
	Where results make sense

Topic: Affine geometry

Related Topics

Affine transformation

Vector space

Finite geometry

Projective geometry

Geometry of numbers

Geometer

Topology glossary

Projective geometry

Synthetic geometry

Vector calculus

Parallel Postulate

Centroid

Poisson summation formula

	Affine geometry - Wikipedia, the free encyclopedia
		In geometry, affine geometry is geometry not involving any notions of origin, length or angle, but with the notion of subtraction of points giving a vector.
		Affine geometry can be explained as the geometry of vectors, not involving any notions of length or angle.
		An example from the plane geometry of triangles is the theorem about the concurrence of the lines joining each vertex to the mid-point of the opposite side (at the centroid or barycentre).
	en.wikipedia.org /wiki/Affine_geometry (746 words)

	Affine geometry - Wikipedia, the free encyclopedia
		It is the geometry of affine space, of a given dimension n over a field K.
		Affine space is distinguished from a vector space of the same dimension by 'forgetting' the origin 0.
		An affine space A for a vector space V is just such a principal homogeneous space; one then has to recover scalar multiplication on A as a well-defined concept.
	www.wikipedia.org /wiki/Affine_geometry (746 words)

	affine geometry
		Circles, angles, and distances are altered by affine transformations and so are of no interest in affine geometry.
		Affine transformations do, however, preserve collinearity of points: if three points belong to the same straight line, their images under affine transformations also belong to the same line and, in addition, the middle point remains between the other two points.
		Similarly, under affine transformations, parallel lines remain parallel, concurrent lines remain concurrent (images of intersecting lines intersect), the ratio of lengths of line segments of a given line remains constant, the ratio of areas of two triangles remains constant, and ellipses, parabolas, and hyperbolas continue to be ellipses, parabolas, and hyperbolas.
	www.daviddarling.info /encyclopedia/A/affine_geometry.html (232 words)

	Various Geometries
		Affine Geometry is not concerned with the notions of circle, angle and distance.
		Affine transformations preserve collinearity of points: if three points belong to the same straight line, their images under affine transformations also belong to the same line and, in addition, the middle point remains between the other two points.
		Analytically, affine transformations are represented in the matrix form f(x) = Ax + b, where the determinant det(A) of a square matrix A is not 0.
	www.cut-the-knot.org /triangle/pythpar/Geometries.shtml (2183 words)

	Affine geometry -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		Affine geometry can be explained as the geometry of (A variable quantity that can be resolved into components) vectors, not involving any notions of length or angle.
		Affine space is distinguished from a (Click link for more info and facts about vector space) vector space of the same dimension by 'forgetting' the origin 0.
		An example from the plane geometry of triangles is the theorem about the concurrence of the lines joining each vertex to the mid-point of the opposite side (at the (The center of mass of an object of uniform density) centroid or (Click link for more info and facts about barycentre) barycentre).
	www.absoluteastronomy.com /encyclopedia/a/af/affine_geometry.htm (977 words)

	PlanetMath: affine transformation
		Compositions of affine maps is again an affine map.
		In affine geometry, the set of affine transformations on the affine space becomes a group under compositions of maps.
		This is version 10 of affine transformation, born on 2004-10-24, modified 2005-02-15.
	planetmath.org /encyclopedia/AffineTransformation.html (117 words)

	Affine Geometry, Curve Flows, and Invariant Numerical Approximations - Calabi, Olver, Tannenbaum (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		A new geometric approach to the affine geometry of curves in the plane and affine-invariant curve shortening is presented.
		We describe methods of approximating the affine curvature with discrete finite difference approximations, based on a general theory of approximating differential invariants of Lie group actions by joint invariants.
		Calabi, P. Olver, and A. Tannenbaum, "Affine geometry, curve flows, and invariant numerical approximations," TR, Department of EE, University of Minnesota, June 1995.
	citeseer.ist.psu.edu /72427.html (783 words)

	COMP 290-001: Lecture Notes: Design Example for an Affine-Geometry Kernel (Site not responding. Last check: 2007-11-06)
		An affine transformation is a linear mapping from an affine space to an affine space that preserves affine combinations.
		This implies that the affine transformation of the difference of two points (a vector) is the same as the difference of two affine transformed points.
		Assuming the reference frame for the new space is given in the reference frame of the points, the affine transformation is easily specified as follows; the base point of the reference frame maps to the origin and each vector in the basis of the reference frame maps to the unit vectors of the standard basis.
	www.mpi-sb.mpg.de /~kettner/courses/lib_design_03/notes/gkernel.html (1098 words)

	Transformation of coordinates (Projective; Affine; Metric)
		Affine properties are for instance collinearity of points; parallel to; ideal point; concurrency of lines; vector; midpoint.
		The affine properties are a subset of the metric properties.
		By affine axes is meant a general basis in the plane.
	www.ping.be /~ping1339/coortf.htm (1901 words)

	[No title]
		Affine space: We will usually be rather informal in our presentation of geometry, but it is good to start things off on a somewhat formal footing.
		The approach in computational geometry texts is to say nothing about geometric representations, but to rely upon an intuitive understanding of geometric concepts.
		An affine geometry consists of a set of scalars (the real numbers), a set of points, and a set of free vectors (or simply vectors).
	www.cs.wustl.edu /~pless/506/l2.html (1841 words)

	Affine Geometry (Site not responding. Last check: 2007-11-06)
		Although the geometry we get is not Euclidean, they are not called non-Euclidean since this term is reserved for something else.
		Two parallel lines are lines in an affine plane which do not meet.
		Since affine transformations preserve planes and incidence, their images lie in an affine plane and do not meet.
	www-groups.dcs.st-and.ac.uk /~john/geometry/Lectures/L13.html (229 words)

	Affine and Projective Geometry
		In the second part, geometry is used to introduce lattice theory, and the book culminates with the fundamental theorem of projective geometry.
		Throughout, the text explores geometry's correlation to algebra in ways that are meant to foster inquiry and develop mathematical insights whether or not one has a background in algebra.
		Affine and Projective Geometry's broad scope and its communicative tone make it an ideal choice for all students and professionals who would like to further their understanding of things mathematical.
	www.allbookstores.com /book/0471113158 (302 words)

	Math 371 Affine Geometry Notes
		Then the correspondence of the points on P0P1 to those on m gives a coordinate system for m in which the points for large values of x (either positive or negative) will be close to the point R, so we can think of R as being a point on m that represents infinity.
		We use to denote the ordinary point on the affine line which corresponds to the homogeneous coordinates (a,b) when b is not 0.
		A line in the Affine plane is still determined by exactly two points.
	www.humboldt.edu /~mef2/Courses/AffineGeometryNotes.html (1692 words)

	Math 371 Geometry Notes on Line
		The impact of this on geometry was that one could not presume that all of geometry could be handled by using simple ratios of whole numbers for measurements.
		A (set theoretic) interpretation for a geometry structure is a set in which the words of the geometry stucture's language are identified with set elements and sets and relations and operations on sets.
		The axioms for projective geometry in a plane uses two basic objects: points and lines, and a relation between those: a point is on a line, or a line passes through a point.
	www.humboldt.edu /~mef2/Courses/m371notes03.html (11762 words)

	Geometry of the I Ching (Site not responding. Last check: 2007-11-06)
		This arrangement was discovered during an investigation of the six-dimensional affine space over the two-element field by S. Cullinane on January 6, 1989.
		A (Lewis) Carroll diagram - predecessor of the 1953 Karnaugh maps
		The I Ching as affine coordinates in a 4x4x4 cube:
	m759.freeservers.com /PHiching.html (642 words)

	Learn more about Geometry in the online encyclopedia. (Site not responding. Last check: 2007-11-06)
		Geometry is the branch of mathematics dealing with spatial relationships.
		In Euclidean geometry, two figures are said to be congruent if they are related by a series of reflections, rotations, and translationss.
		The latter point of view is called the Erlanger program.
	www.onlineencyclopedia.org /g/ge/geometry.html (374 words)

	Affine 2D Analogue
		This flow is what all the the literature on affine evolutions claim to be the affine analogue of the curve shortening flow in Euclidean space.
		Now, in affine geometry, the claimed analogue is the functional is affine length and the fastest way to shrink it is the evolution in the affine normal direction which corresponds to an Euclidean curvature deformation in the Euclidean normal direction.
		We have argued that the affine curve evolution is not necessarily the fastest way to shrink the affine perimeter.
	www.cim.mcgill.ca /~mdesco/shape_report/report/node4.html (794 words)

	PDEs and Submanifolds, Affine Differential Geometry (Site not responding. Last check: 2007-11-06)
		Research cooperation with the geometry group of Peking University (C.P. Wang) and the geometry group Sichuan University (in Chengdu, A.-M. Li, G.S. Zhao).
		Teaching cooperation (DAAD-Sonderprogramm "Akademischer Neuaufbau Südosteuropa") with the geometry group of the University of Belgrade.
		Simon, K. Polthier) with the geometry group of the TU Budapest (M. ilvási-Nagy).
	www.math.tu-berlin.de /geometrie/adg (194 words)

	Applications of Spectral Geometry to Affine and Projective Geometry (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		We use the asymptotics of the heat equation to construct spectral invariants in affine and projective geometry.
		x1 Introduction One of the first global affine results is Blaschke's characterization of the ellipsoid within the class of ovaloids by the constancy of its equiaffine mean curvature [2, p.
		1 Viesel: Introduction to the affine differential geometry of..
	citeseer.ist.psu.edu /bokan94applications.html (376 words)

	Amazon.com: Books: Geometry (Site not responding. Last check: 2007-11-06)
		During the last third of the book (the chapters on hyperbolic and spherical geometry), some basic familiarity with trigonometric functions and hyperbolic functions is assumed (cosh, sinh, tanh, and their inverses).
		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 /exec/obidos/tg/detail/-/0521597870?v=glance (2775 words)

	References (Site not responding. Last check: 2007-11-06)
		Dillen and L. Vrancken, Affine differential geometry of hypersurfaces, 1990, pp.
		Magid and M. Nomizu, On affine surfaces whose cubic forms are parallel relative to the affine metric, Proc.
		Pedersen and A. Swann, Einstein-Weyl geometry, the Bach tensor and conformal scalar curvature, J. Reine Angew.
	www.math.tu-berlin.de /geometrie/adg/bib-1995/node1.html (4332 words)

	The Math Forum - Math Library - Geometry (Site not responding. Last check: 2007-11-06)
		A collection of handouts for a two-week summer workshop entitled 'Geometry and the Imagination', led by John Conway, Peter Doyle, Jane Gilman and Bill Thurston at the Geometry Center in Minneapolis, June 17-28, 1991.
		A short article designed to provide an introduction to geometry, including classical Euclidean geometry and synthetic (non-Euclidean) geometries; analytic geometry; incidence geometries (including projective planes); metric properties (lengths and angles); and combinatorial geometries such as those arising in finite group theory.
		Some notes on a most general definition of "geometry," first elucidated by Felix Klein, which is based on a set of geometric invariants under a group of transformations.
	mathforum.org /library/topics/geometry (2304 words)

	The Math Forum - Math Library - Affine Geom. (Site not responding. Last check: 2007-11-06)
		An introduction to ordinary analytic geometry as studied in secondary school.
		Interactive proofs (elementary geometry, barycentric coordinates, complex numbers, Ceva's theorem, affine geometry) of the fact that the three medians of a triangle meet at a point called the centroid.
		Basics, path curves, counter space, pivot transforms, and some people involved in the development of projective geometry, which is concerned with incidences: where elements such as lines planes and points either coincide or not.
	forum.swarthmore.edu /library/topics/affine_g (296 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Abstract: Projective geometry and affine geometry are two of the many sources of inspiration for matroid theory.
		This talk will start with projective and affine geometries, especially those based on finite fields, and will use these to motivate matroid theory.
		which equality is attained in certain inequalities among matroid parameters; these geometries are also completely determined by a small number of enumerative facts about flats of certain ranks; and lastly, certain invariants associated with matroids (notably, the Tutte polynomial) allow us to distinguish projective and affine geometries from all other matroids.
	mathweb.mathsci.usna.edu /faculty/colloquia/Math/Abstracts/colloqannouncementoct04200.htm (149 words)

	Affine Geometry, Curve Flows, and Invariant Numerical Approximations - Calabi, Olver, Tannenbaum (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Affine Geometry, Curve Flows, and Invariant Numerical Approximations (1997)
		A new geometric approach to the a#ne geometry of curves in the plane and a#ne-invariant curve shortening is presented.
		We describe methods of approximating the a#ne curvature with discrete #nite di#erence approximations, based on a general theory of approximating di#erential invariants of Lie group actions by jointinvariants.
	citeseer.ist.psu.edu /241831.html (488 words)

	Affine Differential Geometry (Site not responding. Last check: 2007-11-06)
		In affine differential geometry a main point of research is the investigation of special models of surfaces in threedimensional affine space as, for instance, affine spheres, ruled surfaces, surfaces of translation and affine surfaces of rotation.
		The convolution of a paraboloid and a parametrized surface.
		Surfaces with affine rotational symmetry and flat affine metric in R
	www.geometrie.tuwien.ac.at /geom/bibtexing/affdg.html (148 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		In particular, we are interested in the interplay of geometry and PDEs.
		In affine differential geometry one often encounters fourth order nonlinear PDEs which are far from being well understood.
		In recent years, there has been significant progress with the solution of some famous conjectures and problems in affine differential geometry related to higher order PDEs; we would like to mention the solutions of the famous affine Bernstein conjectures of Calabi and Chern, resp., and contributions to centroaffine Bernstein problems.
	www.math.tu-berlin.de /geometrie/adg/Bedlewo (463 words)

	Wilson Stothers' Cabri Pages (Site not responding. Last check: 2007-11-06)
		We therefore define conics in affine geometry as the loci with equations quadratic
		They turn out to be very useful in the study of plane conics in affine geometry.
		In euclidean geometry, we saw that the symmetry group of a plane conic C is of
	www.maths.gla.ac.uk /~wws/cabripages/klein/aconics.html (298 words)

Try your search on: Qwika (all wikis)