Projection (mathematics)

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	Manual.doc
		While the mathematical details of the projection process can be rather complex, the student will interact with specially designed software that takes a simple approach to the teaching the mathematics behind map projections and their equations.
		Aside from the mathematics of the equations, the student will also become familiar with the various parameters that are often associated with projection equations.
		This transformation is commonly called a map projection because the latitude and longitude values are "projected" (a mathematical term) into a Cartesian coordinate system (x and y).
	faculty.frostburg.edu /geog/kessler/MaPED/Manual.doc (3585 words)

	Amazon.com: CAD/CAE Descriptive Geometry: Books: Daniel L. Ryan
		Modem descriptive geometry is a mixture of plane geometry (mathematics), third quadrant orthographic projection (engineering drawing), and high-speed communication methods (digital computing).
		CAD/CAE Descriptive Geometry provides a sound foundation in the fundamentals of plane geometry (mathematics), orthographic projection (technical drawing), and high-speed communication methods (digital computing).
		Subjects > Science > Mathematics > Geometry & Topology > General Geometry
	www.amazon.com /exec/obidos/tg/detail/-/0849342732?v=glance (572 words)

	Mercator_Gerardus
		Realising that Mercator wanted to learn mathematics to apply it to cosmography, Gemma Frisius gave him advice on the best route into learning the mathematics he needed to know, giving him books to study at home.
		Mercator returned to Louvain in 1534 where he now studied mathematics under Gemma Frisius.
		Mercator's break from the methods of Ptolemy was as important for geography as was Copernicus for astronomy.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Mercator_Gerardus.html (572 words)

	Projection - Wikipedia, the free encyclopedia
		In mathematics a projection is a linear transformation which remains unchanged if applied twice (p(u) = p(p(u))) (in other words, it is idempotent), such as that taking (x, y, z) in three dimensions to (x, y, 0) in the plane or generalisations of this in other dimensions.
		In mathematics a projection is also a mapping that takes an element to the equivalence class it is in, and mapping an element (x,y) of a Cartesian product to x or y, mapping a function f to the value f (x) for a fixed x, etc.
		In cinematography, projection is the display of a movie in a theater using a movie projector, which is likely to be replaced by digital projection
	en.wikipedia.org /wiki/Projection (246 words)

	Projection - Wikipedia, the free encyclopedia
		In mathematics a projection is a linear transformation which remains unchanged if applied twice (p(u) = p(p(u))) (in other words, it is idempotent), such as that taking (x, y, z) in three dimensions to (x, y, 0) in the plane or generalisations of this in other dimensions.
		In mathematics a projection is also a mapping that takes an element to the equivalence class it is in, and mapping an element (x,y) of a Cartesian product to x or y, mapping a function f to the value f (x) for a fixed x, etc.
		See orthogonal projection, projection (linear algebra), projection operator.
	en.wikipedia.org /wiki/Projection (244 words)

	Projection - Wikipedia, the free encyclopedia
		In mathematics a projection is a linear transformation which remains unchanged if applied twice (p(u) = p(p(u))) (in other words, it is idempotent), such as that taking (x, y, z) in three dimensions to (x, y, 0) in the plane or generalisations of this in other dimensions.
		In mathematics a projection is also a mapping that takes an element to the equivalence class it is in, and mapping an element (x,y) of a Cartesian product to x or y, mapping a function f to the value f (x) for a fixed x, etc.
		In cinematography, projection is the display of a movie in a theater using a movie projector, which is likely to be replaced by digital projection
	en.wikipedia.org /wiki/Projection (244 words)

	Mathematics of Computation
		R. Rannacher, On Chorin's projection method for the incompressible Navier-Stokes equations, Springer-Verlag, Lecture Notes in Mathematics 1530 (1992), 167-183.
		Abstract: The gauge formulation of the Navier-Stokes equations for incompressible fluids is a new projection method.
		Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
	www.ams.org /mcom/2005-74-250/S0025-5718-04-01687-4/home.html (244 words)

	fibdegen.txt
		Degenerate fibres in the Stone-Cech compactification of the universal bundle of a finite group: An application of homotopy theory to general topology David Feldman Department of Mathematics University of New Hampshire Alexander Wilce Department of Mathematics University of Pittsburgh at Johnstown 1 Introduction If p : E !
		It is well-known that if E = B x F where F is a finite set and p is projection on the first factor, then fiE = fiB x fiF, and fip is again projection on the first factor.
		We emphasize that the action of G on EG is not part of the data available to the Stone-Cech functor; rather the compactification process directly detects the symmetry of the bundle.
	hopf.math.purdue.edu /Feldman-Wilce/fibdegen.txt (5513 words)

	RubricGeom
		It is evident that the mathematics of the project is clearly understood.
		The main thrust of the project and the mathematics behind it is understood., but there may be some minor misunderstanding of content, errors in computation, or weakness in presentation.
		An attempt is made to extend the project, but because their work on the solids is incomplete, the extension of the project is also incomplete.
	www.glade.net /~rfletcher/RubricGeom.htm (3620 words)

	Student understanding of topics in linear algebra
		Since course content is dependent on the needs of a number of disciplines for a linear algebra course, the math department must maintain a constant liaison with these departments in order to monitor their needs in higher level mathematics.
		The Linear Algebra Curriculum Study Group Recommendations for the First Course in Linear Algebra, College Mathematics Journal, 12 (1) 41-46.
		He also cites that textbooks in linear algebra are written based on the assumptions "that students recognize models and solve problems by translating them to isomorphic but abstract structures; and that they can apply the principles of abstract setting to solve problems.
	www.physics.umd.edu /perg/plinks/linalg.htm (3620 words)

	Registration & Records - Course Catalog
		Topics selected from areas such as mathematics of finance, probability, statistics, linear programming and theory of games, intuitive topology, recreational math, computers and applications of mathematics.
		Linear equations operations with matrices, row echelon form, determinants, vector spaces, linear independence, bases, dimension, orthogonality, eigenvalues, reduction of matrices to diagonal forms, applications to differential equations and quadratic forms.
		Algebra review, functions, graphs, limits, derivatives, integrals, logarithmic and exponential functions, functions of several variables, applications in management, applications in biological and social sciences.
	www2.acs.ncsu.edu /reg_records/crs_cat/MA.html (3620 words)

	Mathematics Archives - Topics in Mathematics - Geometry
		A Project to Transform 6-12 Mathematics Education: Vertically-Integrated, Inquiry-Based Geometry
		Spherical Geometry, Logic and the Axiomatic Method Incidence Geometry, Betweenness Axioms, Congruence Theorems, Axioms of Continuity, Neutral Geometry, Hyperbolic Geometry, Classification of Parallels, Inversion in Euclidean Circles, Models of Hyperbolic Geometry, Hypercycles and Horocycles, The Pseudosphere, Hyperbolic Trigonometry, Hyperbolic Analytic Geometry
		Geometry Building Blocks, Geometry words, Coordinate geometry, Pairs of lines, Classifying angles, Angles and intersecting lines, Circles, Polygons, Triangles, Quadrilaterals, Area of polygons and circles, Congruent figures, Similar figures, Squares and square roots, The Pythagorean Theorem and right triangle facts, Three-dimensional Figures, Prisms, Pyramids, Cylinders, cones, and spheres
	archives.math.utk.edu /topics/geometry.html (1366 words)

	The Math Forum - Math Library - Matrices
		The toy game of Merlin is not quite trivial and the mathematics is simple enough to provide an entertaining exercise in a Linear Algebra class.
		This website looks at some areas of mathematics that are not familiar to most people, such as Ramsey theory and set theory, but introduces them in an uncomplicated manner.
		Notes from the University of Pennsylvania course "Numerical Methods on Parallel Computing." Features common linear algebra examples, such as solving for x in [A]x=b, where the matrix [A] and the vector b are known.
	mathforum.org /library/topics/matrices (1366 words)

	Mathematics Archives - Topics in Mathematics - Linear Algebra
		Linear Transformations in 2-Dimensions, Products of Linear Transformations in 2-Dimensions, Linear Transformations in 3-Dimensions, Products of Linear Transformations in 3-Dimensions, Eigenvalues and the Characteristic Polynomial, Effect of a Linear Transformation on its Eigenvectors, Change of Basis
		Linear Algebra - An Introduction to Linear Algebra for Pre-Calculus Students
		Course materials, lecture notes, linear functions, linear algebra review, orthonormal vectors and QR factorization, least-squares methods, regularized least-squares and minimum norm methods, autonomous linear dynamical systems, eigenvectors and diagonalization, Jordan canonical form, aircraft dynamics, symmetric matrices, quadratic forms, matrix norm, and SVD, quantum mechanics, controllability and state transfer
	archives.math.utk.edu /topics/linearAlgebra.html (1366 words)

	Student Projects in Linear Algebra
		Each of the following is an article on linear algebra selected from The College Mathematics Journal.
		The College Mathematics Journal is produced by the Mathematical Association of America and contains article accessible by community college students.
		Kevin Yokoyama and I are compiling a list of articles and links on this page that might find potential use as student projects in linear algebra.
	online.redwoods.cc.ca.us /instruct/darnold/laproj/index.htm (1366 words)

	Map Projections
		In general, and this is true for the projections in the three basic aspects of cylindrical, conic, and azimuthal, scale going away from the center of a map increases for a conformal projection, and decreases for an equal-area projection.
		Map projections can be grouped together in two basic ways; and a third characteristic, although it divides different way of using the same projection, is sometimes considered important enough that different versions of the same projection varying only in this characteristic are given different names.
		There is the transverse (or, in the case of an azimuthal projection, equatorial) case, in which the globe has been shifted by 90 degrees before the map is drawn, and there is the oblique case where the globe is shifted by a lesser amount.
	www.hypermaths.org /quadibloc/maps/mapint.htm (1366 words)

	Mercator's Projection
		In 1599 the English mathematician Edward Wright explained the mathematics of exactly how Mercator's projection should be done.
		The Mercator projection was invented by Gerardus Mercator, a Flemish mapmaker.
		The property of the Mercator projection map that made it useful to navigators is that it preserves angles.
	www.math.ubc.ca /~israel/m103/mercator/mercator.html (1366 words)

	Supported map projections and datums
		A map projection is the mathematical function used to plot a point on an ellipsoid on to a plane sheet of paper.
		map projections try to preserve areas so that a coin of any size, for example, covers exactly the same area of the earth’s surface no matter where it is placed on the map.
		map projection correctly shows scale throughout the map, although for many projections there are one or more lines on the map where scale is correct.
	map.sdsu.edu /IWSDoc/html/Append_B_datumprojection_ecwactivex.htm (1366 words)

	Telegraph News Arthur Robinson
		Arthur Robinson, who died on October 19 aged 89, was a cartographer known for the Robinson projection, a two-dimensional map of the world which minimised the distortions of the Mercator projection.
		Robinson's approach to global cartography was down to earth and he used mathematics only after preparing a rough sketch: "Take an orange and draw something on it – say, a human face," Robinson explained in 1989.
		Robinson drew a map which gave a more accurate picture of the world's most populous temperate zone.
	www.opinion.telegraph.co.uk /news/main.jhtml?xml=/news/2004/11/19/db1903.xml (447 words)

	The Mathematics of Face Recognition
		A 2D face is represented by its projection onto this space.
		Face recognition, the art of matching a given face to a database of faces, is a nonintrusive biometric method that dates back to the 1960s.
		There are fundamental mathematical results in the literature that try to address these questions and have not yet been fully exploited for face recognition.
	www.siam.org /siamnews/04-03/face.htm (1302 words)

	Mathematics and Art
		The main theorem in Taylor's theory of linear perspective is that the projection of a straight line not parallel to the plane of the picture passes through its intersection and its vanishing point.
		There are two aspects to the problem, namely how does one use mathematics to make realistic paintings and secondly what is the impact of the ideas for the study of geometry.
		Leonardo distinguished two different types of perspective: artificial perspective which was the way that the painter projects onto a plane which itself may be seen foreshortened by an observer viewing at an angle; and natural perspective which reproduces faithfully the relative size of objects depending on their distance.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Art.html (4272 words)

	Dr. Matrix at the Scientium: Mathematics Links
		The mathematical basis for the Mercator projection is an 1851 law of geometry known as the Riemann mapping theorem (although the 16th-century cartographer himself wasn't aware of it, of course).
		CAMSED - Coalition of Automated Mathematics and Science Education Databases - The Coalition of Automated Mathematics and Science Education Databases (CAMSED) is an emerging collaborative effort, striving to increase access to materials useful to individuals in the mathematics and science education community through a system of federated electronic information services.
		Kids Web - Mathematics Page - The Mathematics Page of the Kids Web is a collection of sites which are very simple to navigate and contain information about mathematics of interest to K-12 students.
	www.scientium.com /drmatrix/sciences/math.htm (1413 words)

	The Peters Projection.
		Not only is the projection area-preserving (a detail which I believe Archimedes knew, albeit he wasn't using it in map-making); it is also easy to visualise (as cylindrical projection of a sphere) and produces a `flat' image of the sphere (by unrolling the cylinder).
		One of the simplest to describe is an axial projection, made famous since the 1970s as the Peters projection, though it has a longer history.
		Since R on the cylinder and r on the sphere are equal, this is equal to the r.dz.dφ/radian we had before projection: hence the projection preserves area.
	www.chaos.org.uk /~eddy/math/peters.html (1413 words)

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

	ipedia.com: Projection Article
		In mathematics a projection is a linear transformation which remains unchanged if applied twice (p(u) = p(p(u))) (in other words, it is idempotent), such as that taking (x, y, z) in three dimensions to (x, y, 0) in the plane or generalisations of this in other dimensions.
		In cartography, a map projection In psychology, psychological projection 3D projection, orthographic projection and isometric projection are ways of r...
		In cinematography, projection is the display of a movie in a theater using a film projector, which is likely to be replaced by digital projection
	www.ipedia.com /projection.html (238 words)

	Know it all Inc. (I)
		The opposite of an injection function is a {projection} function which extracts a component of a constructed object, e.g.
		<mathematics> The image (or range) of a {function} is the set of values of that function applied to all elements of its {domain}.
		<mathematics> A {function}, f : A -> B, is injective or one-one, or is an injection, if and only if for all a,b in A, f(a) = f(b) => a = b.
	artikbre.synchro.net /docs/I.html (238 words)

	Mercator Conformal Projection
		In mathematics, a projection is a function which takes points on one surface and transforms them into points on another surface.
		The Mercator projection is often illustrated as a projection from the center of a globe onto a cylinder wrapped around the globe.
		The insight which is key to the Mercator projection (and indeed all conformal projections) is that if you want to avoid distorting angles (like courses and bearings), you have to keep the scale the same in both dimensions (north-south and east-west) on the map.
	www.ualberta.ca /~norris/navigation/Mercator.html (238 words)

	A Chart of the World on Mercator's Projection
		Mercator expected that his projection would be a valuable tool for navigators but he neglected to provide practical guidelines on how use it.
		Wright published “A Chart of the World on Mercator’s Projection”— in 1600 based on his projection of a globe engraved by the English globe maker Emery Molyneux in 1592.
		Edward Wright (1558?-1615), a professor of mathematics at Cambridge University, modified Mercator’s system and published his results, The Correction of Certain Errors in Navigation, in 1599 and again in an improved edition entitled Certaine errors in navigation, detected and corrected (London, 1610).
	www.lib.virginia.edu /small/exhibits/lewis_clark/exploring/ch1-5.html (238 words)

	Projection
		In mathematics a projection is a linear transformation which remains unchanged if applied twice (p(u) = p(p(u))) (in other words, it is idempotent), such as that taking (x, y, z) in three dimensions to (x, y, 0) in the plane or generalisations of this in other dimensions.
		3D projection, orthographic projection and isometric projection are ways of representing 3D scenes in 2D; see graphical projection.
		In cinematography, projection is the display of a movie in a theater using a film projector, which is likely to be replaced by digital projection
	www.brainyencyclopedia.com /encyclopedia/p/pr/projection.html (238 words)

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