Compute the Frenet-Serret system (curvatures and orthonormal frame) for a parametric curve.
FrenetSerretSystem[{x1, …, xn}, t]
FrenetSerretSystem[{x1, …, xn}, t, chart]
FrenetSerretSystem[{Cos[t], Sin[t]}, t]
→ {{1/Sqrt[Cos[t]^2 + Sin[t]^2]}, {{-(Sin[t]/Sqrt[Cos[t]^2 + Sin[t]^2]), Cos[t]/Sqrt[Cos[t]^2 + Sin[t]^2]}, {-(Cos[t]/Sqrt[Cos[t]^2 + Sin[t]^2]), -(Sin[t]/Sqrt[Cos[t]^2 + Sin[t]^2])}}}FrenetSerretSystem[{1, a t, b t}, t]
→ {{0, 0}, {{0, a/Sqrt[a^2 + b^2], b/Sqrt[a^2 + b^2]}, {0, 0, 0}, {0, 0, 0}}}FrenetSerretSystem[a x ^ 2 + b x + c, x]
→ {{(2*a)/(1 + (b + 2*a*x)^2)^(3/2)}, {{1/Sqrt[1 + (b + 2*a*x)^2], (b + 2*a*x)/Sqrt[1 + (b + 2*a*x)^2]}, {-((b + 2*a*x)/Sqrt[1 + (b + 2*a*x)^2]), 1/Sqrt[1 + (b + 2*a*x)^2]}}}