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Topic: Conformal map

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	Conformal map - Biocrawler
		The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation.
		The Riemann mapping theorem, one of the profound results of complex analysis, states that any non-empty open simply connected proper subset of C admits a bijective conformal map to the open unit disk in C.
		Any conformal map from Euclidean space of dimension at least 3 to itself is a composition of a homothety and an isometry.
	www.biocrawler.com /encyclopedia/Conformal (374 words)

	Map Projections: Conformal Projections
		conformality, is the most fundamental requisite: the angle between any two lines on the sphere must be the same between their projected counterparts on the map; in particular, each parallel must cross every meridian at right angles.
		Conformality is a strictly local property: angles (therefore shapes) are not expected to be preserved much beyond the intersection point; in fact, straight lines on the sphere are usually curved in the plane, and vice versa.
		Peirce's projection is conformal everywhere except at the corners of the inner hemisphere (thus the midpoints of edges in the whole map), where the Equator breaks abruptly.
	www.progonos.com /furuti/MapProj/Normal/ProjConf/projConf.html (1583 words)

	PlanetMath: conformal mapping
		It is clear from the definition that the composition of two such maps and the inverse of any such map is again an invertible conformal mapping, so the set of such mappings forms a group.
		This notion of conformal mappings can be generalized to any setting in which it makes sense to speak of angles between curves or angles between tangent vectors.
		This is version 13 of conformal mapping, born on 2003-04-28, modified 2006-11-03.
	planetmath.org /encyclopedia/ConformalMapping.html (233 words)

	Schwarz-Christoffel mapping
		A conformal map of a region in the complex plane is an analytic (smooth) function whose derivative never vanishes within the region.
		Because of the preservation of angles, a conformal map of a square grid in the plane results in a curvilinear orthogonal grid.
		Furthermore, the transplantation by a conformal map of a solution to Laplace's equation is still a solution in the image region, so that conformal maps can be used in heat transfer, electrostatics, steady fluid flows, and other phyiscal applications in which Laplace's equation arises.
	www.math.udel.edu /~driscoll/research/conformal.html (503 words)

	Map Projections: From Spherical Earth to Flat Map
		Map projections allow us to represent some or all of the Earth's surface, at a wide variety of scales, on a flat, easily transportable surface, such as a sheet of paper.
		Each map projection has advantages and disadvantages; the appropriate projection for a map depends on the scale of the map, and on the purposes for which it will be used.
		A map projection may combine several of these characteristics, or may be a compromise that distorts all the properties of shape, area, distance, and direction, within some acceptable limit.
	nationalatlas.gov /articles/mapping/a_projections.html (2154 words)

	Kids.Net.Au - Encyclopedia > Conformal map
		In cartography, a map projection is called conformal if it preserves the angles at all but a finite number of points.
		It is impossible for a map projection to be both conformal and equal-area[?].
		An important statement about conformal maps is the Riemann mapping theorem.
	www.kids.net.au /encyclopedia-wiki/co/Conformal_map (148 words)

	Map Projections
		Maps showing adjacent areas can be joined at their edges only if they have the same standard parallels (parallels of no distortion) and the same scale.
		USGS maps of the conterminous 48 States, if based on this projection have standard parallels 29 ½°N and 45 ½°N. Such maps of Alaska use standard parallels 55°N and 65°N, and maps of Hawaii use standard parallels 8°N and 18°N. Map is not conformal, perspective, or equidistant.
		Map projection—A map projection is a systematic representation of a round body such as the Earth or a flat (plane) surface.
	www.carolinamapdistributors.com /articles/mapprojections.htm (3307 words)

	Map Projection Overview
		Map projections are attempts to portray the surface of the earth or a portion of the earth on a flat surface.
		When a map portrays areas over the entire map so that all mapped areas have the same proportional relationship to the areas on the Earth that they represent, the map is an equal-area map.
		A Lambert Conformal Conic Projection was proposed with an origin at 31:10 North, 100:00 West and with standard parallels at 27:25 North and 34:55 North.
	www.colorado.edu /geography/gcraft/notes/mapproj/mapproj.html (1829 words)

	Map Projections Poster
		Maps showing adjacent areas can be joined at their edges only if they have the same standard parallels (parallels of no distortion) and the same scale.
		USGS maps of the conterminous 48 States, if based on this projection have standard parallels 29 ½°N and 45 ½°N. Such maps of Alaska use standard parallels 55°N and 65°N, and maps of Hawaii use standard parallels 8°N and 18°N. Map is not conformal, perspective, or equidistant.
		Map projection—A map projection is a systematic representation of a round body such as the Earth or a flat (plane) surface.
	erg.usgs.gov /isb/pubs/MapProjections/projections.html (3453 words)

	Reference.com/Encyclopedia/Conformal map
		Any conformal map on a portion of Euclidean space of dimension greater than 2 can be composed from three types of transformation: a homothetic transformation, an isometry, and a special conformal transformation.
		Conformal mappings are invaluable for solving problems in engineering and physics that can be expressed in terms of functions of a complex variable but that exhibit inconvenient geometries.
		However, by employing a very simple conformal mapping, the inconvenient angle is mapped to one of precisely pi radians, meaning that the corner of two planes is transformed to a straight line.
	www.reference.com /browse/wiki/Conformal_map (824 words)

	[No title]
		The conformal map can be applied to a variety of problems involving a polyg- onal problem domain: for example, solving Laplace's equation with Dirichlet, Neumann, or mixed boundary conditions; solving Poisson's equation; finding eigenvalues of the Laplace opera- tor; hodograph computations for ideal free-streamline flow; grid generation.
		For high-accuracy conformal mapping of a polygon, the Schwarz- Christoffel approach is extremely satisfactory because it handles the singularities at corners exactly and reduces the map to a finite number of parameters.
		Third, map indi- vidual points as desired from the disk to the polygon with routine WSC and from the polygon to the disk with routine ZSC.
	www.netlib.org /conformal/scdoc (2243 words)

	Math Forum Discussions
		mapping is Holder continuous(umlaut over the o), with exponent depending on the way one quantifies being a quasicircle.
		Re: Summary: Boundary maps in the Riemann mapping theorem
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/thread.jspa?messageID=1684811&tstart=0 (685 words)

	Conformal Maps
		However, if the map is at all complicated, then it could be a little difficult to match up a line in the pre-image with its corresponding curve in the image.
		The exponential map is actually implemented as exp(aaa z), where aaa is a complex parameter, say aaa = a + i b.
		Note that this amounts to precomposing z ---> exp(z) with the map that stretches z by a factor r = sqrt(a^2 + b^2) and rotate it by an angle theta = arctan (b/a).
	3d-xplormath.org /Downloads/3DFSdocs/DocumentationPages/ConformalMaps.html (1369 words)

	Maps that shape the world
		Some projections aim to produce maps in which the angles, and therefore the shapes on the paper, are as close as possible to those on the ground.
		These "equal-area" maps, such as the Peters projection, became popular in the 1980s because of their attribute of depicting country sizes in true proportion- unlike the Mercator projection, which minimises the areas of most Third World countries, as they tend to be near the equator, and maximises areas near the poles.
		This is not the case on a map because of the distortions caused when you flatten the curved surface: on any map the scale factor varies from point to point.
	members.fortunecity.com /templarser/maps.html (1890 words)

	Mercator
		The problem of creating maps that could be used for navigation became critical in the 16th century with the voyages of discovery, since Ptolemy's map was not well-adapted to compass navigation and did not include enough of the earth.
		The Mercator map is a conformal map with the scale decreasing toward the poles.
		The basic map has unity scale on the standard meridian, but the scale can be changed slightly to make it smaller than unity on the meridian, and unity a certain distance east and west of the meridian, so that the scale is closer to unity over a wider band of the map.
	www.du.edu /~jcalvert/math/mercator.htm (3839 words)

	Lambert's Map
		Conformal maps preserve the shapes of small areas exactly, although the scale of the map may vary from point to point.
		Conformality is an extremely valuable property for maps that are to be used critically, and not just for general orientation or decoration.
		The mapping by this function of a typical small area dydx is shown in the diagram at the right, where the change of scale and the rotation are clearly shown.
	www.du.edu /~jcalvert/math/lambert.htm (3407 words)

	August's Conformal Projection of the Sphere on a Two-Cusped Epicycloid
		Then, the circle is conformally mapped to the inside of the epicycloid that forms the boundary of the map using complex numbers.
		Functions over the complex numbers are an inexhaustible source of conformal mappings, since functions are extended to complex numbers in the fashion that permits them to be differentiable over the complex plane, and the condition for differentiability and for forming a conformal mapping are one and the same.
		For world maps in conventional aspect, this will appear to be more important, but for applications where every part of the globe is equally important, the Eisenlohr would be advantageous, despite being somewhat more complicated to calculate.
	www.quadibloc.com /maps/mcf0702.htm (1353 words)

	NOAA ARL Conformal Map Functions (CMAPF)
		Most models are based on a rectangular lattice of mesh points drawn on some specific map, the choice of which depends on a number of considerations, including the size and location of the region to be modeled.
		The geometry of the map introduces subtle changes in the equations describing the physics of the model, whose terms are provided by calls to these subroutines.
		Version 1.0 covers the standard conformal map projections centered at the North and South Pole, namely the Polar Stereographic, the Mercator, and the Lambert Conformal projections.
	www.arl.noaa.gov /ss/models/cmapf.html (424 words)

	Map projection Summary
		The map from O to P that send each point of O to the point in P closest to it, is called the projection map from O to P. For example, suppose that O is a knot - that is a loop of string with its ends glued together.
		However, in understanding the concept of a map projection it is helpful to think of a globe with a light source placed at some definite point with respect to it, projecting features of the globe onto a surface.
		Sinusoidal: the north-south scale is the same everywhere at the central meridian, and the east-west scale is throughout the map the same as that; correspondingly, on the map, as in reality, the length of each parallel is proportional to the cosine of the latitude.
	www.bookrags.com /Map_projection (3524 words)

	American Scientist Online - Conformal Mappings (Site not responding. Last check: )
		Conformal maps are used by mathematicians, cartographers and physicists to deform regions in a way that preserves shape on a small scale.
		Difficult problems in heat and fluid flow can be transformed by a conformal map into simple problems whose answers are already known—in effect, tailoring the solution to fit the problem.
		New methods using circle packing are ideally suited for programming on a computer, and they are being applied to the challenge of making a conformal map of the human brain—important for interpreting functional information from brain imaging.
	www.americanscientist.org /amsci/articles/99articles/Krantz.html (110 words)

	Lambert's Conformal Conic
		Conformal maps are particularly suited to depictions of extended areas, since they avoid the shape distortions most noticeable to the eye.
		Although this world map definitely depicts the continental areas with a pleasing appearance, no projection of the globe on a flat piece of paper is perfect, not even this one.
		Then, the scale of the map can be given as the scale on those parallels; although the projection is the same, this is known as the version with two standard parallels, since at those two parallels the scale is correct, even if the curvature of the parallel is correct for another parallel between them.
	www.quadibloc.com /maps/mco0301.htm (865 words)

	General Projections Concepts
		A map projection is a device for reproducing all or part of a round body on a flat sheet.
		Even a carefully constructed globe is not the best map for most applications because its scale is by necessity too small, a straightedge cannot be satisfactorily used on it for measurement of distance, and it is awkward to use in general.
		Direction - While conformal maps give the relative local directions correctly at any given point, there is one frequently used group of map projections, called azimuthal or zenithal, on which the directions or azimuths of all points on the map are shown correctly with respect to the center.
	exchange.manifold.net /manifold/manuals/5_userman/mfd50General_Projections_Concepts.htm (1699 words)

	DIVERSOPHY.COM - using the Peters Map
		When this map was first introduced by historian and cartographer Dr. Arno Peters at a Press Conference in Germany in 1974 it generated a firestorm of debate.
		The first English-version of the map was published in 1983, and it continues to have passionate fans as well as staunch detractors.
		This particular upside down map is colorful, visually engaging, and has all the flags of the countries of the world around the border.
	www.diversophy.com /petersmap.htm (1469 words)

	Search Results for Conformal
		The conformal mappings described in the book are by and large arranged according to the analytic functions giving rise to them, the author having found that this permits a more systematic classification than an arrangement according to geometric properties of domains.
		After considering the problem of conformal mapping on the half-plane of finite polygonal regions bounded by straight lines and circular arcs she applied these ideas to the physical problem of the two-dimensional seepage flow of ground water in an earth dam of a particular shape.
		Other non-trivial conformal representations of a plane area on a second plane area are obtained by comparing the various stereographic projections of the spherical earth which correspond to different positions of the centre of projection on the earth's surface.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Conformal&CONTEXT=1 (2229 words)

	GeoSystems: Map Projections (Site not responding. Last check: )
		Maps are flat, and the process by which geographic locations (latitude and longitude) are transformed from a three-dimensional sphere to a two-dimensional flat map is called a projection.
		Visualize the properties of a map projection by comparing the arrangement of its meridians and parallels with the characteristics of the graticule on the globe: Parallels are spaced equal distances apart.
		Though few maps are truly the result of such projection (most are derived from mathematical formulas), it is a useful way to visualize and understand the transformation process.
	www-personal.umich.edu /~sarhaus/tigdd27proj.html (641 words)

	Riemann mapping theorem Summary
		This means that the map accurately represents small circles as circles (rather than as ellipses or ovals), and that it accurately represents angles.
		The fact that f is a bijective and holomorphic (or "biholomorphic") implies that it is a conformal map and therefore angle-preserving.
		Even though the class of continuous functions is infinitely larger than that of conformal maps, it is not easy to construct a one-to-one function onto the disk knowing only that the domain is simply connected.
	www.bookrags.com /Riemann_mapping_theorem (1845 words)

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