men, women and children before capture—India" Wild tigers that have had no prior contact with humans actively avoid interactions with humans. However, tigers cause more human deaths through direct attack than any other wild mammal. Attacks are occasionally provoked, as tigers lash out after being injured while they themselves are hunted. Attacks can be provoked accidentally, as when a human surprises a tiger or inadvertently comes between a mother and her young, or as in a case in rural India when a postman startled a tiger, used to seeing him on foot, by riding a bicycle.
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Clearly this procedure is reversible – given any point on the surface of the sphere that is not the north pole, we can draw a straight line connecting that point to the north pole and intersecting the flat plane in exactly one point. Under this stereographic projection the north pole itself is not associated with any point in the complex plane. We perfect the one-to-one correspondence by adding one more point to the complex plane – the so-called point at infinity – and identifying it with the north pole on the sphere. This topological space, the complex plane plus the point at infinity, is known as the extended complex plane.
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The angle-preserving effect of stereographic plots is even more obvious in the figure at right, which subtends a full 180° of the orientation space of a face-centered or cubic close packed crystal e.g. like that of Gold or Aluminum. The animation follows {220} fringe-visibility bands of that face-centered cubic crystal between <111> zones, at which point rotation by 60° sets up travel to the next <111> zone via a repeat of the original sequence. Fringe-visibility bands have the same global geometry as do Kikuchi bands, but for thin specimens their width is proportional (rather than inversely proportional) to d-spacing.
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She was the first photographer to take stereographic images of Malmö. She is not known to be active after 1870. She left Malmö in 1884, and moved to Hörby with her likewise unmarried sister in 1910. Hilda Sjölin belonged to the pioneer generation of female professional photographers in Sweden after Brita Sofia Hesselius: the same time as she became active, Hedvig Söderström in Stockholm (1857), Emma Schenson in Uppsala and Wilhelmina Lagerholm in Örebro (1862), among others, became the first professional photographers of their respective cities: during the 1860s, they were at least 15 confirmed female photographers in Sweden, three of whom, Rosalie Sjöman, Caroline von Knorring and Bertha Valerius belonging to the elite of their profession.
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Since leaving ILM, Berton has been digital effects supervisor on I, Robot (2004) and visual effects supervisor on the 2006 film Charlotte's Web. Producer Jordan Kerner explains that Berton was hired for Charlotte's Web because the filmmakers needed someone "who had been involved with field that had a tremendous amount of fully 3-D, computer-generated characters who have to convey thoughts and emotions." The film's visual effects were produced by five different companies, but it was Berton, according to Kerner, who "drew those elements together." After Charlotte's Web Berton worked with Director/Screenwriter David Goyer on a proposed stereographic thriller for Walt Disney Pictures, followed by supervising the visual effects on Bedtime Stories, also for Walt Disney Pictures.
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Since the edge of the plate represents the effective horizon, its centre identifies the pilot's nadir. Mounted behind the plate is a star-planisphere, based on a north pole stereographic zenithal projection (a projection from the north pole onto a plane passing through the south pole and perpendicular to the solar axis). This rotates once in a sidereal day on an axis passing through its south celestial pole and located some 13 cm above the centre of the horizon plate. For decoration it carries a few basic star patterns (considerably distorted owing to the projection used) and an eccentric zodiac/ecliptic/calendar ring faced with silver, and restricted in width to the distance between the solstitial points.
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Tesseract, in stereographic projection, in double rotation A 4D Clifford torus stereographically projected into 3D looks like a torus, and a double rotation can be seen as in helical path on that torus. For a rotation whose two rotation angles form a rational number, the paths will eventually reconnect, while for an irrational ratio they will not. An isoclinic rotation will form a Villarceau circle on the torus, while a simple rotation will form a circle parallel or perpendicular to the central axis. For each rotation of 4-space (fixing the origin), there is at least one pair of orthogonal 2-planes and each of which is invariant and whose direct sum is all of 4-space.
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The five codes together constitute a way of interpreting the text which suggests that textuality is interpretive; that the codes are not superimposed upon the text, but rather approximate something intrinsic to the text. The analogy Barthes uses to clarify the relationship of codes to text is to the relationship between a performance and the commentary that can be heard off-stage. In the “stereographic space” created by the codes, each code becomes associated with a voice. To the proairetic code Barthes assigns the Voice of Empirics; to the semic the Voice of the Person; to the cultural the Voice of Science; to the hermeneutic the Voice of Truth; and to the symbolic the Voice of Symbol.
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Lambina DFN Station: a typical outback fireball observatory (with some unrelated equipment in the background) The DFN observatories use consumer still photographic cameras (specifically DSLRs) with 8mm stereographic fish- eye lenses covering nearly the entire sky from each station. The cameras are controlled via an embedded Linux PC using gPhoto2 and images are archived to multiple hard disk drives for storage until the observatories are visited for maintenance (every 8–18 months depending on the storage capacity). The observatories take one long exposure image every 30 seconds for the entire night. After capture, automated event detection searches the images for fireballs, and events are corroborated on the central server using images from multiple stations.
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U. In the two days following the unveiling of the Wii U, Nintendo's stock fell nearly 10% to levels not seen since 2006. Some analysts expressed skepticism in regards to the addition of a touch-screen, expressing concern that the controller would be less affordable and less innovative than the original Wii Remote. When asked about whether or not the Wii U was going to support stereographic 3D via 3D televisions, Iwata stated that it was "not the area we are focusing on." On January 26, 2012, Iwata announced that the Wii U would be launched by the end of the 2012 shopping season in all major regions, and that its final specifications would be revealed at E3 2012.
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Among Hayden's party were Jackson, Moran, geologist George Allen, mineralogist Albert Peale, topographical artist Henry Elliot, botanists, and other scientists who collected numerous wildlife specimens and other natural data. William Henry Jackson, as a member of the U. S. Geological Survey exploring the Teton country in 1872 Jackson worked in multiple camera and plate sizes, under conditions that were often incredibly difficult. His photography was based on the collodion process invented in 1848 and published in 1851 by Frederick Scott Archer. Jackson traveled with as many as three camera-types—a stereographic camera (for stereoscope cards), a "whole-plate" or 8x10" plate- size camera, and one even larger, as large as 18x22".
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The two grids covering the Arctic and Antarctic The universal polar stereographic (UPS) coordinate system is used in conjunction with the universal transverse Mercator (UTM) coordinate system to locate positions on the surface of the earth. Like the UTM coordinate system, the UPS coordinate system uses a metric-based cartesian grid laid out on a conformally projected surface. UPS covers the Earth's polar regions, specifically the areas north of 84°N and south of 80°S, which are not covered by the UTM grids, plus an additional 30 minutes of latitude extending into UTM grid to provide some overlap between the two systems. In the polar regions, directions can become complicated, with all geographic north–south lines converging at the poles.
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By the 1980s, centralized printing of standardized topographic maps began to be superseded by databases of coordinates that could be used on computers by moderately skilled end users to view or print maps with arbitrary contents, coverage and scale. For example, the Federal government of the United States' TIGER initiative compiled interlinked databases of federal, state and local political borders and census enumeration areas, and of roadways, railroads, and water features with support for locating street addresses within street segments. TIGER was developed in the 1980s and used in the 1990 and subsequent decennial censuses. Digital elevation models (DEM) were also compiled, initially from topographic maps and stereographic interpretation of aerial photographs and then from satellite photography and radar data.
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Until recently, the two major government mapping authorities in Romania have been the Military Topographic Department (Directia Topografica Militara (DTM)), and the Institute for Geodesy, Photogrammetry, Cartography and Land Management (Institutul de Geodezie, Fotogrammetrie, Cartografie, si Organizarea Teritoriului (IGFCOT)). This situation has recently changed, following a decision in 1996 by the Romanian Government to establish a combined civilian National Office of Cadastre, Geodesy and Cartography (Oficiul National de Cadastru, Geodezie si Cartografie (ONCGC). Maps continued to be published under the imprint of the previous organizations into the late 1990s. From 1958, a number of town maps at scales of 1:5,000 or 1:10,000 were also made, initially on the Gauss-Krϋger projection, but after 1970 on a stereographic projection.
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George Coates (born March 19, 1952) is an American theatre director most notable for his work with George Coates Performance Works (GCPW), which he founded in 1977 in San Francisco, CA. The company produced over 20 multi-media live performances over a span of 25 years, winning a multitude of awards for its international performances, earning critical acclaim in Asia, Europe and South America and gaining North American attention at Brooklyn Academy of Music's Next Wave Festival. In the 1990s, he was the first to merge live performers within stage environments created by computer generated graphics in real time live theatre. Coates became known as a pioneer of experimental live theatre using stereographic projections and 3-D glasses populated by live actors and musicians.
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The related truncated icosidodecahedral prism is constructed from two truncated icosidodecahedra connected by prisms, shown here in stereographic projection with some prisms hidden. The spherinder is related to the uniform prismatic polychora, which are cartesian product of a regular or semiregular polyhedron and a line segment. There are eighteen convex uniform prisms based on the Platonic and Archimedean solids (tetrahedral prism, truncated tetrahedral prism, cubic prism, cuboctahedral prism, octahedral prism, rhombicuboctahedral prism, truncated cubic prism, truncated octahedral prism, truncated cuboctahedral prism, snub cubic prism, dodecahedral prism, icosidodecahedral prism, icosahedral prism, truncated dodecahedral prism, rhombicosidodecahedral prism, truncated icosahedral prism, truncated icosidodecahedral prism, snub dodecahedral prism), plus an infinite family based on antiprisms, and another infinite family of uniform duoprisms, which are products of two regular polygons.
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In 1909, Mason sailed for Karachi and was posted to the Survey of India. 1910-1912 saw him engaged on triangulation in Kashmir, where he learned climbing techniques, taught himself to ski and went on to make a stereographic land survey. RGS Founders Medal 250px Mason returned to India after the First World War and began preparing for his most important scientific project, the exploration of the Shaksgam Valley, in 1926. At that time the only westerner to see the valley had been Francis Younghusband, whose book The heart of a continent : a narrative of travels in Manchuria, across the Gobi Desert, through the Himalayas, the Pamirs, and Chitral, 1884-1894 had first inspired Mason as a schoolboy to pursue a career in geography.
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Part of the engraving on the back-side de Maricourt's universal astrolabe The Nova Compositio Astrolabii Particularis (found in only 4 manuscripts) describes the construction and use a universal astrolabe which could be used at a variety of latitudes without changing the plates. Unlike al-Zarqālī’s more famous universal astrolabe in which vertical halves the heavens were projected onto a plane through the poles, this one had both the northern and southern hemispheres projected onto a plane through the equator (which was also the limit of projection). There are no known surviving astrolabes based on this treatise. The use of such an astrolabe is very complicated, and since it is probable that most sophisticated users were not frequent travelers, they were more likely happier with the traditional (and simpler) stereographic planispheric astrolabe.
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Stereographic card showing an MIT mechanical drafting studio, 19th century (photo by E.L. Allen, left/right inverted) Rogers Building, Back Bay, Boston, 19th century In 1859, a proposal was submitted to the Massachusetts General Court to use newly filled lands in Back Bay, Boston for a "Conservatory of Art and Science", but the proposal failed. A charter for the incorporation of the Massachusetts Institute of Technology, proposed by William Barton Rogers, was signed by John Albion Andrew, the governor of Massachusetts, on April 10, 1861. Rogers, a professor from the University of Virginia, wanted to establish an institution to address rapid scientific and technological advances. He did not wish to found a professional school, but a combination with elements of both professional and liberal education,Lewis 1949, p. 8.
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In an attempt to understand the origin of the Isua Greenstone Belt, scientists have used several different methods. These include enlisting U-Pb zircon dating that measures the decay of uranium to lead in zircons using sensitive high- resolution ion microprobe (SHRIMP), analysing elemental chemistry and composition, rendering three-dimensional features on paper using the stereographic projections that geologists call "stereonets", and assessing lithologic associations. In addition to information gathered directly from the rocks, scientists have also used observations of the placement of the rocks and how they are separated into units: this is a more kinematic approach to the area. In addition, zoned garnets from different areas of the Isua Greenstone Belt have been used in garnet-biotite geothermometry, which has been used to determine the timing of metamorphism.
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Distortion is unavoidable in a lens that encompasses an angle of view exceeding 125°, but Hill and Beck claimed in the patent that stereographic or equidistant projection were the preferred mapping functions. The three-element, three- group lens design uses a highly divergent meniscus lens as the first element to bring in light over a wide view followed by a converging lens system to project the view onto a flat photographic plate. The Hill Sky Lens was fitted to a whole sky camera, typically used in a pair separated by for stereo imaging, and equipped with a red filter for contrast; in its original form, the lens had a focal length of and cast an image in diameter at 8. Conrad Beck described the camera system in an article published in 1925.
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A "little astrolabe", or "plane astrolabe", is a kind of astrolabe that used stereographic projection of the celestial sphere to represent the heavens on a plane surface, as opposed to an armillary sphere, which was globe-shaped. Armillary spheres were large and normally used for display, whereas a plane astrolabe was portable and could be used for practical measurements. The statement from Synesius's letter has sometimes been wrongly interpreted to mean that Hypatia invented the plane astrolabe herself, but the plane astrolabe is known to have been in use at least 500 years before Hypatia was born. Hypatia may have learned how to construct a plane astrolabe from her father Theon, who had written two treatises on astrolabes: one entitled Memoirs on the Little Astrolabe and another study on the armillary sphere in Ptolemy's Almagest.
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The plane of dual numbers has a projective line including a line of points .Corrado Segre (1912) "Le geometrie proiettive nei campi di numeri duali", Paper XL of Opere, also Atti della R. Academia della Scienze di Torino, vol XLVII. Isaak Yaglom has described it as an "inversive Galilean plane" that has the topology of a cylinder when the supplementary line is included.Isaak Yaglom (1979) A Simple Non-Euclidean Geometry and its Physical Basis, Springer, , Similarly, if A is a local ring, then P(A) is formed by adjoining points corresponding to the elements of the maximal ideal of A. The projective line over the ring M of split-complex numbers introduces auxiliary lines and Using stereographic projection the plane of split-complex numbers is closed up with these lines to a hyperboloid of one sheet.
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Near the end of World War II, the Universal Transverse Mercator (UTM) coordinate system extended this grid concept around the globe, dividing it into 60 zones of 6 degrees longitude each. Circa 1949, the US further refined UTM for ease of use (and combined it with the Universal Polar Stereographic system covering polar areas) to create the Military Grid Reference System (MGRS), which remains the geocoordinate standard used across the militaries of NATO counties. In the 1990s, a US grass-roots citizen effort led to the Public X-Y Mapping Project, a not-for- profit organization created specifically to promote the acceptance of a national grid for the United States. The Public XY Mapping Project developed the idea, conducting informal tests and surveys to determine which coordinate reference system best met the requirements of national consistency and ease of human use.
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The year is usually represented by the 12 signs of the zodiac, arranged either as a concentric circle inside the 24-hour dial, or drawn onto a displaced smaller circle, which is a projection of the ecliptic, the path of the sun and planets through the sky, and the plane of the Earth's orbit. The ecliptic plane is projected onto the face of the clock, and, because of the Earth's tilted angle of rotation relative to its orbital plane, it is displaced from the center and appears to be distorted. The projection point for the stereographic projection is the North pole; on astrolabes the South pole is more common. The ecliptic dial makes one complete revolution in 23 hours 56 minutes (a sidereal day), and will therefore gradually get out of phase with the hour hand, drifting slowly further apart during the year.
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Portrait of a Siamese woman, c. 1861\. Albumen silver print. In 1861, Rossier was in Siam, where he assisted the French zoologist Firmin Bocourt by taking ethnographic portraits for the latter's scientific expedition of 1861–1862, and in 1863, Negretti and Zambra issued a series of 30 stereographic portraits and landscapes taken in Siam that are almost certainly the work of Rossier. In February 1862, Rossier was again in Shanghai, where he sold his cameras and other photographic equipment before embarking for Europe.The equipment listed in the advertisement include: a patent mahogany folding camera, a Ross portrait lens, and a Ross landscape lens – all in a portable case, a portable mahogany tripod, a travelling case "with all the necessary apparatus," a large fresh supply of chemicals ("just received from London"), and two practical works on photography (Bennett PiJ, 49).
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The typical tourist generally did not carry a camera and much of the Notman studio's images were taken with the tourist's needs in mind. Visitors would look through Notman's picture books and chose views, to buy individually mounted or perhaps made up into an album, and have a portrait taken as well. Street scenes in the burgeoning cities of Canada, the magnificence of modern transportation by rail and steam, expansive landscapes and the natural wonders, were all in demand either as 8" x 10" print, or in the popular stereographic form, and were duly recorded by the many staff photographers working for the Notman studio. He was a regular contributor to the photographic journal Philadelphia Photographer and in partnership with its editor, Edward Wilson, formed the Centennial Photographic Company for the Centennial Exhibition in Philadelphia, held in honour of the 100th anniversary of the United States in 1876.
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Gunter's quadrant is an instrument made of wood, brass or other substance, containing a kind of stereographic projection of the sphere on the plane of the equinoctial, the eye being supposed to be placed in one of the poles, so that the tropic, ecliptic, and horizon form the arcs of circles, but the hour circles are other curves, drawn by means of several altitudes of the sun for some particular latitude every year. This instrument is used to find the hour of the day, the sun's azimuth, etc., and other common problems of the sphere or globe, and also to take the altitude of an object in degrees. A rare Gunter quadrant, made by Henry Sutton and dated 1657, can be described as follows: It is a conveniently sized and high-performance instrument that has two pin-hole sights, and the plumb line is inserted at the vertex.
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In geology, especially in the study of glacial till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of clasts in a soil sample, which can only be compared graphically such as in a Tri-Plot (Sneed and Folk) diagram,Graham, D., and Midgley, N., 2000. Earth Surface Processes and Landforms (25) pp 1473-1477 or as a stereographic projection. The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. Eigenvectors output from programs such as Stereo32 Stereo32 are in the order E1 > E2 > E3, with E1 being the primary orientation of clast orientation/dip, E2 being the secondary and E3 being the tertiary, in terms of strength.
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In 1611, he started a special school of mathematics in Antwerp, fulfilling a dream of Christopher Clavius for a Jesuit mathematical school; in 1616, he was joined there by Grégoire de Saint- Vincent.. The notable geometers educated at this school included Jean-Charles della Faille, André Tacquet, and Theodorus Moretus. Illustration by Rubens for Opticorum Libri Sex demonstrating how the projection is computed. His book, Opticorum Libri Sex philosophis juxta ac mathematicis utiles, or Six Books of Optics, is useful for philosophers and mathematicians. It was published by Balthasar I Moretus in Antwerp in 1613 and illustrated by the famous painter Peter Paul Rubens.. It included one of the first studies of binocular vision... It also gave the names we now use to stereographic projection and orthographic projection, although the projections themselves were likely known to Hipparchus.... This book inspired the works of Desargues and Christiaan Huygens.. He died in Antwerp, aged 50.. Footnote 41, p. 38.
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According to the Koebe–Andreev–Thurston circle-packing theorem, any planar graph may be represented by a packing of circular disks in the plane with disjoint interiors, such that two vertices in the graph are adjacent if and only if the corresponding pair of disks are mutually tangent. As show, for such a packing, there exists a circle that has at most 3n/4 disks touching or inside it, at most 3n/4 disks touching or outside it, and that crosses O(√n) disks. To prove this, Miller et al. use stereographic projection to map the packing onto the surface of a unit sphere in three dimensions. By choosing the projection carefully, the center of the sphere can be made into a centerpoint of the disk centers on its surface, so that any plane through the center of the sphere partitions it into two halfspaces that each contain or cross at most 3n/4 of the disks.
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When Robertson died in 1777 William Wales decided to revise the book and under the same title an edition was published in 1780 attributed to Robertson and Wales. In 1750 Robertson published A Translation of De La Caille's Elements of Astronomy and he published nine papers in the Philosophical Transactions between 1750 and 1772. These were On Logarithmic Tangents, On Logarithmic Lines on Gunter's Scale, On Extraordinary Phenomena in Portsmouth Harbour, On the Specific Gravity of Living Men, On the Fall of Water under Bridges, On Circulating Decimals, On the Motion of a Body deflected by Forces from Two Fixed Points, and On Twenty Cases of Compound Interest After losing his position at the Royal Naval Academy in 1766 Robertson was appointed as a clerk and librarian to the Royal Society, positions which he held until his death. He continued his scientific practice and was the first to be show that stereographic projection from the sphere is a conformal map projection.
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It describes the representation of circles by 2\times 2 Hermitian matrices, the inversion of circles, stereographic projection, pencils of circles (certain one-parameter families of circles) and their two-parameter analogue, bundles of circles, and the cross-ratio of four complex numbers. The chapter on Möbius transformations is the central part of the book, and defines these transformations as the fractional linear transformations of the complex plane (one of several standard ways of defining them). It includes material on the classification of these transformations, on the characteristic parallelograms of these transformations, on the subgroups of the group of transformations, on iterated transformations that either return to the identity (forming a periodic sequence) or produce an infinite sequence of transformations, and a geometric characterization of these transformations as the circle-preserving transformations of the complex plane. This chapter also briefly discusses applications of Möbius transformations in understanding the projectivities and perspectivities of projective geometry.
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Benjamin Franklin was the primary founder, benefactor, President of the Board of Trustees and a trustee of the Academy and College of Philadelphia, which merged with the University of the State of Pennsylvania to form the University of Pennsylvania in 1791 (Joseph Duplessis, c. 1785). Academy and College of Philadelphia (c. 1780), 4th and Arch Streets, Philadelphia, home of what became the University from 1751 to 1801 "House intended for the President of the United States" from "Birch's Views of Philadelphia" (1800), home of the College of Philadelphia/University of Pennsylvania from 1801 to 1829 Ninth Street Campus (above Chestnut Street) in stereographic image: Medical Hall (left) and College Hall (right), both built 1829–1830 The University of Pennsylvania considers itself the fourth-oldest institution of higher education in the United States, though this is contested by Princeton and Columbia Universities. The university also considers itself as the first university in the United States with both undergraduate and graduate studies.
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Each choice of such a projection point results in an image that is congruent to any other. But because such a projection point lies on the Möbius band itself, two aspects of the image are significantly different from the case (illustrated above) where the point is not on the band: 1) the image in R3 is not the full Möbius band, but rather the band with one point removed (from its centerline); and 2) the image is unbounded – and as it gets increasingly far from the origin of R3, it increasingly approximates a plane. Yet this version of the stereographic image has a group of 4 symmetries in R3 (it is isomorphic to the Klein 4-group), as compared with the bounded version illustrated above having its group of symmetries the unique group of order 2. (If all symmetries and not just orientation-preserving isometries of R3 are allowed, the numbers of symmetries in each case doubles.) But the most geometrically symmetrical version of all is the original Sudanese Möbius band in the three-sphere S3, where its full group of symmetries is isomorphic to the Lie group O(2).
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