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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
Integrate speed for length, read 1 over R for circles, and split acceleration into throttle plus steering.
2 · Watch
Take it off screen
Where it sits
Learn first
This opens up
Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.