Lemma 2 shows every Jordan curve could be approximated uniformly by a sequence of Jordan polygons. 22 Parity Function for Polygons The Jordan curve theorem for polygons is well known.
The Jordan curve theorem states that every simple closed pla nar curve separates the plane into a bounded interior region and an unbounded exterior.
Jordan curve theorem for polygons. The complement of a polygon consists of two non-empty planar regions and with the following properties. The Jordan curve theorem 5 often abbreviated as JCT in the literature was one of the starting points in the modern development of topology originally called Analysis Situs. Tverberg AProofoftheJordanCurveTheorem Bulletin of the London Math Society 1980 pp 34-38.
The Jordan curve theorem for polygons A simple closed polygon Q divides the Euclidean plane into three non-empty disjoint sets the polygon itself and two open. A number of intersections of a polygon in question by an ray coming from a p point to point p laying in the same domain depends on the domain A or B to which the points belongs to. To prove that it cannot be any other integer is the intrinsic core of the Jordan curve theorem.
See this post for an elementary proof of the Jordan curve theorem for polygons. It is a polygonal arc if it is 11. The famous Jordan Curve Theorem says that for any Jordan Curve Γ the complement R2 Helge Tverberg wrote a paper which gives a very clean elementary proof.
The order of a point q with respect to a polygon P not passing through q is defined as the number of revolutions made by an arrow joining q to a point p on P as p is moved once around along the boundary of P. Thu Fs is a closed polygon without self intersections. The proof the technical part consists of 4 pages is self-contained except for the Jordan theorem for polygons taken for granted.
Or one can prove the Jordan curve theorem for polygons first and then approximate the Jordan curve with polygons. Jordan curve theorem holds for Jordan polygons as well. One hundred years ago Oswald Veblen declared that this theorem is justly regarded.
Jordan curve theorem 1 Jordan curve theorem In topology a Jordan curve is a non-self-intersecting continuous loop in the plane and another name for a Jordan curve is a simple closed curve. The Jordan curve theorem abbreviated as JCT A simple closed continuous curve K in the plane separates its complement into two open sets of which. Jordan Curve Theorem 165 of P.
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. The theorem states that any simple closed curve partitions the plane into two connected subsets exactly one of which is bounded. We can now easily define the winding number of a polygon around a point in the following way.
If the point is outside the polygon the winding number is 0. Then if q has an even order with respect to P it is outside and if its order is an odd number it is inside P. Courant described the outline of an elementary proof of the Jordan curve theorem for polygons using the order of points.
Here is the polygonal Jordan Curve Theorem Theorem 01 Let J be any polygon. A new elementary nonstandard proof of the Jordan curve theorem is given. Any two points of Bcan be joined by a polygonal path in B.
Jordan Curve Theorem Pro only the first part of the full theorem. Otherwise you will be even and it is outside the polygon. Then R2 J consists of exactly two sets Aand Bsuch that Any two points of Acan be joined by a polygonal path in A.
01 R2 that is a subset of a ﬁnite union of lines. Then P k 6ˇP as otherwise Kwould have a self-intersection. Paradoxically the first way which appears to be more the indirect one is.
Denote edges of Γ to be EE E 12. A polygon is a Jordan curve that is a subset of a ﬁnite union of lines. Captions Summary edit.
Proof of the Jordan Curve Theorem. To see that the other arc is not included in the monad consider any t k which is 6ˇany of t i01. A proof of the following theorem can be found in 1.
The following theorem and other easily established facts about polygons will be used without proof in this paper. Jordan 5 considered the assertion to be evident for simple polygons. It is comparatively easy to prove that the Jordan curve theorem holds for every Jordan polygon in Lemma 1 and every Jordan curve can be approximated arbitrarily well by a Jordan polygon in Lemma 2.
Deﬁnition 3 Orientation A Jordan polygon is said to be oriented counter-clockwise if as you move along the edges from to to et cetera the inside of the curve is to your left. Otherwise the polygon is oriented clockwise. A Jordan curve is a subset of R2 which is homeomorphic to the circle S1.
Cases can not happe ton a Jordan curve. I wish to thank Larry Francis for his excellent editorial assistance. This result is considered diﬃcult to prove at least compared to its intuitive evidence.
A polygonal path is a continuous function P. Lemmas 3 and 4 provide certain metric description of Jordan polygons which helps to evaluate the limit. There is a simple polygon which in nitesimally approximates K.
Then Lemma 3 and Lemma 4 deal with the situation in limiting processes to prevent the cases from the polygons that may thin to zero somewhere. It assumes the theorem without proof in the important special case of a simple polygon and of the argument from that point on one must admit at. 1 The Jordan Polygon Theorem The Jordan Curve Theorem and its generalizations are the formal foundations of many results if not every result in surface topology.
Although this statement is intuitively clear. The order of a point p_0 is defined by the net number of complete revolutions made by an arrow joining the point p_0 to an moving point p as p traverses the polygon P once. No point of Acan be joined to a point of Bby a polygonal path that avoids J.
The Jordan curve holds theorem for every Jordan polygon f. The theorem is super simple and the idea is like this given an nth closed polygon you draw an arbitrary line if your point is inside you line will intersect with the edges an odd number of times. We will only need a weak.
The Jordan curve theorem holds for every Jordan polygon Γ with realisation γΘ. Ycost sint Xt fi pt a with constants H pa. For the understanding of a convex polygon we will need the following Jordan Curve Theorem for PolygonswhichisTheorem413onpage195ofRNLE.