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Topic: Homothetic transformation

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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)

	Homothetic transformation - Wikipedia, the free encyclopedia
		In mathematics, a homothety (or homothecy) is a transformation of space which dilates distances with respect to a fixed point A called the origin.
		Such a transformation is also called an enlargement.
		A homothety is an affine transformation (if the fixed point is the origin: a linear transformation) and also a similarity transformation.
	en.wikipedia.org /wiki/Homothety (153 words)

	PlanetMath: affine transformation
		An affine property is a geometry property that is preserved by an affine transformation.
		This is version 24 of affine transformation, born on 2004-10-24, modified 2006-07-18.
		I think the linear transformation in the definition of affine transformation is required to be invertible.
	planetmath.org /encyclopedia/AffineTransformation.html (379 words)

	kig: Transformation Class Reference (kig) (Site not responding. Last check: 2007-10-21)
		Construct a Transformation by using one of its static members, and use it either with its Transformation::apply method, or the ObjectImp::transform method.
		Returns a Transformation that transforms points in such a way that it appears to cast a shadow, given a certain light source, and a line indicating a plane.
		Returns a Transformation that transforms points in such a way that it appears to cast a shadow, given a certain light source (center), and a line (axis) indicating a plane.
	developer.kde.org /documentation/library/cvs-api/kdeedu-apidocs/kig/html/classTransformation.html (345 words)

	Affine transformation Did You Mean affine_transformation
		The invertible affine transformation form the affine group, which has the general linear group of degree n as subgroup and itself is a subgroup of the general linear group of degree n+1.
		Iff the determinant of A is 1 or -1 then the transformation preserves area; these also form a subgroup.
		To visualise the general affine transformation of the Euclidean plane, take labelled parallelograms ABCD and A?B?C?D? Whatever the choices of points, there is an affine transformation T of the plane taking A to A?
	www.did-you-mean.com /Affine_transformation.html (1023 words)

	Similarity (mathematics) Did You Mean similarity_(mathematics)
		One of the meanings of the terms similarity and similarity transformation (also called dilation) of a Euclidean space is a function f from the space into itself that multiplies all distances by the same positive scalar r, so that for any two points x and y we have
		A special case is a homothetic transformation or central similarity: it neither involves rotation nor taking the mirror image.
		One of the meanings of the term "similarity transformation" is such a transformation of a matrix A into a matrix B.
	www.did-you-mean.com /Similarity_03mathematics04.html (1137 words)

	Math Forum - Mathematics Teacher Bibliography: Transformational Geometry
		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.
	www.mathforum.org /mathed/mtbib/transformations.html (1004 words)

	An generator equation for mumeral series as measurement langage mathched to the universe matter fineness. (Site not responding. Last check: 2007-10-21)
		In a homothetic transformation, the irrational uncertainty is transmitted to the system and is amplified the more is it when in recursive changes.
		On the figure, a length [BC] known as an irrationnel one is transformed by homothety of factor [k] from the center [A].
		Because the basic knowledges of matter transformations that are the four addition, substraction, multiplication and division operations, are not known accurately on the Earth, particularly the multiplication and the division.
	www.dakhi.com /somen61.htm (9332 words)

	Generalization of Napoleon's Theorem
		Therefore, A' is obtained from A by a spiral similarity with center N, angle ANA' and ratio NA:NA'.
		Therefore, C is obtained from A by a spiral similarity with center N, angle ANC and ratio NA:NC.
		This similarity transforms the circle R into one traced by C. Remark
	www.cut-the-knot.org /Generalization/napoleon.shtml (883 words)

	All Elementary Mathematics - Study Guide - Analytic geometry - Transformations of coordinates... (Site not responding. Last check: 2007-10-21)
		Consider some transformations tied with a transition from one coordinate system to another.
		A homothetic transformation with a center O (a, b) and a coefficient k
		An affine transformation transfers straight lines to straight lines, intersecting lines to intersecting lines, parallel straight lines to parallel straight lines.
	www.bymath.com /studyguide/angeo/sec/angeo1.htm (188 words)

	Reconstruction of Forms by Linear Algebra by Elena Marchetti and Luisa Rossi Costa in the Nexus Network Journal vol. 7 ...
		Another important aspect of linear algebra to be pointed out to the students is its application in graphics software packages, which work with transformations that change the position, orientation and size of objects in a drawing.
		Now the students are ready to apply the right matrices to the crucial vectors, realizing the sequential steps (that is, the plane transformations) necessary to rebuild the shape virtually so that the whole object gradually emerges, step by step.
		The same plane transformation is also evident in Fig.2, the virtual reconstruction of Itten's drawing entitled The white man's house (1920), shown in Fig.3.
	www.univie.ac.at /EMIS/journals/NNJ/MarRos-v7n1.html (2559 words)

	Homotecia (Site not responding. Last check: 2007-10-21)
		When k>0 the transformation is called direct homothety and when k<0 the transformation is called inverse homothety.
		When the centre of enlargement is at the origin it is easy to find the relation between the coordinates of points and their homothetic images.
		When two homothetic transformations with the same centre of enlargement are combined the result is one transformation whose ratio or scale factor is the product of the other two ratios.
	descartes.cnice.mecd.es /ingles/4th_year_secondary_educ_B/Similarity_&_homothety/Homote1.htm (285 words)

	examples
		, which is a monotonically increasing transformation of a homogeneous-of-degree-1 function is called homothetic.
		So ``homothetic'' is a property of the underlying Cobb-Douglas preferences.
		This propperty, that the slope of an indifference curve does not vary as all elements of the consumption bundle are increased by the same prroportion, will hold for any homothetic preferences.
	dept.econ.yorku.ca /~sam/4000/examples/examples.html (1068 words)

	Homothetic preferences on star-shaped sets
		This paper describes properties of homothetic preferences on a subset X of a vector space which is star-shaped with respect to 0 (e.g., a cone).
		We prove that a preference relation on X is homothetic, greedy and calibrated if and only if there exists a positively homogeneous function that represents it.
		As a corollary, we find that, if X is contained in a topological vector space, then ⪰ is homothetic and continuous if and only if there exists a positively homogeneous and continuous function that represents it.
	ideas.repec.org /a/spr/decfin/v24y2001i1p41-47.html (289 words)

	Homothety
		Homothety, dilation, central similarity are all interchangeable terms used to describe a geometric transformation H
		leads to a trivial transformation and is usually excluded from consideration.
		Two homotheties with a common center commute as a matter of course.
	www.cut-the-knot.org /Curriculum/Geometry/Homothety.shtml (269 words)

	The demonstration of the Fermet's theorem
		Thus, it can be seen that this is a real recurrence process with a defined Unit Object and the uniform transformation appears as homothetic ones using the given variables (a) and (b).
		Finally, all the transformations are only two dimensions homothetic ones, exhaustively in the whole series.
		It means that the property of this relationship belong to space in the homothetic transformation.
	www.dakhi.com /somen1.htm (4453 words)

	EconPapers: An Experimental Test for Stability of the Transformation Function in Rank-Dependent Expected Utility Theory ... (Site not responding. Last check: 2007-10-21)
		The main feature of interest in the model is the "payment transformation function," which operates on proportions of a fixed total of payments just as the probability weighting function in RDEU operates on probabilities.
		This implies a strictly concave transformation function, and is at odds witht the finding of an "inverted S" shaped function that many researchers have suggested for the RDEU model.
		A disaggregated analysis in which we classify subjects according to which transformation function is most consistent with their revealed choice behavior shows that a linear and a strictly concave transformation function are the most common for both risky choice and for intertemporal choice.
	econpapers.repec.org /paper/rutrutres/199826.htm (415 words)

	Computer Graphics : Exercises : Exercice 2
		The area of the frame should be four times the area of the picture so long as it remains equal to or less than half the area of your screen display.
		The dimension of the frame should be kept homothetic(*) to that of the picture.
		A homothetic map preserves within a global scaling factor relative distances and angles.
	escience.anu.edu.au /lecture/cg/exercises/ex2/index.en.html (407 words)

	Virus World (Site not responding. Last check: 2007-10-21)
		This new structure is not a homothetic transformation of the icosadeltahedron because it now contains pentagones AND hexagones.
		The uniform hexagonal net helped to visualize the formation of icosahedral structures by transforming some hexagones into pentagones.
		Pentagones and hexagones can both be considered to be constructed by the same building block represented by the equilateral triangular units.
	rhino.bocklabs.wisc.edu /cgi-bin/virusworld/htdocs.pl?docname=triangulation.html (1322 words)

	John Quah's Page
		Homothetic or Cobb-Douglas behavior through aggregation, Contributions to Theoretical Economics, Vol.
		Shows how consumers' (heterogeneous) preferences in a market could be distributed in such a way that aggregate market demand takes on exact homothetic or Cobb-Douglas properties.
		Early version: Homothetic Preferences, Homothetic Transformation, and the Law of Demand in Exchange Economies, Working Paper No. 93-210, Department of Economics, UC Berkeley.
	www.economics.ox.ac.uk /Members/john.quah/homepage.htm (505 words)

	Nomizu, Yano: On infinitesimal transformations preserving the curvature tensor field and its covariant differentials
		Nomizu, Yano: On infinitesimal transformations preserving the curvature tensor field and its covariant differentials
		Nomizu, K. Yano, K. On infinitesimal transformations preserving the curvature tensor field and its covariant differentials.
		KOBAYASHI, A theorem on the affine transformation group of a Riemannian Manifold, Nagoya Math.
	www.numdam.org /numdam-bin/item?id=AIF_1964__14_2_227_0 (91 words)

	Citations: A Library for Doing Polyhedral Operations - Wilde (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Two particular problems have to be considered: ffl Pn has to be homothetic (or bordered) according to definitions 2 and 3, i.e.
		Pn (or its kernel) can be deduced from P1 (or the kernel of P1) by the n homothetic transformation.
		It is obtained by applying the transformation x 0, to the inhomogeneous system.
	citeseer.ist.psu.edu /context/119408/94120 (1403 words)

	Integral Polynomial Arithmetic
		Integral polynomial binary homothetic transformation, leading variable.
		Integral polynomial binary homothetic transformation, main variable.
		B <- IUPNT(A) Integral univariate polynomial negative transformation.
	www.mcs.drexel.edu /~krandick/saclib/node33.html (399 words)

	Apparatus and method for picture representation by data compression - Patent 5510838
		Process according to claim 10, wherein the window is a square having a side of a few pixels and the grid is constituted by the pixels themselves, if the side is an uneven number of pixels, and is constituted by the central points between the pixels, if the side is an even number of pixels.
		Clarke, Transform Coding of Images, Academic Press, 1985, from page 248) may be carried out, and the three original monochrome signals, corresponding to the RGB system, can be transformed into one (monochrome) luminance signal and two reduced bandwidth colour-information carrying signals (sometimes referred to collectively, hereinafter, as "colour information signals).
		Alternatively, transform coding of color data, by methods known in the art, may be carried out and the original monochrome signals corresponding to the RGB system can be completed by transform coding into one monochrome luminance signal and two reduced bandwidth color-information carrying signals.
	69.64.183.131 /5510838.html?highlight=4715005 (17652 words)

	PlanetMath:
		homothetic transformation (in affine transformation) owned by matte
		Hotelling transform (=principal components analysis) owned by akrowne
		Huffman tree (in Huffman coding) owned by Logan
	planetmath.org /encyclopedia/H (1464 words)

	Curves in a plane
		In statics it is proved that the equation of the catenary, in a suitable coordinate system, is
		If we take the function y = cosh(x), and we transform this curve with the (homothetic) transformation y = y'/a and x = x'/a, then we have a catenary.
		All catenaries are homothetic transformations of the graph of the cosh function.
	www.cartage.org.lb /en/themes/sciences/Mathematics/Algebra/foci/topics/Curves/Curves.htm (2438 words)

	Gmsh 1.65: Geometry module
		Geometrical transformations can be applied to elementary entities, or to copies of elementary entities (using the
		s giving the X, Y and Z direction of the homothetic transformation.
		Transforms all elementary entities (points, lines or surfaces) symmetrically to a plane.
	www.geuz.org /gmsh/doc/texinfo/gmsh_4.html (1612 words)

	Process and system for the visual display of moving targets - United States Patent 4,308,535
		In other words, the corrected vector would have the same amplitude and phase as the ideal vector.
		In case the weighting factor is real, the vector in question undergoes no homothetic transformation.
		The device in which weighting takes place is advantageously positioned between two filtering devices which cancel out the zero frequency.
	xrint.com /patents/us/4308535 (2331 words)

	kde-cvs digest
		a circle does not map onto a circle), - an example of a projective transformation that is not an affine map: it depends on a ray and an angle.
		The addition of the transformations is extremely localized in those files: changes are present only in kig_part.cpp in order to add the new types (one per object that can be transformed).
		Right now there is no new entry in kigpartui.rc, anyway the transformations can be reached by means of the contextual menu (right click) of any object.
	members.shaw.ca /dkite/dec27.html (3777 words)

	Gmsh 1.65:
		The sixth case permits to retrieve the coordinates of a given geometry point (see section 3.1.1 Points).
		The last two cases permit to retrieve the indices of entities created through geometrical transformations and extrusions (see 3.1.6 Transformations, and 3.1.5 Extrusions).
		To see the practical use of such expressions, have a look at the first couple of examples in 7.
	www.geuz.org /gmsh/doc/texinfo/gmsh-full.html (5209 words)

	[CMEC] Secondary Education in Canada: A Student Transfer Guide
		They explore triangles, polygons, and geometric solids and calculate their area and volume.
		Students also study homothetic transformation for enlarging and reducing geometric figures.
		In the field of statistics and probabilities, students cover data analysis, the average, median, and mode in statistics, data dispersion, probabilities and sampling, graphical representation of probabilities, and the reading and interpreting of tables, charts, and graphs.
	www.cmec.ca /tguide/2004/ns-fr.en.html (9619 words)

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