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Topic: Glide reflection

	Glide Reflection
		Glide reflection is a composite transformation which is a translation followed by a reflection in line parallel to the direction of translation.
		Reflection is isometry: a glide reflection preserves distances.
		The importance of the glide reflection lies in the fact that it is one of the four isometries of the plane.
	www.cut-the-knot.org /Curriculum/Geometry/GlideReflection.shtml (269 words)

	The Seventeen Wallpaper Groups
		The length of the glide vector is equal to the length of one of the edges parallel to the mirror of reflection.
		The length of the glide vector is the length of one of the edges parallel to its mirror.
		Let a and b be translations and g1 and g2 be glide reflections, g1 parallel to a with lenght 1/2 a and g2 parallel to b with lenght 1/2 b, and r1, r2, r3, and r4 be rotations of 180 degrees at each vertex of the fundamental region numbered as before.
	jwilson.coe.uga.edu /EMT668/EMAT6680.F99/McCallum/WALLPA~1/SEVENT~1.HTM (3603 words)

	Symmetry in Oriental Carpets
		A glide reflection is a combination of a translation and a reflection.
		Glide reflections are unique from the other three basic symmetries because they involve two steps.
		The seven border groups are translation, horizontal reflection, vertical reflection, reflection and reflection, glide reflection, rotation, reflection and glide reflection.
	www.ccds.charlotte.nc.us /2005hpc12/symmetry_in_oriental_carpets.htm (2212 words)

	Glide reflection - Wikipedia, the free encyclopedia
		In geometry, a glide reflection is a type of isometry of the Euclidean plane: the combination of a reflection in a line and a translation along that line.
		Combining two equal glide reflections gives a pure translation with a translation vector that is twice that of the glide reflection, so the even powers of the glide reflection form a translation group.
		If the translation vector of a glide reflection is itself an element of the translation group, then the corresponding glide reflection symmetry reduces to a combination of reflection symmetry and translational symmetry.
	en.wikipedia.org /wiki/Glide_reflection (395 words)

	Glide Reflection and other Combined Transformations
		Glide Reflection is one of the four (translation, rotation, reflection and glide reflection) symmetrie transformations we use to classify the regular divisions of the plane.
		In a way glide reflection is somewhat different from the other three, because it’s not a simple tranformation.
		The shapes of the light and dark horseman are the same apart from the operation of reflection.
	www.mi.sanu.ac.yu /vismath/roelofs/index.html (1035 words)

	Glide reflection (via CobWeb/3.1 planetlab1.tamu.edu) (Site not responding. Last check: 2007-10-22)
		In geometry, a glide reflection is a type of isometry of the Euclidean plane.
		On the other hand the product of a reflection and a translation is usually not a reflection, but can produce a transformation with no everyday name: a glide reflection.
		If one considers the effect of a reflection combined with any translation, it is a glide reflection with respect to a line parallel to the line of the reflection, as one sees by resolving the translation into components parallel and orthogonal to that line.
	publicliterature.org.cob-web.org:8888 /en/wikipedia/g/gl/glide_reflection.html (227 words)

	Isometries As Products of Reflections (Site not responding. Last check: 2007-10-22)
		The four Euclidean isometries, reflection, translation, rotation and glide reflection, can be expressed as compositions of reflections.
		As glide reflections is a combination of translation and reflection we can see from the above discussion that a glide reflection can be expressed as the composition of three reflections.
		The first two reflections compose the translation step of the glide reflection map, and the third reflection is the same reflection as the second step of the glide reflection map.
	www.geom.uiuc.edu /~crobles/hyperbolic/eucl/iref (347 words)

	Wallpaper Groups: the 17 plane symmetry groups (Site not responding. Last check: 2007-10-22)
		The direction of the glide reflection is parallel to one axis of translation and perpendicular to the other axis of translation.
		The lattice is square, and an eighth, a triangle, of a fundamental region for the translation group is a fundamental region for the symmetry group.
		The axes of the reflections are again inclined at 60° to one another, but for this group all of the centers of rotation lie on the reflection axes.
	www.clarku.edu /~djoyce/wallpaper/seventeen.html (970 words)

	Glide Reflection Exercise - Math 356 (via CobWeb/3.1 planetlab1.tamu.edu) (Site not responding. Last check: 2007-10-22)
		A glide reflection can be described by a glide mirror line together with a translation by a vector in the direction of this line.
		The glide reflection then can also be thought of as reflection in the glide mirror line followed by this translation.
		The glide mirror line of this glide reflection is shown in purple on the sketch.
	www.mathlab.cornell.edu.cob-web.org:8888 /computer_and_portfolio/geometry/refthree (537 words)

	Glide plane - Wikipedia, the free encyclopedia
		In crystallography, a glide plane is symmetry operation describing how a reflection in a plane, followed by a translation parallel with that plane, may leave the crystal unchanged.
		In geometry, a glide plane operation is a type of isometry of the Euclidean space: the combination of a reflection in a plane and a translation in that plane.
		The glide plane operation in the strict sense and the pure reflection are two of the four kinds of indirect isometries in 3D.
	en.wikipedia.org /wiki/Glide_plane (413 words)

	Plane Isometries (Site not responding. Last check: 2007-10-22)
		An improper isometry is either a reflection or a glide reflection [Coxeter, Yaglom].
		The axis of reflection is simply the perpendicular bisector of the segment joining P and f(P).
		The product of two reflections is either a reflection or a rotation; for each of these two reflections can be found (and not even uniquely) whose successive applications result in a given transformation.
	www.cut-the-knot.org /pythagoras/Transforms/index.shtml (871 words)

	Pattern Lesson 3 Art Part
		Mirror reflection is a special case of glide reflection.
		Mirror reflection is glide reflection with zero glide, as translation is a special case of rotation.
		Thus, the four symmetry operations are reduced to two operations--rotation and glide reflection.
	www.dartmouth.edu /~matc/math5.pattern/lesson3art.html (1935 words)

	Reflect and Glide
		A reflection of a neutral plane is an isometry.
		The inverse of a reflection of a neutral plane is the reflection itself.
		A glide reflection of a Euclidean plane is an isometry.
	www.mnstate.edu /peil/geometry/C3Transform/4Indirectw.htm (618 words)

	Illuminations: Symmetries IV
		Glide reflections are very tricky to identify in a tiling or wallpaper pattern.
		A glide reflection is a symmetry transformation that consists of a translation followed by a reflection across the translation line.
		In a glide reflection, the translation is always a translation along the mirror line of the reflection.
	illuminations.nctm.org /index_d.aspx?id=475 (799 words)

	Isometries of the plane
		A glide reflection (or glide) is a reflection in a line followed by a translation in a direction parallel to that line.
		Note that glide reflections are really the most general form of an opposite symmetry.
		A reflection could be considered as a glide where the translation happens to be trivial.
	www-groups.dcs.st-and.ac.uk /~john/geometry/Lectures/L6.html (456 words)

	Math Forum: Types of Symmetry in the Plane
		A glide reflection combines a reflection with a translation along the direction of the mirror line.
		The reflection of a figure in the plane about a line moves its reflected image to where it would appear if you viewed it using a mirror placed on the line.
		That is, pick a reflection line and a translation in a direction parallel to the reflection line.
	mathforum.org /sum95/suzanne/symsusan.html (946 words)

	Reflection in a Line (via CobWeb/3.1 planetlab1.tamu.edu) (Site not responding. Last check: 2007-10-22)
		In a glide reflection, the line of reflection is parallel to the direction of the translation.
		Since translations and reflections are both isometries, a glide reflection is also an isometry.
		A glide reflection is a transformation in the plane that is the composition of a line reflection and a translation through a line (a vector) parallel to that line of reflection.
	www.regentsprep.org.cob-web.org:8888 /Regents/math/geometry/GT6/greflect.htm (179 words)

	Glide reflection, 3dfx glide, glide xp (via CobWeb/3.1 planetlab1.tamu.edu) (Site not responding. Last check: 2007-10-22)
		Reflection in a Line; A Glide Reflection: translation parallel to a line glide reflection.(Any plane symmetry is a translation, a reflection, a glide-reflection or a.
		Glide reflections are the transformations used to describe the symmetry we seein footprints¡ªfirst translate and then reflect glide reflection.
		It consists of glide reflections and translations and is generated by a minimumglide reflection.
	www.teensage.com.cob-web.org:8888 /glide/glide_reflection.html (323 words)

	Some Glide Reflection Facts (Site not responding. Last check: 2007-10-22)
		Let the glide transformation with invariant line m and glide vector AB be the transformation G that takes any point C to a point C'' defined as follows: let C' be the reflection of C across m and C'' be the translation of C' by vector AB.
		Proposition: If G is a glide reflection, for any point C, the midpoint of the segment from C to G(C) is on the invariant line m.
		However, knowing the 3 line reflections still leaves quite a bit of work to find the invariant line and the glide vector (Brown, problem 19, page 54 and class discussion). If a transformation G is known to be a glide reflection, then the midpoint method above is one way to find this defining data directly.
	www.math.washington.edu /~king/coursedir/m444a02/class/12-12-glideref.html (378 words)

	Transformation (mathematics) - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab1.tamu.edu) (Site not responding. Last check: 2007-10-22)
		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.
		Reflection sometimes is considered as a special case of inversion with infinite radius of the reference circle.
		A glide reflection is a type of isometry of the Euclidean plane: the combination of a reflection in a line and a translation along that line.
	en.wikipedia.org.cob-web.org:8888 /wiki/Transformation_(mathematics) (621 words)

	Survey of Math
		In glide reflection, each successive element in the design in a reflection of the previous one.
		The first reflection symmetry is a line joining corners B and C. Notice how the shape looks the same after the reflection--we could not tell that a reflection had been performed except that the labelling of the corners has changed.
		The second reflection symmetry is a line joining corners A and D. These two reflection symmetries all the only reflection symmetries of the object.
	cda.morris.umn.edu /~mcquarrb/SurveyofMath/Resources/Lecture19b.html (1032 words)

	[No title]
		Extending the previously mentioned result on reflections, it is possible to show that the composition of two non-parallel glide reflections is a rotation by twice their angle of intersection...
		More specifically, everything follows from the easily proven fact that the composition of a reflection M and a half turn R at a distance d from M is a glide reflection G passing through R, perpendicular to M and of gliding vector of length 2d.
		While the interactions between (glide) reflection axes and half turn centers that determine this number are best studied in the 'abstract' context of symmetry plans, it is more convenient here to start with a specific pg/pm/cm pattern in the vertical direction and see what the half turn creates out of it in the horizontal direction.
	www.oswego.edu /~baloglou/103/seventeen.html (2841 words)

	Basic Tilings: The 17 Wallpaper Groups
		This kind of tiling has a glide reflection; this means that the tiling can be completely unchanged if it is both reflected and displaced at the same time.
		This tiling has both reflections and glide reflections, the axes of which are parallel.
		As a result, it also has a glide reflection, and the cell must be square or rectangular.
	www.quadibloc.com /math/tilint.htm (713 words)

	Glide Reflection Worksheet
		Given that the two congruent figures below are glide reflection images of each other, draw three lines (m, n and p) of reflection that could be used to accomplish the glide reflection.
		In general, would you expect to be able to rewrite the composition of two different glide reflections as a reflection, a rotation, a translation, or another glide reflection?
		In the glide reflection, do you think it matters whether the translation is performed first, before the reflection, or whether the reflection is performed first, before the translation?
	www.public.asu.edu /~royal/glide_reflection_worksheet.htm (232 words)

	Seminole Patchwork Topics
		The four rigid motions are reflection, rotation, translation, and glide reflection.
		Reflections in strip patterns can exist in two ways, in the direction of the pattern, parallel to the pattern, or perpendicular to the direction of the pattern.
		The third position in the notation denotes whether there is a reflection parallel to the direction of the pattern or a glide reflection.
	www2.austin.cc.tx.us /hannigan/Presentations/NSFMar1398/SPTopics.html (1309 words)

	Your Heading Goes Here
		These types of symmetries are reflection, translation, rotation, and glide reflection.
		The last type of symmetry discussed in this lesson is a glide reflection.
		A glide reflection is a combination of reflecting (flipping) and translating (sliding).
	homepages.ius.edu /TEHURST (414 words)

	Chapter 1.7
		The glide reflection P is a transformation without invariant points, with an invariant line- the axis l of the glide reflection.
		Because this group is generated by a glide reflection P perpendicular to the reflection R
		reflections with reflection lines perpendicular to the translation axis, can be used.
	www.emis.de /monographs/jablan/chap17.htm (1999 words)

	Symmetries of Culture
		It may seem that it is redundant to introduce the idea of a glide reflection, since it is a combination of two of the other isometrics.
		Note again that the phrase "horizontal reflection" is used merely as an abbreviation for "reflection in the central axis of the infinite band", and "vertical reflection" refers to reflection in a line perpendicular to the direction of the band.
		For practice the reader may verify that the pattern of Figure 9a is of type pmg (reflections in vertical lines, and glide reflections in horizontal lines), while Figure 9b has type p4m, the same as an ordinary checkerboard.
	members.tripod.com /vismath6/crowe1 (6239 words)

	paperaddressDuality and Dual States: Glossary Definition
		Because the rigid motions require centers of rotation and axes of repetition or reflection for symmetry to be present.
		If this wave equation remains identical after simultaneous reflection of all spatial coordinates of the particle through the origin of the coordinate system, then it is said to have even parity.
		If such simultaneous reflection results in a wave equation that differs from the original wave equation only in sign, then the particle is said to have odd parity.
	www.math.vt.edu /people/gao/symmetry/gao.html (1449 words)

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