# Why is the Jordan curve theorem hard to prove?

## Why is the Jordan curve theorem hard to prove?

The difficulty arises when you try to handle the general case. This includes nowhere-differentiable curves like the boundary of the Koch snowflake, and even wilder curves which can’t even be drawn by hand, like Mariano says.

## Why is the Jordan Curve Theorem important?

In computational geometry, Jordan curve theorem can be used for testing whether a point lies inside or outside a simple polygon. The method is basically as follow. From a given point, trace a ray that does not pass through any vertex of the polygon (all rays but a finite number are convenient).

What is Jordan curve in graph theory?

Jordan curve theorem, in topology, a theorem, first proposed in 1887 by French mathematician Camille Jordan, that any simple closed curve—that is, a continuous closed curve that does not cross itself (now known as a Jordan curve)—divides the plane into exactly two regions, one inside the curve and one outside, such …

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What is rectifiable Jordan curve?

dist(γ(ti),γ(ti−1)) : k ∈ N and 0 = t0 < t1 < ··· < tk = 1. } A rectifiable curve is a curve with finite length. In general, the length of a (computable) Jordan curve can be infinite.

### How do you prove a curve is rectifiable?

A rectifiable curve is a curve having finite length (cf. Line (curve)). More precisely, consider a metric space (X,d) and a continuous function γ:[0,1]→X. γ is a parametrization of a rectifiable curve if there is an homeomorphism φ:[0,1]→[0,1] such that the map γ∘φ is Lipschitz.

What is the meaning of rectifiable?

Definition of rectifiable : capable of being rectified especially : having finite length a rectifiable curve.

What is smooth curve in complex analysis?

A curve γ is called a smooth curve if γ is differentiable and γ is continuous and nonzero for all t. A contour/piecewise smooth curve is a curve that is obtained by joining finitely many smooth curves end to end.

## What meant by rectifiable curve?

General definition A rectifiable curve is a curve having finite length (cf. Line (curve)). We can think of a curve as an equivalence class of continuous maps γ:[0,1]→X, where two maps γ and γ′ are equivalent if and only if there is an homeomorphism φ of [0,1] onto itself such that γ′=γ∘φ.