turning angle

For a plane curve, the tangential angle ϕ is defined by ρ d ϕ = d s, where s is the arc length and ρ is the radius of curvature. The tangential angle is therefore given by ϕ = integral_0^t s'(t) κ(t) d t, where κ(t) is the curvature. For a plane curve r(t), the tangential angle ϕ(t) can also be defined by (r'(t))/( left bracketing bar r'(t) right bracketing bar ) = [cos[ϕ(t)] sin[ϕ(t)]]. Gray calls ϕ the turning angle instead of the tangential angle.

arc length | curvature | plane curve | radius of curvature | torsion