Calculus › Section M · Vector-valued functions › Topic M.5
Arc length and curvature
How much wire does it take to wind a spring, and how sharply does a railway curve bend at its tightest point? Arc length measures a curve in units of distance, curvature measures how fast its direction swings per unit travelled, and together they explain why speed multiplies the sideways shove you feel in a turning car.
In this topic
- Measuring a curve in space
- Worked example: the length of a helix
- The arc length function
- Parametrising by arc length
- Curvature: how fast direction changes
- Worked example: curvature of a parabola
- Circles, lines, and the circle of curvature
- Curvature and the sideways shove
- Applied: friction on a bend
- Common mistakes to avoid
- Key takeaways
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Standards: Beyond AP