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Topic: Isometry

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	Isometry - Wikipedia, the free encyclopedia
		In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces.
		Isometries are often used in constructions where one space is embedded in another space.
		A path isometry or arcwise isometry is a map which preserves the lengths of curves (not necessarily bijective).
	en.wikipedia.org /wiki/Isometry (367 words)

	Euclidean plane isometry - Wikipedia, the free encyclopedia
		An isometry (or rigid motion) of the Euclidean plane is a distance-preserving transformation of the plane.
		The identity isometry is also an identity for composition, and composition is associative; therefore isometries satisfy the axioms for a semigroup.
		In terms of complex numbers, the isometries of the plane are addition of a complex constant (translation), multiplication by a complex constant with modulus 1 (rotation), complex conjugation (reflection in the real axis), and combinations.
	en.wikipedia.org /wiki/Euclidean_plane_isometry (2433 words)

	PlanetMath: isometry (Site not responding. Last check: 2007-10-07)
		Warning: some authors do not require isometries to be surjective (and in this case, the isometry will not necessarily be a homeomorphism).
		Cross-references: homeomorphic, topologies, induced, homeomorphism, between, injective, surjective, Isometry, function, metric spaces
		This is version 3 of isometry, born on 2002-02-13, modified 2003-02-08.
	planetmath.org /encyclopedia/Isometry.html (126 words)

	Isometry
		The image of a line segment (ray, angle, or triangle) under an isometry of a neutral plane is a line segment (ray, angle, or triangle).
		The image of a triangle under an isometry of a neutral plane is a congruent triangle.
		The image of a circle under an isometry of a neutral plane is a congruent circle.
	www.mnstate.edu /peil/geometry/C3Transform/2isometry.htm (1013 words)

	Plane Isometries (Site not responding. Last check: 2007-10-07)
		(We shall be mainly concerned with the isometries of the plane.) In the context of plane transformation "dist" is the common Euclidean metric.
		The difference between the two kinds of isometries is that the proper ones preserve the orientation while the improper isometries change it.
		An improper isometry is either a reflection or a glide reflection [Coxeter, Yaglom].
	www.cut-the-knot.com /pythagoras/Transforms/index.shtml (871 words)

	Symmetry - Wikipedia, the free encyclopedia
		Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa).
		In 2D geometry the main kinds of symmetry of interest are with respect to the basic Euclidean plane isometries: translations, rotations, reflections, and glide reflections.
		For a common notion of symmetry in Euclidean space, G is the Euclidean group E(n), the group of isometries, and V is the Euclidean space.
	en.wikipedia.org /wiki/Symmetry (2824 words)

	Math 5337 Symmetry Definitions (Site not responding. Last check: 2007-10-07)
		Many more exotic transformations exist, but for the sake of simplicity we are going to restrict ourselves to isometries that can be described in terms of rubber stamping out many copies of the same image.
		An isometry of an object or space is any contortion or movement of the object or space which doesn't change the distances between the points of that object or space.
		A rotation is another isometry, determined by a center and an angle.
	www.geom.uiuc.edu /~math5337/Symmetry/sym.1.1.5.html (371 words)

	Wikinfo \| Isometry
		A path isometry or arcwise isometry is a map which preserve lengths of curves (not nesesury bijective).
		Metric spaces X and Y are called isometric if there is an isometry X→ Y. The set of isometries from a metric space to itself form a group with respect to compositon (called isometry group).
		For example, there is an isometry consisting of the reflection on the x-axis, followed by translation of one unit parallel to it.
	www.wikinfo.org /wiki.php?title=Isometry (495 words)

	Isometries
		Definition: A transformation of the plane is an isometry if, for all points X and Y, the distance between the image points X' and Y' equals the distance between X and Y. In other words, an isometry is a transformation which preserves distance.
		The inverse of an isometry is an isometry: For all transformations F, if F is an isometry and G is its inverse, then G is an isometry.
		Definition: An isometry F is orientation preserving if, for all noncollinear points A, B, C, the proper angle measures of the angles ABC and A'B'C' have the same sign.
	www.math.uga.edu /~clint/2005/5210/isom.htm (580 words)

	Introduction to Isometries (Science U)
		Isometries are transformations the plane which don't distort shapes.
		The most familiar isometry is probably just translation -- shifting a shape in a straight line.
		This is worth pointing out, since it suggests a way of classifying any kind of isometry, by considering their fixed point behavior.
	www.scienceu.com /library/articles/isometries/rotation.html (157 words)

	Euclidean similarities
		A congruence or isometry (for ``equal distance'') is a transformation that leaves all distances exactly the same (i.e.
		A similarity with scaling factor 1 is called an isometry (for ``equal distance'') or congruence.
		Finally, we are left to consider isometries T that fix both the points (0,0) and (1,0).
	www.math.okstate.edu /mathdept/dynamics/lecnotes/node25.html (945 words)

	Partial isometry: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-07)
		In the mathematical discipline of geometry and mathematical analysis, an isometry, isometric isomorphism or congruence mapping is a distance-preserving...
		Partial isometries are also characterized by the condition that either W W* or W* W is a projection.
		A pair of projections one of which is the initial projection of a partial isometry and the other a final projection of the same isometry are said to be equivalent.
	www.absoluteastronomy.com /encyclopedia/p/pa/partial_isometry.htm (681 words)

	Euclidean Group
		An isometry which preserves size and sense of angles is
		It is by no means clear that each isometry is either direct or indirect.
		Suppose that t is an isometry which fixes the points O(0,0) and X(1,0).
	www.maths.gla.ac.uk /~wws/cabripages/klein/euclid.html (376 words)

	The Learning Toolbox - If It Fits
		The reflected square WXYZ is the isometry of square ABCD.
		The context clues in the sentence tell me that somehow a figure and its reflection combine to make an isometry.
		"Isometry is a transformation in which a figure and its image are equal reflections of each other.
	coe.jmu.edu /Learningtoolbox/ifitfits2.html (296 words)

	Isometry (Site not responding. Last check: 2007-10-07)
		The notion of isometry comes in two main flavors: global isometry and a weaker notion path isometry orarcwise isometry.
		Apath isometry or arcwise isometry is a map which preserve lengths of curves (not nesesury bijective).
		The set of isometries from a metric space to itself form a groupwith respect to compositon (called isometry group).
	www.therfcc.org /isometry-199279.html (271 words)

	NationMaster.com - Encyclopedia: Metric space
		The set of all (isometry classes of) compact metric spaces form a metric space with respect to Gromov-Hausdorff distance.
		They are called isometrically isomorphic if there exists a bijective isometry between them.
		In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some real number k < 1 such that, for all x and y in M, The smallest such value of k is called the...
	www.nationmaster.com /encyclopedia/metric-space (4221 words)

	Isometry (Site not responding. Last check: 2007-10-07)
		The notion of isometry comes in two flavors: global isometry and a weaker notion path isometry or arcwise isometry.
		Metric spaces X and Y are called isometric if there is an isometry $X\to The set of$ isometries from a metric to itself form a group with respect compositon (called isometry group).
		In Euclidean space with the usual distance function the isometries can be characterized: there are no than the 'expected' examples generated by rotations and translations.
	www.freeglossary.com /Euclidean_transformation (474 words)

	Re: C*-algebras
		>>I is an isometry iff II* is a projection and I*I is the identity.
		As I was writing that little chart, I thought: "He won't be able to resist asking about the missing fourth entry." The thing you're asking about is usually called "the adjoint of an isometry", but if you insist on a snappier name, we can call it a "coisometry".
		You've already answered my puzzle in one way: you've noticed that the free C-algebra on an isometry* is the same as the C*-algebra of bounded operators on l^2 generated by the "left shift operator".
	www.lns.cornell.edu /spr/2000-02/msg0021620.html (965 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		In a more formal language: A transformation of the plane is an isometry if, for all points X and Y, the distance between the image points X' and Y' equals the distance between X and Y. In other words, an isometry is a transformation which preserves distance.
		Products The product of two isometries is an isometry: For all transformations F and G, if F and G are isometries, then GF is an isometry.
		Orientation Definition: An isometry F is orientation preserving if, for all noncollinear points A, B, C, the proper angle measures of the angles ABC and A'B'C' have the same sign.
	www.cst.cmich.edu /users/manou1a/251/251.VIII.doc (1075 words)

	Body (Site not responding. Last check: 2007-10-07)
		An isometry is a one-to-one transformation from one geometric space onto itself which preserves all intrinsic properties such as length, angle measure, and area.
		Here is one very important property of isometries of a plane: If two isometries act on three non-colinear points of the plane in the same way, then they are the same isometry.
		Every isometry of the plane or a sphere or a hyperbolic plane is either a reflection, a translation, a rotation, a glide reflection, or the identity.
	www.math.cornell.edu /~dwh/books/eg99/Ch18/Ch18.html (2141 words)

	kirupa.com - Isometric Transformation
		These tutorials are all about isometry, or, more specifically, about creating a "filmation" isometric game similar to Knight Lore.
		I have to point out that Senocular's tutorials are excellent for the introduction to isometry and it's problems, but there are many things that cannot be done by using this (as I call it) "static isometry" approach.
		With "static isometry" I assume that the majority of the objects are "fixed" to the floor, meaning they cannot be moved.
	www.kirupa.com /developer/actionscript/isometric_transforms.htm (588 words)

	Re: Symmetries of complex projective space
		The isometry group for CP^n is what I was actually looking for.
		These are called "isometries", >and they form a group called the "isometry group".
		These are called "isometries", > and they form a group called the "isometry group".
	www.lns.cornell.edu /spr/2003-09/msg0053710.html (948 words)

	isometry - any good? - Indiegamer Developer Discussion Boards
		There are some games which use isometry just for a better look (eg Sim City) and games which are somewhat annyoing to handle (eg Solstice [NEW/Famicom]), because isometry lacks any depth - accurate jumping is really tough (watching the shadows isn't fun).
		That game for example used isometry very well, becaue you could see everything and because the height was an important factor.
		I would like to know if you know any other games were the use of isometry actually added something to the game and - of course - why it worked for you.
	forums.indiegamer.com /showthread.php?t=176 (763 words)

	Any planar isometry is the product of at most 3 reflections.
		Any planar isometry is the product of at most 3 reflections.
		Any plane isometry is a reflection or the product of two or three reflections.
		Proof: Since any plane isometry is determined by its transformation of three noncolinear points, suppose the isometry T has T(A)=A', T(B)=B' and T(C)=C' where A,B,C are the vertices of a triangle.
	www.humboldt.edu /~mef2/Courses/m371_3reflections.html (284 words)

	Citebase - The Weil-Petersson Isometry Group (Site not responding. Last check: 2007-10-07)
		We prove that every Weil-Petersson isometry of the Teichmuller space T(g,n) is induced by an element of the extended mapping class group; here 3g-3+n > 1 and (g,n) is not (1,2).
		Our method follows Ivanov's proof of the Royden's analogous theorem for the Teichmuller metric: we study the action of an isometry on the frontier of the metric completion of the Teichmuller space, and show that the isometry then induces an automorphism on the relevant complex of curves.
		As a consequence one obtains a classification of the elements of the mapping class group as Weil-Petersson isometries whic...
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0008065 (652 words)

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