Mr. Integral
Math notes/Calculus III/Vectors and Geometry in Space

Arc Length, Curvature and Turning — Cheat sheet

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Core rule

T=r′/∥r′∥,κ=∥r′×r′′∥/∥r′∥3,aT=d∥r′∥/dt,aN=κ∥r′∥2.T=r'/\|r'\|,\quad \kappa=\|r'\times r''\|/\|r'\|^3,\quad a_T=d\|r'\|/dt,\quad a_N=\kappa\|r'\|^2.

Watch for

Curvature is geometric and has inverse-length units. These formulas require nonzero speed; a normal direction may fail where curvature is zero.