Given a parametrised curve \alpha\colon [t_0,t_f] \to \mathbb{R}^n,

• Norm: |\alpha|=\sqrt{\alpha_1^2+\dots+\alpha_n^2}
• Exterior product: given v,w \in \mathbb{R}^n, v\wedge w \in \mathbb{R}^n is the only vector orthogonal to v and w with |v\wedge w|=|v||w|.
• Curvature and torsion:

k=|\alpha''|;\quad \tau=-\frac{\det(\alpha',\alpha'',\alpha''')}{k^2}

• Frenet frame (n=3): t=\alpha', n=t'/k, b=t\wedge n
• Rotation index (n=2): 2\pi I=\int^{t_1}_{t_0}k
• Evolute: \beta=\alpha+n/k
• Arc-length: s(t)=\int^{t_f}_{t_0}|\alpha|
• Local canonical form (n=3; arc-length):

\alpha(s)=\left(s-\frac{1}{6}k^2s^3,\frac{1}{2}ks^2+\frac{k's^3}{6},-\frac{1}{6}k\tau s^3\right)