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	Affine transformation - Wikipedia, the free encyclopedia
		The invertible affine transformations form the affine group, which has the general linear group of degree n as subgroup and is itself a subgroup of the general linear group of degree n+1.
		For example, describing a transformation as a rotation by a certain angle with respect to a certain axis is easier to get an idea of the overall behavior of the transformation than describing it as a combination of a translation and a rotation.
		A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations.
	en.wikipedia.org /wiki/Affine_transformation (1084 words)

	PlanetMath: geometry
		Geometry, or literally, the measurement of land, is among the oldest and largest areas of mathematics.
		An approach to geometry first formulated by Felix Klein in his Erlangen lectures is to describe it as the study of invariants under certain allowed transformations.
		The discovery of intrinsic geometry led thoughtful geometers such as Riemann (who was a student of Gauss), Clifford, and Mach to the conclusion that a “right and natural” approach to geometry should regard surfaces as geometrical spaces in their own right on a par with Euclidean and projective space.
	planetmath.org /encyclopedia/Geometry.html (4223 words)

	Transformation geometry - Wikipedia, the free encyclopedia
		In mathematics, transformation geometry is a name for a pedagogic theory for teaching Euclidean geometry, based on the Erlangen programme.
		In the end, the reform of geometry teaching came simultaneously with the New Math movement.
		To do real transformation geometry requires some work with symmetry groups; if not group theory, matrix computations are required.
	en.wikipedia.org /wiki/Transformation_geometry (132 words)

	Category:Geometry - Wikipedia, the free encyclopedia
		Geometry is the branch of mathematics dealing with spatial relationships.
		From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry.
		Such axioms are insusceptible to proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions.
	en.wikipedia.org /wiki/Category:Geometry (135 words)

	Math Forum - Mathematics Teacher Bibliography: Transformational Geometry (Site not responding. Last check: 2007-11-01)
		Transformation Geometry: An Application of Physics Ken A. Dunn An analytic approach to transformations in the Euclidean plane and in the Minkowski plane.
		Transformation Geometry and The Artwork Of M.C. Escher Sheila Haak Analyzing the symmetries and transformations in Escher's drawings.
		A Transformation Proof Of The Collinearity Of The Circumcenter, Orthocenter and Centroid Of A Triangle Thomas W. Shilgalis Using a dilation and a half-turn.
	mathforum.org /mathed/mtbib/transformations.html (1004 words)

	numerical identity, object identity, apparent motion, ventriloquism effect, prism adaptation, priming, Gestalt grouping
		The resulting geometries could be ordered based on the number of properties left unchanged by the defining group of transformations: the more radical a transformation, the more properties that are altered, the fewer the properties that remain in the geometry, and the larger/more general the resulting geometry.
		A geometric transformation relates the visual form (or space) to the proprioceptive form (or space) For instance, in a standard prism adaptation experiment, the primary distortion of a 20 diopter prism with the base facing left is a rigid 11.3 degree leftward shift of the visual image.
		That the least radical transformation (and smallest geometry) in the situation be preferred for resolving object identity seems intuitively right, yet precisely why the different levels of transformations and geometries, generated by systematic removal of nested geometric properties, correspond to ordered psychological levels remains a deep mystery.
	www.u.arizona.edu /~bedford/papers/targetarticle.htm (19277 words)

	MC244 Geometry
		Finally, using affine transformations the module examines affine properties of the plane and deals with the affine classification of conics.
		To identify transformations in Euclidean and non-Euclidean plane geometries.
		Define the operation of inversion in a given circle as a transformation of the extended complex (inversive) plane.
	www.mcs.le.ac.uk /Modules/Year4_98-99/MC244.html (877 words)

	The stratification of 3D geometry
		This concept of stratification is closely related to the groups of transformations acting on geometric entities and leaving invariant some properties of configurations of these elements.
		Attached to the projective stratum is the group of projective transformations, attached to the affine stratum is the group of affine transformations, attached to the metric stratum is the group of similarities and attached to the Euclidean stratum is the group of Euclidean transformations.
		The associated groups of transformations, their invariants and the corresponding invariant structures are presented.
	www.cs.unc.edu /~marc/tutorial/node25.html (322 words)

	Bibliography Related to Geometry
		At the same time, he uncovers new geometric principles and new applications of geometry that may have a major influence on the state of architecture today.
		Hersey, George L. Architectgure and Geometry in the Age of the Baroque.
		The beginning of the book contains a discussion the geometry needed for the construction of the altars this beginning section is apparently the oldest surviving geometry textbook.
	www.math.cornell.edu /~dwh/biblio (1492 words)

	Amazon.com: Geometry: Books: David A. Brannan,Matthew F. Esplen,Jeremy J. Gray (Site not responding. Last check: 2007-11-01)
		The approach is that of Klein in his Erlangen program: a geometry is a space together with a set of transformations of that space.
		In the eighth chapter all of these geometries are demonstrated to be special cases of the Kleinian vieuw of geometry: that is, every geometry can be seen as consisting of the invariants of a specific group of transformations of the 2 dimensional plane into itself.
		And, by passing to the more abstract Projective geometry, you can express the abstract idea of 'conic' by giving just one quadratic curve, be it a parabola, ellipse or hyperbola, by the pair (Qu, P), whereby P is the group of all projective transformations.
	www.amazon.com /Geometry-David-A-Brannan/dp/0521597870 (2941 words)

	Developing Connections Between Algebra and Geometry,
		In general an affine transformation in an invertible linear transformation which preserves collinearity, betweeness, parallelism and ratio of division of segments.
		Finally, the equations describing affine transformations are linear, allowing for the efficient use of matrix algebra in the analysis of these equations and the subsequent use of technology.
		From the prior description, an affine transformation is a function defined by x'=ax+by+h and y'= cx+dy+k where a,b,h,c,d,k are real numbers and ad-bc does not equal zero.
	faculty.millikin.edu /~damiller/affinepaper.htm (1385 words)

	Geometry and Topology - Cambridge University Press
		Geometry provides a whole range of views on the universe, serving as the inspiration, technical toolkit and ultimate goal for many branches of mathematics and physics.
		The discussion moves from Euclidean to non-Euclidean geometries, including spherical and hyperbolic geometry, and then on to affine and projective linear geometries.
		Topology combines with group theory to yield the geometry of transformation groups,having applications to relativity theory and quantum mechanics.
	www.cambridge.org /catalogue/catalogue.asp?isbn=0521613256 (288 words)

	The Math Forum - Math Library - Transfrmtnl Geom. (Site not responding. Last check: 2007-11-01)
		Geometry software for Euclidean, coordinate, transformational, analytic, and fractal geometry, recommended for students from grade 5 through college.
		MapPoint is a geometry program that combines the traditional geometric constructions of classical Euclidean geometry with the more dynamic geometric transformations of modern geometry.
		The Geometry Project of the Month is a regular feature of the Math Forum, giving middle and high school students an opportunity to answer questions and receive feedback and recognition from the Forum staff.
	mathforum.org /library/topics/transform_g (2192 words)

	CRM: Thematic Programme 2001-2002
		The Escher Fish (by Silvio Levy) is reproduced by courtesy of the Geometry Center of the University of Minnesota.
		The study of 3-manifolds through their fundamental groups and symmetries has turned out to be a particularly rich vein withapplications to such topics as the tabulation of knots, geometrization problems, group actions, and surgery theory.
		Geometry intervenes in a crucial fashion in the proof of such results as the Kazhdan-Lusztig conjecture, the construction of canonical bases for representations, and the work of Beilinson-Drinfeld on the Geometric Langlands program.
	www.crm.umontreal.ca /act/theme/theme_2001-2002_an.html (1592 words)

	Re: Klein's Definition of Geometry
		Symplectic geometry turns out be essential for important parts\nof dynamical systems (we are back to differential equations here!) as well\nas many parts of physics.\n\nA few remarks about the sketch diagram of the hierarchy of plane\ngeometries might help convey the flavor.
		In euclidean geometry we have of course an obvious two\npoint invariant, and given m, the ring of m point invariants is generated\nby the pairwise squared distances.
		We can\nsay that -euclidean distance geometry is "more rigid than" euclidean\nsimilarity geometry-, because -it admits more invariants-.\n\nThis was one of Klein\'s greatest insights: we have a whole partially\nordered hierarchy of geometries, with the increasing "rigidity"\ncorresponding to inclusion and quotient relations among the releveant\ninvariant rings of the symmetry groups.
	www.physicsforums.com /showthread.php?t=87833 (8488 words)

	[No title]
		Modern Geometry was written with a particular audience in mind: Undergraduate mathematics education majors and practicing secondary school mathematics teachers.
		The Modern Geometry Technology Resources have been provided because of my strong belief that having convenient access to modeling tools and information resources such as these empowers students to think, act, and communicate mathematically in new and powerful ways.
		Modern Geometry was written to empower undergraduate mathematics education majors in this manner and to encourage teachers of mathematics to make similar opportunities available to their own students.
	www.math-ed.com /Resources/mgnew/mgscript.htm (464 words)

	Geometry
		31-04-01/02 Geometry AB The major purpose of geometry is to present patterns which are important to the development of thinking skills and problem-solving skills.
		The students will work with the body of geometry theorems, including geometry of two and three dimensions.
		The major purpose of geometry is to present patterns which are important to the development of thinking skills and problem-solving skills.
	www.lausd.k12.ca.us /Jefferson_HS/lessons/elittrel/Geometry.htm (777 words)

	Joseph Malkevitch: Geometry in Utopia
		In light of this, and the fact that unlike many other parts of mathematics, geometry often requires relatively little background to get started, it sad how little awareness there is either within the mathematics community and certainly outside of it of the vast domains of geometry.
		Geometry can be thought of as being built up from many parts (not all incorporated below) even though these parts overlap greatly.
		My goal is to encourage mathematicians at all levels to explore the exciting world of geometry and show more of its wonders to their students.
	www.york.cuny.edu /~malk/utopia.html (5458 words)

	Geometry Glossary
		- a type of composite transformation where a figure is reflected and then translated in a parallel direction
		- a transformation that keeps the same size and shape of a figure but moves it to a new location; see reflection, rotation, translation, glide reflection
		- a transformation that is the composite of size changes and/or reflections
	library.thinkquest.org /2647/geometry/glossary.htm (3915 words)

	Computing Papers on Geometry (Site not responding. Last check: 2007-11-01)
		The equivalence of rst-order rigidity in Euclidean, hyperbolic and spherical geometries, with W. Whiteley.
		The Geometry of a river bed surface is determined by the arrangement and size of the particles that form the bed.
		Regular Geometry to which the nodes are mapped, this Geometry guides which nodes of the level are to be joined.
	computing.breinestorm.net /Geometry (1879 words)

	Numerical Identity, Object identity, transformation geometry, Felix Klein
		As discussed, and reiterated by Sharon, the greater the amount of transformation (as defined by the Klein hier-archy) between the two modalities, the greater the amount of learning there was in an adaptation-like paradigm, even though there was only one pair of samples on each trial.
		For instance, a mapping in which a visual square was turned into a proprioceptive rectangle lead to less adaptation than a mapping in which the visual square became a smaller square in proprioceptive space.
		The man-woman transformation Sharon mentions, for example, can be judged as two people and not one with help from such specialized knowledge that, except in rare medical facilities, men do not turn into woman and surely cannot even cross-dress so quickly.
	www.u.arizona.edu /~bedford/papers/proofs.htm (4647 words)

	NLVM 3 - 5 - Geometry Manipulatives
		Virtual manipulatives related to the NCTM Geometry standard for grades 3 - 5.
		– Explore the effect of applying a composition of translation, rotation, and reflection transformations to objects.
		– Dynamically interact with and see the result of a reflection transformation.
	nlvm.usu.edu /en/nav/category_g_2_t_3.html (264 words)

	Transformation Geometry (Site not responding. Last check: 2007-11-01)
		For this reason, the metricity condition (13) holds also after a geometry transformation ...
		Motivate students and improve math grades with puzzler a game teaching transformation geometry.
		A mapping is a transformation if and only if it is a one-to-one mapping of...
	www.transformationgeometry.info (269 words)

	BRUNNERMATH -> your math connection
		Geometry quizzes and tests from a geometry class in Ohio
		From the University of Mississippi, Geometry challenge problems for students to try and solve.
		Interesting array of geometric activities that are on the fringes of normal geometry.
	www.brunnermath.com /geometry.htm (1331 words)

	Mathematics 138 (Classical Geometry) Home Page
		An introduction to spherical, Euclidean and hyperbolic geometry in two and three dimensions, with an emphasis on the similarities and differences between these flavors of geometry.
		The most important tool in analyzing these geometries will be a study of their symmetries; we will see how this leads naturally to basic notions in group theory and topology.
		Homework 3 - spherical geometry; available as a.tex file, or as a.pdf file.
	www.its.caltech.edu /~dannyc/courses/138/138.html (417 words)

	Basic Library List-Geometry
		Geometry, Relativity, and the Fourth Dimension Mineola, NY: Dover, 1977.
		Transformations and Geometries Englewood Cliffs, NJ: Appleton-Century-Crofts, 1969.
		Combinatorial Geometry in the Plane New York, NY: Holt, Rinehart and Winston, 1964.
	www.maa.org /BLL/geometry.htm (2473 words)

	Xah: Special Plane Curves: Curve Bibliography
		Casey, J., A Treatise on the Analytic Geometry of the Point, Line, Circle, and Conic Section, London (1883).
		Zwikker, C.; The Advanced Geometry of Plane Curves and their Applications; Dover Publications, N.Y.; 1963; Jeffrey, A., "On the determination of the envelope of a family of epitrochoids with applications" SIAM J. Applied Math.
		Bridge, Treatise on the Construction, Properties, and Analogies of the Three Conic Sections (from the 2nd London edition), jointly published by Durrie and Peck, New Haven and Collins, Keese, and Co., New York, 1839.
	xahlee.org /SpecialPlaneCurves_dir/Intro_dir/curveBibliography.html (1192 words)

	geometrystuff (Site not responding. Last check: 2007-11-01)
		Neutral geometry: the consistency of the hyperbolic parallel postulate and the inconsistency of the elliptic parallel postulate with neutral geometry.
		A graduate level class.In this course we propose to study first the basics of symmetry groups in geometry and especially crystallographic groups and their relationship to physical crystals, in particular, the symmetry restrictions on lattices in Euclidean three-space, and at least an introduction to the classification of crystallographic symmetries in two and three space dimensions.
		The Computational Geometry in C FTP Site contains the source that accompanies Joseph O'Rourke's book, the GEOMPACK FTP Site contains Fortran source, the Graphics Gems FTP Site has source for 2D and 3D geometry in C and C++, and The Stanford GraphBase is a set of C literate programming sources.
	www.math.tamu.edu /~bsong/geometry.htm (1534 words)

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