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Topic: Coordinate rotations and reflections

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Coordinate rotation

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	Geometry and Trigonometry (Site not responding. Last check: 2007-10-18)
		Coordinates are used to quantify distance, slope of lines, and to express the numeric representation of the relation of the slopes of two perpendicular lines.
		Coordinates are further used to model isometries and size transformations and their compositions.
		Coordinate models of points allow matrices to be introduced as another way to represent a polygon and a transformation that leaves the origin fixed.
	www.wmich.edu /~coreplus/parentsupport/geometry.html (980 words)

	Reflections
		Examples: right/left hands (by laying their hands on the table in front of them, this example of a reflection is something that students could easily look at to visualize a reflection), isomers of drugs (this example could be used if students wanted to know a “real world” application).
		Introduction of reflections on the coordinate plan (may not be until day 2) through use of technology.
		From our generalization that during a reflection over the line y=x the x and y coordinates are just switched, we know that two points (a,b) and (c,d) have images (b,a) and (d,c) respectively.
	www.msu.edu /~samarasi/project/reflections.html (623 words)

	Rigid Rotations
		In other words, the first coordinate is mapped onto the top row of m, the second coordinate is mapped onto the second row of m, and so on.
		We already showed the inverse of such a matrix is its transpose, and the inverse of a rigid rotation preserves lengths and angles, hence the inverse, or transpose, of an orthonormal matrix is orthonormal.
		The group of rotations and reflections defines the dihedral group on the circle.
	www.mathreference.com /la-det,rot.html (1171 words)

	NRaD TD 2780 Text (2)
		Rotations should never be used when there are segments on the z-axis or when crossing the z-axis, since overlapping segments would result.
		Rotations produce the same effect on the structure as the Transformations if the Number in Array is equal to the Number of New Structures + 1, and if the Z Rotation is equal to 360/(Number of New Structures + 1) degrees.
		Reflections should never be used when there are segments located in the plane about which reflection would take place or when crossing this plane.
	www.spawar.navy.mil /sti/publications/pubs/td/2870/nradtd2870txt2.html (6103 words)

	Rotation (mathematics) - Wikipedia, the free encyclopedia
		Rotations about the origin are most easily calculated using a 3×3 matrix transformation called a rotation matrix.
		Rotations about another point can be described by a 4×4 matrix acting on the homogeneous coordinates.
		In special relativity a Lorentzian coordinate rotation which rotates the time axis is called a boost, and, instead of spatial distance, the interval between any two points remains invariant.
	en.wikipedia.org /wiki/Coordinate_rotation (568 words)

	Coordinates and similar figures
		Coordinates are pairs of numbers that are used to determine points in a plane, relative to a special point called the origin.
		Point D has coordinates (9,-2.5); it is 9 units to the right, and 2.5 units down from the origin.
		Point E has coordinates (-4,-3); it is 4 units to the left, and 3 units down from the origin.
	www.mathleague.com /help/geometry/coordinates.htm (591 words)

	Coordinate rotations and reflections - Wikipedia, the free encyclopedia
		On the other hand, the composition of a reflection and a rotation, or of a rotation and a reflection (composition is not commutative), will be equivalent to a reflection.
		Let these rotations and reflections operate on all points on the plane, and let these points be represented by position vectors.
		Rotation matrices have a determinant of +1, and reflection matrices have a determinant of −1.
	en.wikipedia.org /wiki/Coordinate_rotations_and_reflections (379 words)

	Tilted and/or decentered surfaces
		It is useful to think of each surface having two (right-handed) coordinate systems: a base coordinate system, and a local coordinate system that is tilted and possibly decentered with respect to the base coordinate system.
		The base coordinate system normally has its origin on the z-axis of the previous system and is separated from it by the thickness of the previous surface.
		The bend command automatically sets the optical axis after reflection to be coincident with the ray that connects the previous vertex with the current vertex, i.e.
	www.sinopt.com /software1/usrguide54/enterdat/tiltdec.htm (976 words)

	Antenna Apherical Coordinate Systems
		The rotation of the sphere with the AUT for this coordinate system is apparent in Figure 4 where the AUT has been rotated in both Azimuth and Elevation.
		In the specific case of a rotated coordinate system, there is a compelling reason to use a specific coordinate system and a specific set of vector components as the "initial" or "natural" set.
		This was done for all three of the azimuth beam rotations and the four far-field measurement results for each beam rotation were averaged to reduce the effect of the multiple reflections.
	www.nearfield.com /amta/Amta99_0_an-gh.htm (2537 words)

	Group Theory and Physics
		Systems were also seen to be described by functions of position that are subject to the usual symmetry operations of rotation and reflection, as well as to others not so easily described in concrete terms, such as the exchange of identical particles.
		If we had taken the coordinate axes in some arbitrary position, we would have obtained six rather full matrices and the reducibility with a fortunate selection of axes would not be at all obvious.
		In fact, the Hamiltonian is invariant under the four-dimensional rotation group (actually, a group isomorphic to the four-dimensional rotation group), and its irreducible representations explain the added degeneracy, which is really not accidental at all.
	www.du.edu /~jcalvert/phys/groups.htm (5735 words)

	Reflection (mathematics) - Wikipedia, the free encyclopedia
		In order to reflect a planar figure one needs the "mirror" to be a line ("axis of reflection"), while for reflections in the three-dimensional space one would use a plane for a mirror.
		Geometrically, to find the reflection of a point one drops a perpendicular from the point onto the line (plane) used for reflection, and continues the same distance on the other side.
		Since these reflections are isometries of Euclidean space fixing the origin they may be represented by orthogonal matrices.
	en.wikipedia.org /wiki/Reflection_(mathematics) (457 words)

	Math B - Reviewing Transformations
		When you reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite.
		When you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is transformed into its opposite.
		While any point in the coordinate plane may be used as a point of reflection, the most commonly used point is the origin.
	regentsprep.org /Regents/mathb/3D1/reviewTranformations.htm (317 words)

	Math Tools Browse
		Investigate reflections, translations, glide reflections, dilations, rotations of 90, 180, 270 or any degree, and composition of reflections.
		A rotation around a point is one of three types of rigid transformations.
		Students will predict the effect of a rotation through a given angle and even the effect of two or more rotations performed one after the other and find angles that leave a figure unchanged.
	mathforum.org /mathtools/cell/m7,9.12.9,ALL,ALL (480 words)

	WULFFMAN - CTCMS
		Each rotation axis is defined as a rotation by 360/n degrees, where the user supplies n and the direction around which the rotation occurs.
		If facets are entered in HKL coordinates, a coordinate transformation must be performed on the facets to change them into their appropriate Cartesian values.
		Rotations in such a group are restricted to being of order 1, 2, 3, 4 and 6.
	www.ctcms.nist.gov /wulffman/docs_1.2 (4384 words)

	book2mod4
		rotation - A turn of a figure about a fixed point, the center of rotation, a certain number of degrees either clockwise or counterclockwise.
		rotational symmetry - When a figure can be rotated less than 360 about its center and fit exactly on itself, the figure has rotational symmetry.
		rotational symmetries - The number of degrees less than 360 that a figure can be rotated and fit exactly on itself.
	www.kent.k12.wa.us /staff/DavidChesley/book2mod4.htm (524 words)

	History
		Changed subroutine aptaxis to find center of symmetry only after all coordinate rotations and reflections, to correct bug in analysis of certain parabolic and hyperbolic cylinders.
		Added aptrois to recalculate the coefficients of the implicit equation of a quadric surface due to operating with a 3x3 matrix operator, representing reflection, rotation, inversion, and/or scaling, relative to an invariant point.
		Added aptpoly to generate regular polygons in a major plane, given the number of edges and the coordinates of the center, one vertex, and another point in the plane of the polygon.
	nuclear.llnl.gov /CNP/apt/aptLog.html (7983 words)

	Chapter 5
		On the plane or on spheres rotations and reflections are both intrinsic and extrinsic (in the sense that they are also symmetries of the plane or sphere), and thus they are particularly easy to see.
		In addition, on the plane and sphere all rotations and reflections are global in the sense that they are symmetries of the whole space (the plane or sphere).
		On cylinders and cones, intrinsic rotations and reflections exist locally because cones and cylinders are locally isometric with the plane.
	www.math.cornell.edu /~dwh/books/eg00/00EG-05 (3493 words)

	4.2.4 Rotations and Euclidean Motions (Site not responding. Last check: 2007-10-18)
		Thus in coordinates this linear mapping is given by the unitarian matrix
		is a reflection at the plane spanned by
		Consider an airplane or an hang-glider: We have the basis given by the axes of airplane: the direction from the left to the right wing, the vertical direction, and the direction from back to front.
	www.mat.univie.ac.at /~kriegl/Skripten/CG/node64.html (386 words)

	Welcome to Florida Virtual School
		To specify points by their coordinates in the coordinate plane.
		To write and use the equation of a circle in the coordinate plane.
		including reflections in the coordinate plane, and their properties.
	www.flvs.net /students_parents/course_descr/cd_geometry.php (759 words)

	CS184 Lecture 6 summary
		So R is an explicit representation for a coordinate rotation, and vice versa.
		That follows because applying R to both the p and the coordinates [X' Y' Z'] moves the latter to normal coordinates [X Y Z], from which point the coordinates of p can be read off.
		A general 3D coordinate frame can be described by the directions of its axes X' Y' Z' and the position of its origin t.
	www.cs.berkeley.edu /~jfc/cs184f98/lec6/lec6.html (398 words)

	The Identity element, Reflections to Rotations.
		Any reflection can be seen as an reflection trough a coordinate plane if the an adequate coordinate system is chosen.
		This means that rotations and reflections differ, and could not ever be performed in just one step using the other.
		If the reflection is making the new set coincide with the old then it will be a set of exchanges pairs of points.
	hemsidor.torget.se /users/m/mauritz/math/alg/refrot.htm (577 words)

	ACT Test Preparation - Mathematics Test (Site not responding. Last check: 2007-10-18)
		The questions are designed to measure your achievement of the mathematical knowledge, skills, and reasoning techniques; they cover a full range of math topics, from pre-algebra and elementary algebra through intermediate algebra, coordinate geometry, plane geometry, and even trigonometry.
		Coordinate geometry questions deal with the real number line and the (x,y) coordinate plane.
		Included are the properties and relations of plane figures; angles, parallel lines, and perpendicular lines; translations, rotations, and reflections; proof techniques; simple three-dimensional geometry; and measurement concepts like perimeter, area, and volume.
	www.onlinetestprep.com /engine/actmath.asp (571 words)

	Writing Rules for Rotations (Site not responding. Last check: 2007-10-18)
		Motivation: Have the class brainstorm real-life applications of rotations such as clocks, wheel of fortune, car wheels, and water wheels.
		Give some examples where the point of rotation is the origin and then make the problems progressively harder with the angle of rotation.
		Emphasize direction of rotation such as positive is counterclockwise.
	www.tcnj.edu /~michlik2/writing_rules_for_rotations.htm (320 words)

	Exploring Transformations
		use informal concepts of congruence to describe images after translations, rotations, and reflections.
		Create, analyse, and describe translations, rotation, and reflections of 2-D shapes.
		Draw 2-D shapes using ordered pairs (in all four quadrants) together with their translation and reflection images.
	argyll.epsb.ca /jreed/math7/strand3/3303.htm (238 words)

	[No title]
		A point group is a group of symmetries of an object where the elements of the group are restricted to rotations and reflections (elements which hold a point fixed).
		A space group is a group of object symmetries where the elements of the group include translations, glide reflections, rotations, and reflections.
		With the irreducible forms shown, it is easily seen that the x and y coordinates are associated with the irreducible representation Eand the z coordinate the trivial representation, A1.
	www.math.harvard.edu /archive/126_fall_98/papers/kuruvill.doc (1672 words)

	Gale Schools - Lesson Plans - Secondary: Math - Ecology
		Students will create a 10 foot X 10 foot model rectangular (Cartesian) coordinate axis system in sample area outside using string. Students will toss several cut-out polygons into the coordinate area. Students will collect sample species from the area where the polygon lye. All results will be sketched, labeled and recorded.
		At Level 2, the student is able to: Comprehend and Apply these concepts: translations, rotations, reflections, Cartesian Systems, polygons, coordinate geometry, area estimates using graph paper, sampling types, congruency, scale factors and scale drawings.
		At Level 3, the student is able to: Synthesize new ideas related to and Evaluate concepts: translations, rotations, reflections, Cartesian Systems, polygons, coordinate geometry, area estimates using graph paper, sampling types, congruency, scale factors and scale drawings.
	www.galeschools.com /lesson_plans/secondary/math/ecology.htm (1349 words)

	[No title]
		Students should understand how multiple reflections behave when mirrors are parallel or when mirrors intersect.MaterialsBlank paper Markers Rulers Lab handouts (see enclosed) Geometry software such as Geometer Sketchpad or Cabri Computers or TI92 calculator TextbookContext for Unit and Course Reflections and rotations are two of the key transformations studied in this unit.
		There are numerous opportunities in this unit to review and reflect upon theorems learned earlier in the course.Progression of the lesson activities Introduction to Reflections Lab: Reflections in Parallel Mirrors (2 parts) Lab: Rotations See handouts enclosed.Assessment Informal: The teacher will circulate among the students during seatwork and labs to observe their progress.
		NCTM Geometry Standard for Grades 9—12: Apply transformations and use symmetry to analyze mathematical situations Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices; Use various representations to help understand the effects of simple transformations and their compositions.
	www.tjhsst.edu /~jlynn/TCubed2005.doc (1296 words)

	Untitled Document
		Summarizing Paragraph: This ‘Reflections and Translations’ lesson plan was designed to be implemented in both middle school mathematics classrooms and high school geometry classes.
		All students will understand the relationship between transformations in a coordinate plane and their application in real life.
		Students should be able to: Specify locations and describe spatial relationships using coordinate geometry and other representational systems: Students are required to describe the spatial similarities and differences among transformed figures and images.
	filebox.vt.edu /k/kimiller/Technologylesson.htm (812 words)

	Quantum Gravity Concept Map - Symmetries
		A field configuration (or field equations) are "symmetric" under a "transformation" when the transformation leaves the configuration (or form of the equations) unchanged.
		For example, a field is "symmetric with respect to rotations in three dimensions" or "spherically symmetric" if it is unchanged when the spatial coordinates are rotated in any direction.
		Manifolds with multiple differential structures (like the 7-dimensional sphere) have multiple, mutually-exclusive equivalence classes of metrics, which are characterized by different, independent definitions of volume (since the volume form changes by a factor of the Jacobian under coordinate transformations).
	www.rwc.uc.edu /koehler/qg/sym.html (593 words)

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