##  [Length](/length-0) 

 Definition

A one-dimensional measurement of the extent of a curve or segment between two points within a geometric setting, computed by integrating the speed (norm of the velocity) of a parametrized curve with respect to a chosen metric.

 

 

 

 

 

 





## Principle

Principle

Accumulate infinitesimal distances: the length of a differentiable curve is the integral over the parameter interval of the norm of its derivative with respect to the ambient metric, which yields a coordinate-independent scalar when the metric is specified.

 

 

 

 

 





## Demonstration

Demonstration

In Euclidean space, the length of a smooth curve γ(t) for t in [a,b] is ∫_a^b |γ'(t)| dt; for a straight line segment between two points this reduces to the Euclidean distance between the endpoints.

 

 

 

 

## Misapplication

Misapplication

Using coordinate differences directly as lengths on a manifold without accounting for the metric; such computations can give incorrect results when the ambient metric is non‑Euclidean or when parametrization is not by arc length.

 

 

 

 

 





## Consequence

Consequence

Provides a metric notion of distance along curves and underlies definitions of geodesic, perimeter, and arc length; it is invariant under reparametrization by orientation‑preserving diffeomorphisms and depends on the chosen metric structure.

 

 

 

 

## Reversal

Reversal

Instead of integrating the norm of the velocity, one could consider other curve functionals (e.g., energy ∫|γ'|^2), which are related but favor different analytical properties; minimizing energy often implies minimizing length but with different regularity behavior.

 

 

 

 

 





## Boundary

Boundary

Applies to differentiable (or rectifiable) curves in a metric or Riemannian space; excludes non‑rectifiable continuous curves whose total variation (length) is infinite and discrete point sets where length is not defined without additional structure.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Nearby concepts include 'distance' (shortest length between points) and 'energy' (integral of squared speed); length is a geometric, reparametrization‑invariant integral, while energy depends on parametrization and analytic convenience.

 

 

 

 

 





## Synthesis

Synthesis

Length measures the accumulated infinitesimal displacements of a curve according to a specified metric, giving the canonical one‑dimensional size of segments and paths that underpins distance, geodesics, and perimeter.