Arc Length and Rectifiable Curves

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1 · Prerequisites

Abstract Algebra

Linear Algebra

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Definition

A C-Lipschitz map multiplies path length by at most C; isometries preserve length and scalar dilation multiplies it by the absolute scale

Theorem

Arc length is lower semicontinuous under uniform convergence of paths

Theorem

If γ:[a,b]→ℝⁿ is continuous, differentiable on (a,b), and γ' extends continuously to [a,b], then L(γ)=∫ₐᵇ‖γ'(t)‖₂ dt

Corollary

If f is continuous on [a,b], differentiable on (a,b), and f' extends continuously to [a,b], then the graph of f has length ∫ₐᵇ√1+f'(t)² dt

Corollary

A continuous piecewise-C¹ path is rectifiable and its length is the sum of the speed integrals over its pieces

Definition

The arc-length function s_γ(t)=L(γ|_[a,t]) of a rectifiable path

Lemma

The arc-length function is continuous and nondecreasing, with increments equal to subpath lengths; it is strictly increasing exactly when no nondegenerate subpath is constant

Theorem

Every rectifiable path factors through its arc-length function as a unit-speed path on [0,L]

Corollary

A regular C¹ path has a C¹ arc-length reparametrization with derivative of Euclidean norm one

Sources

Arc Length and Rectifiable Curves

3 · Items

Paths in ℝⁿ, inscribed polygonal sums, arc length as their supremum, and rectifiability

Every endpoint chord is no longer than the arc: ‖γ(b)-γ(a)‖₂≤ L(γ)

Refining a partition cannot decrease its inscribed polygonal length

A path in ℝⁿ is rectifiable exactly when every coordinate has bounded variation

Arc length is additive across every subdivision point and decreases under restriction

Arc length is invariant under every continuous surjective monotone reparametrization, including pauses and reversal

A C-Lipschitz map multiplies path length by at most C; isometries preserve length and scalar dilation multiplies it by the absolute scale

Arc length is lower semicontinuous under uniform convergence of paths

If γ:[a,b]→ℝⁿ is continuous, differentiable on (a,b), and γ' extends continuously to [a,b], then L(γ)=∫ₐᵇ‖γ'(t)‖₂ dt

If f is continuous on [a,b], differentiable on (a,b), and f' extends continuously to [a,b], then the graph of f has length ∫ₐᵇ√1+f'(t)² dt

A continuous piecewise-C¹ path is rectifiable and its length is the sum of the speed integrals over its pieces

The arc-length function s_γ(t)=L(γ|_[a,t]) of a rectifiable path

The arc-length function is continuous and nondecreasing, with increments equal to subpath lengths; it is strictly increasing exactly when no nondegenerate subpath is constant

Every rectifiable path factors through its arc-length function as a unit-speed path on [0,L]

A regular C¹ path has a C¹ arc-length reparametrization with derivative of Euclidean norm one