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 …

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.

What is Jordan curve in complex analysis?

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.

Is fixable a word?

Capable of being fixed, repairable. Attachable.

Will be rectify or rectified?

verb (used with object), rec·ti·fied, rec·ti·fy·ing. to make, put, or set right; remedy; correct: He sent them a check to rectify his account. to put right by adjustment or calculation, as an instrument or a course at sea.