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A Compendium of Curve Formulae In the following r(t) = x(t) , y(t) , z(t) is a parametrization of a curve. The vectors T̂(t), N̂(t), and B̂(t) = T̂(t) × N̂(t) are the unit tangent, normal and binormal vectors, respectively, at r(t). The tangent vector points in the direction of travel (i.e. direction of increasing t) and the normal vector points toward the centre of curvature. The arc length from time 0 to time t is denoted s(t). Then ◦ the velocity v(t) = dr dt (t) ◦ the acceleration a(t) = d2 r (t) dt2 ◦ the speed ds dt (t) ◦ the arc length = = ds dt (t) T̂(t) d2 s (t) T̂(t) dt2 + κ(t) 2 ds (t) N̂(t) dt = |v(t)| = dr dt (t) R1p Rt (τ ) dτ = x′ (τ )2 + y ′ (τ )2 + z ′ (τ )2 dτ s(t) = 0 ds dt 0 ◦ the curvature κ(t) = ◦ the radius of curvature ρ(t) = |v(t) × a(t)| 3 ds (t) dt 1 κ(t) ◦ the centre of curvature is r(t) + ρ(t)N̂(t) ◦ the torsion (t) v(t) × a(t) · da dt τ (t) = 2 |v(t) × a(t)| ◦ the binormal B̂(t) = T̂(t) × N̂(t) = v(t) × a(t) |v(t) × a(t)| Under arclength parametrization (i.e. if t = s) we have T̂(s) = formulae dT̂ (s) = κ(s) N̂(s) ds dN̂ ds (s) = dB̂ ds (s) = −τ (s) N̂(s) dr (s) ds and the Frenet-Serret τ (s) B̂(s) − κ(s) T̂(s) When the curve lies in the x–y plane dx d2 y (t) 2 (t) − dy (t) d2 2x (t) dt dt dt κ(t) = hdt 2 2 i3/2 dy dx (t) + dt (t) dt When the curve lies in the x–y plane and the parameter is x (so that y is given as a function y(x) of x) d2 y 2 (x) dx κ(x) = h 2 i3/2 dy (x) 1 + dx January 10, 2016 A Compendium of Curve Formulae