Related papers: A Concise Proof of Discrete Jordan Curve Theorem

According to a general definition of discrete curves, surfaces, and manifolds. This paper focuses on the Jordan curve theorem in 2D discrete spaces. The Jordan curve theorem says that a (simply) closed curve separates a simply connected…

This paper presents a formalized proof of a discrete form of the Jordan Curve Theorem. It is based on a hypermap model of planar subdivisions, formal specifications and proofs assisted by the Coq system. Fundamental properties are proven by…

Here is present short proofing of Jordan's theorem about dividing of flat on two disjoint subsets by one closed curve.

A new elementary nonstandard proof of the Jordan curve theorem is given. The proof (the technical part consists of 4 pages) is self-contained, except for the Jordan theorem for polygons taken for granted.

We prove the Jordan curve theorem by generalizing the sweepline algorithm for trapezoidal decomposition of a polygon. Our proof uses Zorn's lemma (or, equivalently the axiom of choice). Though several proofs have been given for the Jordan…

Using a definition of Jordan curve similar to that of Dieudonn\'e, we prove that our notion is equivalent to that used by Berg et al. in their constructive proof of the Jordan Curve Theorem. We then establish a number of properties of…

The Jordan Curve Theorem (JCT) states that a simple closed curve divides the plane into exactly two connected regions. We formalize and prove the theorem in the context of grid graphs, under different input settings, in theories of bounded…

Although there are many simple proofs of Jordan's decomposition theorem in the literature (see [1], the references mentioned there, and [2]), our proof seems to be even more elementary. In fact, all we need is the theorem on the dimensions…

We study compact stable embedded minimal surfaces whose boundary is given by two collections of closed smooth Jordan curves in close planes of Euclidean 3-space. Our main result is a classification of these minimal surfaces, under certain…

We prove that if two subsets ${A}$ and ${B}$ of the plane are connected, ${A}$ is bounded, and the Euclidean distance $\rho({A},{B})$ between ${A}$ and ${B}$ is greater than zero, then for every positive $\varepsilon<\rho({A},{B})$, the…

The classical Jordan curve theorem for digital curves asserts that the Jordan curve theorem remains valid in the Khalimsky plane. Since the Khalimsky plane is a quotient space of $\mathbb R^2$ induced by a tiling of squares, it is natural…

The square peg problem asks whether every Jordan curve in the plane has four points which form a square. The problem has been resolved (positively) for various classes of curves, but remains open in full generality. We present two new…

We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties…

The paper proves a result on the convergence of discrete conformal maps to the Riemann mappings for Jordan domains. It is a counterpart of Rodin-Sullivan's theorem on convergence of circle packing mappings to the Riemann mapping in the new…

We explicitly construct simple, piecewise minimizing geodesic, arbitrarily fine interpolation of simple and Jordan curves on a Riemannian manifold. In particular, a finite sequence of partition points can be specified in advance to be…

We prove that a Jordan $\calc^1$-curve in the plane contains any non-flat triangle up to translation and homothety with positive ratio. This is false if the curve is not $C^1$. The proof uses a bit configuration spaces, differential and…

Theorem converse to Jordan's curve theorem says that {\it if a compact set $K$ has two complementary domains in $R^{2}$, from each of which it is at every point accessible, it is a simple closed curve}. We show that the requirement of this…

We address the question of what is the correct higher dimensional analogue of Jordan curves from the point of view of quantitative rectifiability. More precisely, we show that 'topologically stable' sets can be used as covering objects in…

There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a…

I show that every rectifiable simple closed curve in the plane can be continuously deformed into a convex curve in a motion which preserves arc length and does not decrease the Euclidean distance between any pair of points on the curve.…