Unit 12.4–12.5: Tangent, Normal, Arc Length, Curvature

Unit tangent vector

T=v/|v| velocity direction

Principal unit normal vector

N=T'/|T'| change of direction

Binormal vector

B=T×N

Arc length

s=∫|r'(t)|dt

Curvature using T

κ=|T'(t)|/|r'(t)| T’ is change in direction in respect to time and dividing by r’ just gets speed out of the equation

Curvature using cross product

κ=|r'×r''|/|r'|³

Meaning of curvature

How sharply a curve bends.

Large curvature

Tight turn.

Small curvature

Almost straight.

How to find T

Velocity divided by speed.

How to find N

Differentiate T then normalize.

Exam shortcut

For curvature use κ=|r'×r''|/|r'|³ almost every time.

Common mistake

Forgetting to normalize T before finding N.