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Topic: Jordan curve

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	Jordan curve theorem - Wikipedia, the free encyclopedia
		In topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an "inside" and an "outside".
		A rigorous 200,000-line formal proof of the Jordan curve theorem was produced in 2005 by an international team of mathematicians using the Mizar system.
		There is a generalisation of the Jordan curve theorem called the Jordan-Schönflies theorem which states that any Jordan curve in the plane can be extended to a homeomorphism of the plane.
	en.wikipedia.org /wiki/Jordan_curve_theorem (412 words)

	Curve - Wikipedia, the free encyclopedia
		A rectifiable curve is a curve with finite length.
		Important examples of algebraic curves are the conics, which are nonsingular curves of degree two and genus zero, and elliptic curves, which are nonsingular curves of genus one studied in number theory and which have important applications to cryptography.
		From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis.
	en.wikipedia.org /wiki/Curve (1536 words)

	PlanetMath: curve
		The second notion is geometric; in this sense a curve is an arc, a 1-dimensional subset of an ambient space.
		The two notions are related: the image of a parameterized curve describes the trajectory of a moving particle.
		In algebraic geometry, the term curve is used to describe a 1-dimensional variety relative to the complex numbers or some other ground field.
	planetmath.org /encyclopedia/JordanCurve.html (448 words)

	PlanetMath: Jordan curve theorem
		Informally, the Jordan curve theorem states that every Jordan curve divides the Euclidean plane into an “outside” and an “inside”.
		The two connected components mentioned in each formulation are, of course, the inside and the outside the Jordan curve, although only in the first formulation is there a clear way to say what is out and what is in.
		This is version 6 of Jordan curve theorem, born on 2002-11-11, modified 2004-02-16.
	planetmath.org /encyclopedia/JordanCurveTheorem.html (179 words)

	Jordan curve - Hutchinson encyclopedia article about Jordan curve
		It is one of the family of curves called conic sections that also includes the circle, ellipse, and hyperbola.
		Other common geometrical curves are the ellipse, parabola, and hyperbola, which are also produced when a cone is cut by a plane at different angles.
		Many curves have been invented for the solution of special problems in geometry and mechanics – for example, the cissoid (the inverse of a parabola) and the cycloid.
	encyclopedia.farlex.com /Jordan+curve (189 words)

	BIBLIOGRAPHY
		Simply stated, the Jordan Curve Theorem states that any continuous simple closed curve in a plane separates the plane into two disjoint regions, the inside and the outside.
		The Jordan-Schönflies Curve Theorem states that for any simple closed curve in the plane, there is a homoeomorphism of the plane that takes that curve into the standard circle.
		In topology, Jordan investigated symmetries in polyhedra and of course the Jordan Curve.
	www.facstaff.bucknell.edu /udaepp/090/w3/melissal.htm (2174 words)

	Search Results for Jordan
		Jordan married Marie-Isabelle Munet, the daughter of the deputy mayor of Lyon, in 1862.
		Jordan's use of the group concept in geometry in 1869 was motivated by studies of crystal structure.
		Jordan is best remembered today among analysts and topologists for his proof that a simply closed curve divides a plane into exactly two regions, now called the Jordan curve theorem.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Jordan&CONTEXT=1 (3554 words)

	ScienceWeek - In Focus: Pascual Jordan
		Of the triumvirate Pascual Jordan, Max Born, and Werner Heisenberg that formulated matrix quantum mechanics in 1925, Jordan was the principal architect of the theory.
		Camille Jordan was the foremost specialist in algebra of his time, publishing research in topology, analysis, and particularly in group theory.
		The so-called "Jordan curve" in analysis is the curve of Camille Jordan.
	scienceweek.com /focus001.htm (1380 words)

	Jordan curve theorem: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-03)
		Topology (greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces....
		In mathematics, the jordan-schönflies theorem in geometric topology is a sharpening of the jordan curve theorem in two dimensions....
		In the mathematical field of topology a homeomorphism or topological isomorphism (from the greek words homeos = identical and morphe = shape) is a...
	www.absoluteastronomy.com /encyclopedia/j/jo/jordan_curve_theorem.htm (1029 words)

	[No title]
		The curves in question are approximately the shape of the seam of a tennis ball.
		Such a condition, for example, can be stated as follows: every point on the boundary curve which is not on the convex hull of the boundary curve has the property that every plane through it meets the boundary curve in at most 8 points.
		Note that this generalizes the condition of [15], that the boundary curve lie on the boundary of a convex body.
	www.cs.sjsu.edu /faculty/beeson/Papers/minsurf.html (2320 words)

	[No title]
		In article <34430@sdcc12.ucsd.edu> kwalker@canyon.ucsd.edu (Kevin Walker) writes: > "Conjecture: For any [embedded] curve C in the plane, there exist > points > a, b, c, d on C such that a, b, c and d form a square." Vaguely related question: A room has an uneven floor, but the skirting board is entirely horizontal.
		Curve (C-3) draws therefore all the centre points of circle (A) and circle (B).
		Apparently, if circle (A) with radius psqrt(2) rolls on the outside of the smooth convex curve* C-3, it is possible to draw a new smooth convex curve C-4 that is at the radius distance (psqrt(2)) from the curve* C-3.
	www.ics.uci.edu /~eppstein/junkyard/jordan-square.html (2617 words)

	The Hilbert Space Filling Curve (Site not responding. Last check: 2007-11-03)
		The actual Hilbert Curve is not a member of this family, it is the limit that the sequence approaches.
		The next curve in the sequence is a refinement of this, we consider each of the quarters to be a box with the appropriate orientation, so that the curve is entering and leaving from the bottom left and leaving in the bottom right.
		This apparent riddle is solved as although none of the family of curves leading to the Hilbert curve crosses itself the final curve does cross itself all over the place.
	www.dcs.napier.ac.uk /~andrew/hilbert.html (363 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Classify the points of the annulus as "in" if p(x,y) is inside the Jordan curve, "out" if p(x,y) is outside the Jordan curve, and "on" if it is on the curve.
		Then (for smooth curves) the points near one boundary of the annulus are all classified "in", the points near the other boundary are all classified "out", and the points classified "in" and "out" are both open sets.
		But it doesn't work for all Jordan curves; for instance the equilateral triangle has both "in" and "out" points near both boundaries of the annulus.
	www.math.niu.edu /~rusin/known-math/99/jordan-square (634 words)

	The Jordan Curve Theorem
		Take one of these curves and call it C. An apparently straightforward property is that all curves similar to C represent boundaries for two sets of points, let's call them A and B, the inside and outside set respectively.
		Any simple closed curve C divides the points of the plane not on C into two distinct domains (with no points in common) of which C is the common boundary.
		Therefore, proving the theorem for this particular case (a much simpler task than proving it for the case of general simple closed curves) may be of great use in solving a lot of other important problems.
	www-cgrl.cs.mcgill.ca /~godfried/teaching/cg-projects/97/Octavian/compgeom.html (1554 words)

	March 2002 solution
		Let S be a simple closed curve and let A be a point on S such that the tangent at A is defined.
		The Jordan curve theorem (published by Camille Jordan in 1893) says that such a curve separates the plane into two regions, the interior (which is bounded by the curve) and the exterior (which is unbounded).
		Of course, if one imagines the Jordan curve to be the boundary of a pancake, such a result is obvious - the interior is the part you eat.
	mathcentral.uregina.ca /MP/previous2001/apr02sol.html (1179 words)

	Formalization of Jordan Curve Theorem
		The proof of Jordan curve theorem for special polygons is the first part of the formalization of general Jordan curve theorem for simple closed curves:
		Jordan curve theorem - the theorem that states that every simple closed curve divides a plane into two parts and is the common boundary between them (see The American Heritage Dictionary of the English Language).
		The preliminary work on the proof of general Jordan curve theorem started with defining the external (so called Cages) and internal (so called Spans) approximation of the curve by special polygons in JORDAN9 and JORDAN13 respectively.
	mizar.uwb.edu.pl /jordan (414 words)

	Jordan Curve Theorem
		The result is called the Jordan Curve Theorem in honor of the French mathematician Camille Jordan (1838-1922).
		If we shade the simple closed curve that passes through house 1, water, house 2, and electricity, we find that house 3 is inside this curve and that it has not yet been joined to gas, which is outside the curve.
		According to the Jordan Curve Theorem, it is impossible to connect them without crossing the curve.
	britton.disted.camosun.bc.ca /jbjordan.htm (671 words)

	[No title]
		JORDAN The Jordan Curve Theorem is an intuitively clear theorem that in rough terms states, "A closed curve divides a plane into the inside and outside areas" (see figure below).
		Camille Jordan stated in his publications in 1887 that "this theorem is clearly true", but it was later discovered that the proof of the theorem is in fact quite difficult.
		Since then, many books and papers have been published concerning the Jordan Curve Theorem, but in many cases they introduce references to other books and papers and upon searching the references, these in turn introduce yet further references and at some point the deepest link of references is lost.
	markun.cs.shinshu-u.ac.jp /mizar/jordan/jordancurve-e.html (920 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		This curve is a union of the invariant curves of the equichordal map $T$: \[C=W^s(A_1)\cup W^u(A_2),\] where $A_1=(-1/2,0)$ and $A_2=(1/2,0)$.
		The existence of an equichordal curve would mean that for some parameter value the oscillation would cease, so that the curve can be closed with an analytic piece passing through $A_1$.
		It has also been known since the 1920's \cite{suss} that the equichordal curve would have to be symmetric with respect to the reflection in the line $O_1O_2$ as well as in the bisector of the segment $O_1O_2$.
	www.maths.tcd.ie /EMIS/journals/ERA-AMS/1996-03-002/1996-03-002.tex.html (3772 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		The theorem says that a closed (f(a)=f(b)) continuous curve f > : [a,b] -> R2, with no double points, that is f : (a,b) -> R2 is > injective, will divide the plane into two connected components, one > which is unbounded and another which is bounded.
		> > Books on algebraic topology typically prove the generalized Jordan Curve > theorem: if you remove from the n-sphere S^n a subset C homeomorphic to > S^(n-1), what remains has two components whose common boundary is C. > The proof depends on homology theory.
		It is constructive in that the proof shows how to determine if a point in in the bounded or unbounded regions, and it show how to contruct a path joing two points that are in the same component of the complement.
	www.math.niu.edu /~rusin/papers/known-math/99/jordan_crv (419 words)

	Regular polyhedra
		It's a variant of the Jordan's Theorem that asserts the same property for a more general class of curves.
		For a polygon it means that every closed curve in its interior can be continuously shrunk into a point while the deformation is being carried entirely inside the polygon.
		For a polyhedron it means about the same thing with surface curves shrunk into a point while staying on the surface.
	www.cut-the-knot.org /do_you_know/polyhedra.shtml (1138 words)

	Citebase - On the theorem converse to Jordan's curve theorem (Site not responding. Last check: 2007-11-03)
		Citebase - On the theorem converse to Jordan's curve theorem
		On the theorem converse to Jordan's curve theorem
		We show that the requirement of this theorem that it all points of K were accessible from it both complementary domains is surplus and prove one generalization of this theorem.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0009164 (250 words)

	The Topology of Complex Numbers
		Intuitively, we think of a curve as a piece of string placed on a flat surface in some type of meandering pattern.
		We mentioned earlier that a simple closed curve is positively oriented if its interior is on the left when the curve is traversed.
		The Jordan curve theorem is a classic example of a result in mathematics that seems obvious but is very hard to demonstrate, and its proof is beyond the scope of this book.
	math.fullerton.edu /mathews/c2003/ComplexPlaneTopologyMod.html (1097 words)

	Earliest Known Uses of Some of the Words of Mathematics (J)
		JORDAN CURVE appears in W. Osgood, "On the Existence of the Green's Function for the Most General Simply Connected Plane Region," Transactions of the American Mathematical Society, Vol.
		(July 1900): "By a Jordan curve is meant a curve of the general class of continuous curves without multiple points, considered by Jordan, Cours d'Analyse, vol.
		Jordan curve-theorem is found in D. Woodard, "On two-dimensional analysis situs with special reference to the Jordan curve-theorem," Fundamenta (1929).
	members.aol.com /jeff570/j.html (416 words)

	0pt (Site not responding. Last check: 2007-11-03)
		We will discuss many examples of contours, but they are all familiar as curves in the plane.
		This of course assumes that the arc is differentiable.
		There is a very difficult theorem called the Jordan curve theorem which states that any Jordan curve split the plane into two regions, the inside and the outside.
	www.lehigh.edu /dlj0/Desktop/dlj0/courses/208ss199-30-33.html (603 words)

	Jordan curve theorem (Site not responding. Last check: 2007-11-03)
		In topology, the Jordan curve theorem states that everynon-self-intersecting loop in the plane divides the plane into an "inside" and an "outside".
		The statement of the Jordan curve theorem seems obvious, but it is a very difficult theorem to prove, and an incorrect proofwas originally given by Camille Jordan.
		This is a much stronger statement than theJordan curve theorem.
	www.therfcc.org /jordan-curve-theorem-70816.html (233 words)

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