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Topic: Homothety

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	PlanetMath: homothetic
		The concurrence point of all lines is called the homothety center Notice that a necessary and sufficient condition for two similar polygons to be homothetic is that corresponding sides be parallel.
		is known as homothety ratio, and it coincides with the similarity ratio.
		is positive we speak of direct homothety and not only similarity is preserved, but pictures also have the same orientation (corresponding points to the upper part of one figure are on the upper part of the other).
	planetmath.org /encyclopedia/Homothetic.html (328 words)

	Homothety
		Homothety, translation, central similarity are the interchangeable terms used to describe a geometric transformation H
		Every homothety with k different from 1 has one and only one fixed point - the center O. Every line through O is also fixed although not pointwise.
		Two homotheties with a common center commute as a matter of course.
	www.cut-the-knot.org /Curriculum/Geometry/Homothety.shtml (202 words)

	homothety (Site not responding. Last check: 2007-10-22)
		In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point called the origin.
		The homothety maps any point to a point such that (as vectors).
		A homothety is an affine transformation and more precisely a similarity transformation.
	www.yourencyclopedia.net /Homothety.html (153 words)

	Manufacture of cushions - Patent 4046611
		In the transverse direction of a vehicle to be equipped with a cushion manufactured by the method of the invention, the homothety centre is located in a plane located at equal distances from the longitudinal borders of the cushion, the thickness of this cushion varying very little in the transverse direction.
		12, slots 30 allow the sliding of the cloth between two superimposed frames 8c and 9c to be limited, and all converge to a point 0 which is the homothety centre of two geometric locations 30c and 30d indicated by broken lines of the inner and outer ends 30a and 30b of the slots 30.
		The homothety centre 0 is spaced from the part of the corresponding frame at the front of the cushion which is substantially equal to twice its distance to the corresponding part of the frame corresponding to the rear of this cushion, i.e.
	www.freepatentsonline.com /4046611.html (3703 words)

	Fractal dimension
		This is a figure the Euclidean dimension of which is 1 (it is a broken line) and the fractal dimension of which is greater than 1 and, moreover, is not a whole number.
		The example of the von Koch snowflake is easy to understand because it is simple and because the ratio remains constant in the peculiar case I have chosen.
		True because for the fractal made of lines which are simple, as is the von Koch curve, the dimensions of homothety and of Hausdorff-Besicovitch are equal.
	fractals.iut.u-bordeaux1.fr /jpl/dimension_a.html (1246 words)

	Inversion
		Two touching circles (as in the problem) are related by a single homothety with their common point of tangency as the center.
		All points related by a homothety are collinear with its center B. In particular this is true of the lowest (A and M in the diagram) points of the two circles.
		Indeed, any center of homothety of the two circles could be used as the center of inversion.
	www.maa.org /editorial/knot/Circle.html (1970 words)

	[No title]
		Find the ratio of homothety as the ratio of the distances from the homothetic center EMBED Equation.2 to EMBED Equation.2 and EMBED Equation.2 (note that the second distance equals EMBED Equation.2 ).
		The ratio of homothety is equal to EMBED Equation.2 .
		This transformation is a composition of homothety with center EMBED Equation.2 and ratio EMBED Equation.2 , and symmetry with respect to the bisector of the angle EMBED Equation.2 .
	www.mccme.ru /olympiads/lktg/2002/problem3.en/solmain.doc (2485 words)

	Generalization of Napoleon's Theorem
		Central similarity (homothety with a fixed ratio and a center of homothety)
		In which case, we have a spiral similarity with a given center, a given angle, and a given ratio.
		Of course, rotation and homothety are particular cases of spiral similarity.
	www.cut-the-knot.com /Generalization/napoleon.shtml (891 words)

	The EDIT_MESH application (Site not responding. Last check: 2007-10-22)
		This means that after it is defined, a transformation will either apply to all subdomains or to the original subdomains (in continuous lines) or to explicitly selected subdomains.
		Defines the homothety of center the point and ratio equal to the value.
		This command is used to effectively generate all the transformed subdomains in order to edit them independently of their original.
	dragon.ian.pv.cnr.it /~modulef/GB/Guide3-18/node23.html (1382 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations (Site not responding. Last check: 2007-10-22)
		In a particular case when the transformation is a pure homothety, we find analytic solutions for the curvature and the torsion.
		In the general case, when the transformation is a superposition of a nontrivial rotation and a homothety, the asymptotics of the solutions of the first class are given explicitly and are related to the parameters characterizing the transformation.
		It is found that the solutions of the second class (with decreasing scale) either have asymptotes or are periodic (when the transformation is a pure homothety) or else exhibit chaotic behavior.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=20425444 (357 words)

	Class Plan (Site not responding. Last check: 2007-10-22)
		Euclidean Motions of the Plane: Isometries, Translations, Rotations, Reflections, and Homothety.
		Homothety: Given O a fixed point and k a nonzero number, H(O,k) represents a transformation of the plane that transforms a point P to P’ such that the segments OP and OP’ are related as follows, OP=k OP’.
		Show that the set of all translations in the plane is an abelian group.
	www.msci.memphis.edu /~botelhof/IV.html (232 words)

	Transformation (Site not responding. Last check: 2007-10-22)
		In reflection in a plane, each point P of S is carried into the point P1 of S such that PP1 is perpendicularly bisected by a fixed plane p of space, and the points of p are the invariant points of the transformation.
		In homothety, each point P of S is carried into the point P1 of S collinear with P and a fixed point O of space, and such that OP1/OP=k, where k is a nonzero real number.
		A homology is the product of a homothety and a rotation about an axis passing through the center of the homothety.
	www.cecm.sfu.ca /~hle/polyhedra/transformation.html (805 words)

	AoPS Math Forum :: View topic - Simson Line Property
		is the external homothety center of the circumcircle and the 9-point circle.
		is the external homothety center of the incircle and the 9-point circle.
		In this case, until you find the fixed intersections of the incircle with the diacentral line to be on the Simson lines with the poles identical with the endpoints of the diacentral diameter, it is like trying to nail a piece of slime to the wall.
	www.artofproblemsolving.com /Forum/post-189438.html (1944 words)

	Concyclic Circumcenters: A Sequel
		(It was also the centroid of the original triangle.) The triangles S and D are homothetic with the center of homothety at K. The six circumcenters are found at the intersections of the side lines of these two triangles.
		The bisectors also form two homothetic triangles with the center of homothety at G, the centroid of the base triangle.
		Expand that triangle to twice its size with homothety in K. In the new triangle, the base triangle is then found as the pedal triangle of K. What about the Tucker circles corresponding to the coefficients other than -1/2?
	www.maa.org /editorial/knot/sixcircum2.html (1373 words)

	Pure Mathematics - Steele Papers
		For corrections to this paper see the 1998 preprint Simply-Transitive Homothety Groups and the Ricci tensor, (Compressed Postscript).
		Steele, J.D., On Simply-Transitive Homothety Groups, Proceedings of the fourth Monash Workshop on Relativity, ed.
		Steele, J.D., A formalism for spacetimes with a homothety.
	web.maths.unsw.edu.au /~jds/papers.html (308 words)

	MathLinks Math Forum :: View topic - Sequel of Febuerbach (Site not responding. Last check: 2007-10-22)
		It is not so well formulated (actually the point K could be simply defined as the image of T in the homothety with center G and factor -2, i.
		Then, the point K' lies on the ray TG and on the circumcircle of triangle ABC (in fact, this is because the point T lies on the nine-point circle of triangle ABC, i.
		In other words, our point K is the image of the point T in the homothety with center G and factor -2.
	www.mathlinks.ro /Forum/post-112261.html (653 words)

	Team-ANUBIS (Site not responding. Last check: 2007-10-22)
		In more general geometries the family of surfaces sweeping over the domain may be more complicated.
		We have studied the case of a star shaped domain with a family of surfaces derived by homothety.
		We derived a new Riccati equation for that case which has been used for the computing zoom [29].
	www.inria.fr /rapportsactivite/RA2004/anubis2004/uid35.html (69 words)

	MathLinks Math Forum :: View topic - Thailand TST (Site not responding. Last check: 2007-10-22)
		This homothety then, of course, must map the midpoint of the segment WR to the midpoint of the segment EE'.
		Since the point E" is the center of this homothety, and since the center of a homothety always lies on the line joining a point with its image under that homothety, it thus follows that the point E" lies on the line joining the midpoints of the segments WR and EE'.
		Since the midpoint of a segment is the center of the circle which has this segment as diameter, we can restate this result as follows: The point E" lies on the line joining the centers of the circles with diameters WR and EE'.
	www.mathlinks.ro /Forum/topic-22633.html (3233 words)

	Homotecia y Semejanza (Site not responding. Last check: 2007-10-22)
		This unit deals with some of the basic concepts of homothety (or enlargement) and some formulae connected to Thales' theorem.
		There is an introduction to metric relations in a right-angled triangle: the theorem of the sides adjacent to the right angle and the perpendicular height theorem.
		Understand and know how to apply direct and inverse homothety to simple shapes.
	descartes.cnice.mecd.es /ingles/4th_year_secondary_educ_B/Similarity_&_homothety/index_Homo.htm (158 words)

	GOSSARD PERSPECTOR (Site not responding. Last check: 2007-10-22)
		Let Pa be the point such that line lines Y-to-Pa and BP are parallel and lines Z-to-Pa and CP are parallel.
		Then the triangle A'B'C' formed by the lines Ga-to-Pa, Gb-to-Pb, Gc-to-Pc is congruent and inversely homothetic to ABC, and the center of homothety (i.e., the perspector), is on the line GP.
		If P is the orthocenter in Yiu's generalization, then A'B'C' is the Gossard triangle and the center of homothety is the Gossard perspector.
	faculty.evansville.edu /ck6/tcenters/recent/gosspersp.html (396 words)

	Homothety shoot (Site not responding. Last check: 2007-10-22)
		A homothety of the plane is a zoom (magnification or reduction) of the plane.
		The center of a homothety is the unique point of the plane which stays unchanged after the homothety.
		It presents you two shapes, one being the image of the other after a homothety in the plane.
	wims.unice.fr /wims/it_H5~geometry~homothshoot.en.html (134 words)

	An generator equation for mumeral series as measurement langage mathched to the universe matter fineness. (Site not responding. Last check: 2007-10-22)
		Then, if thereis no homothety, change in matter should need ti be able to change the ever fixed Direction.
		On the figure, a length [BC] known as an irrationnel one is transformed by homothety of factor [k] from the center [A].
		However, of we are still talking about homothety an immutability of precise defined laws, this prove that no irrational length eever exist un the universe.
	www.dakhi.com /somen61.htm (8577 words)

	[No title]
		One is produced from the other by a homothety in the plane.
		To give your reply: click in the figure, at the place where you think is the good one.
		The center of the homothety is a point which lies on any line joining a point and its image by the homothety.
	wims.unice.fr /modules/H5/geometry/homothshoot.en/main.phtml (118 words)

	invcplx
		We will be interested in this article to the geometrical transformations, like complex homotheties, rotations, similarities and inversions.
		b B is a homothety of center O and ratio K. Example with the image by Z=3*z of y=x-1
		For a Rotation or Similarity, it is thus necessary to convert a complex expression into polar format.
	www.ti-cas.org /texas/method/invcpxus.htm (645 words)

	1.5.3 The construction application (Site not responding. Last check: 2007-10-22)
		The application is able to duplicate an object n times by using the classical affine transformations: symmetry, rotation, homothety.
		The application is able to invert arcs and segments.
		applies the homothety of center point and ratio %RATIO to the elements.
	www.math.psu.edu /local_doc/modulef/GB/Guide3-14/node21.html (2242 words)

	[No title]
		ÂÓ²Ø@Ü~II A0 B0 C0ÿÿÿÿÿÿÿÿ={¤ Ò ÿÿÿ.1 @À&ÿÿÿÿÀÿÿÿ¦ÿÿÿæ& MathType`û ÿÿÿÿÿ°ÿÿÿÿÿÿÿÿÿÿÿÿ‘“’”•——¦ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ§¨©ª«®q´þÿÿÿ±²³þÿÿÿµ¶¿¸¹º»¼½¾ÀÁÂÕÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿr of homothety lies on IO.
		The pairwise centers of external homothety of three circles from Problem 5 lie on a line t that is perpendicular to IO.
		Homothety with center EMBED Equation.3 and ratioEMBED Equation.3 is a geometrical transformation taking point M to EMBED Equation.2 by tÿÿÿÿÿÿÿÿýÿÿÿ‘’“”•—— ¡¢£¤¥¦§¨©ª«¬®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõ}÷øùúûüýþÿObjInfoÿÿÿÿÿÿÿÿÿÿÿÿOlePres000XZÿÿÿÿ ÌEquation Native ÿÿÿÿÿÿÿÿÿÿÿÿ\Ole ÿÿÿÿÿÿÿÿÿÿÿÿ&CompObj[^ÿÿÿÿ$fObjInfoÿÿÿÿÿÿÿÿÿÿÿÿ#OlePres000]_ÿÿÿÿÌEquation Native ÿÿÿÿÿÿÿÿÿÿÿÿ\Ole ÿÿÿÿÿÿÿÿÿÿÿÿ1CompObj`cÿÿÿÿ/fObjInfoÿÿÿÿÿÿÿÿÿÿÿÿ.OlePres000bdÿÿÿÿ(DEquation Native ÿÿÿÿÿÿÿÿÿÿÿÿ'4Ole ÿÿÿÿÿÿÿÿÿÿÿÿ?CompObjehÿÿÿÿ=fObjInfoÿÿÿÿÿÿÿÿÿÿÿÿ
	www.mccme.ru /olympiads/lktg/2002/problem3.en/main.doc (447 words)

	HexadivisionSymmetric (Site not responding. Last check: 2007-10-22)
		Draw lines AU, AT and their intersections H, G with BC respectively.
		Define the homothety with center A and ratio k = HG/UT. DEFGHI is the homothetical of QPVTUR under this homothety.
		The minimal hexagon is symmetric and its center of symmetry coincides with the triangle-center X(37).
	www.math.uoc.gr /~pamfilos/eGallery/problems/HexadivisionSymmetric.html (89 words)

	A shift between Dirichlet and Neumann spectrum for generalized linear elasticity (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		A shift between Dirichlet and Neumann spectrum for generalized linear elasticity
		Abstract: Introduction The operator of linear elasticity is a good example of an non scalar operator : its principal symbol is not an homothety.
		In the theory of Elasticity one studies the deformation due to displacement of solid bodies regarded as continuous media.
	citeseer.ist.psu.edu /188866.html (322 words)

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