Jordan Curve Definition
Suppose we have the. If is a simple closed curve in then the Jordan curve theorem also called the Jordan-Brouwer theorem Spanier 1966 states that has two components an inside and outside with the boundary of each.
The Jordan curve theorem is a standard result in algebraic topology with a rich history.
Jordan curve definition. Suppose f is a piece-wise circular simple closed curve and is a piece-wise circular arc. The proof first proves the Jordan curve theorem for polygons and then uses an approximation argument to derive the Jordan curve theorem in general. Thu Fs is a closed polygon without self intersections.
The geometric locus of points M x y of a plane the coordinates of which satisfy the equations x ϕ t and y ψ t where ϕ and ψ are continuous functions of the argument t on some segment a b. Jordan curve – a closed curve that does not intersect itself. In topology a Jordan curve sometimes called a plane simple closed curve is a non-self-intersecting continuous loop in the plane.
Definition of Jordan curve theorem. Cases can not happe ton a Jordan curve. A simple closed curve.
Prelhninary Lemmas We had defined when an arc is said to cross a circle. Theoretically these three definition shall be equivalent. Theorem 21 A conformal map φ of a Jordan domain F onto a Jordan domain G can be extended to a homeomorphism of F onto G.
A Jordan curve is said to be a Jordan polygon if C can be covered by finitely many arcs on each of which y has the form. It is not known if every Jordan curve contains all four polygon vertices of some square but it has been proven true for sufficiently smooth curves and closed convex curves Schnirelman. By Jordan curve we mean the homeomorphic image of T.
I sort of understand what its trying to say. Sorry for the trivial question. A Jordan curve Γ is said to be a Jordan polygon if there is a partition Θθθ θ 01 n of the interval 02π ie 02.
A curve is similar to a line segment. Princetons WordNet 000 0 votes Rate this definition. Stated differently the Jordan curve is a continuous image of the segment.
Closed curve – a curve such as a circle having no endpoints. Though Im not sure what continuous image means. We broaden the definition of crossing as follows.
1950 Joseph Leonard Walsh The Location of Critical Points of Analytic and Harmonic Functions American Mathematical Society page 242 When μ is small and positive the locus 1 consists of a Jordan curve near each of. Jordan who suggested the definition. A homeomorphic image of a circle cf.
Also is the Jordan Curve Theorem just saying that a closed loop partitions the plane into an inside and outside. A continuous simple closed curve or Jordan curve in a topological space or convergence space locale etc X X is the image of a continuous injection to X X from the unit circle S 1 S1. All the pieces to conclude the proof of the Jordan Curve theorem.
A closed curve is similar to a circle. Also HomeomorphismNamed after C. It bounds two Jordan domains.
A fundamental theorem of topology. Also called a simple closed curve. Jordan curve plural Jordan curves topology A non-self-intersecting continuous loop in the plane.
A closed curve that does not intersect itself. Anything with a round or oval shape formed by a curve that is closed and does not intersect itself. Loop – anything with a round or oval shape formed by a curve that is closed and does not intersect itself.
A plane simple closed curve is also called a Jordan curveIt is also defined as a non-self-intersecting continuous loop in the plane. Simple closed curve Jordan curve noun. Jordan Curve Similarly a closed Jordan curve is an image of the unit circle under a similar mapping and an unbounded Jordan curve is an image of the open unit interval or of the entire real line that separates the plane.
This map itself is a continuous parametrization. Jordan curve theorem definition the theorem that the complement of a simple closed curve can be expressed as the union of two disjoint sets each having as. Based on the definition of discrete surfaces we give three reasonable definitions of simply connected spaces.
The Jordan curve theorem states that the set complement in a plane of a Jordan curve consists of two connected components that is the curve divides the plane in two non-intersecting regions that are both connected. We have proved the Jordan curve theorem under the third. See also Line curve.
Of the curve The word continuous is generally assumed so that one speaks simply of a. Jordan Curve Theorem. The Jordan curve holds theorem for every Jordan polygon f.
THE CASE OF POLYGONS. Every simple closed curve divides the plane into two regions for which it is the common boundary First Known Use of Jordan curve theorem. Ycost sint Xt fi pt a with constants H pa.
The Jordan curve theorem says that a simply closed curve separates a simply connected surface into two components. A Jordan curve is a plane curve which is topologically equivalent to a homeomorphic image of the unit circle ie it is simple and closed. To ease our way into the formal proof let us recall Thomassens argument showing that if C is a simple closed polygonal curve in the.