---
title: "Arc Length and Curvature of a Space Curve"
description: "Reparametrise a curve by arc length and measure how fast its direction turns, separating the path from the speed along it."
canonical: https://lightmysky.com/learn/mathematics/arc-length-and-curvature-of-a-space-curve-mt_8mXYE8bVG1
source: https://lightmysky.com/learn/mathematics/arc-length-and-curvature-of-a-space-curve-mt_8mXYE8bVG1.md
retrieved: 2026-09-12
---

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# Arc Length and Curvature of a Space Curve

Reparametrise a curve by arc length and measure how fast its direction turns, separating the path from the speed along it.

Subject: Mathematics · Area: Geometry · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/arc-length-and-curvature-of-a-space-curve-mt_8mXYE8bVG1

## Ready when they can

- Compute the arc length of a space curve and reparametrise a simple one by it
- Compute curvature and interpret it as the reciprocal of the radius of the best-fitting circle
- Split an acceleration into tangential and normal components and say what each means physically

## Lesson: How sharply a space curve bends

A vector valued function glues three ordinary functions into one moving point r of t. Differentiate component by component for velocity, and once more for acceleration. Speed is the length of the velocity vector, and these tools turn curve geometry into derivative algebra.

**Example.** For r of t equal (t, 2t, 2t), velocity is the constant (1, 2, 2) with speed 3, so over t from 0 to 2 the particle travels 6. Arc length is the integral of speed, and measuring distance travelled gives the unit speed parameter s.

Curvature is the turning rate under that unit speed walk. For a circle it equals 1 over the radius: radius 4 gives 0.25 and radius 5 gives 0.2, while a straight line never turns so its curvature is 0. This number belongs to the path, not to how fast it is traced.

**Tip.** Acceleration splits into a tangential part that changes speed and a normal part that changes direction. The normal part points inside the bend with size curvature times speed squared. A car at steady speed on a bend accelerates inward, while a rocket on a straight track shows only the tangential part.

**Recap.** Integrate speed for length, read 1 over R for circles, and split acceleration into throttle plus steering.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Lines and Planes in Space](https://lightmysky.com/learn/mathematics/lines-and-planes-in-space-mt_kIX5ak1GRv)
- [The Cross Product and Oriented Area](https://lightmysky.com/learn/mathematics/the-cross-product-and-oriented-area-mt_QEBM_Ms-SK)
- [Vector-Valued Functions and Motion Along a Curve](https://lightmysky.com/learn/mathematics/vector-valued-functions-and-motion-along-a-curve-mt_Xcjt6exMy3)