Frenet-Serret Bases#

Definition: Tangent Vector (Velocity)

Let \(\gamma: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}^n\) be a vector-valued function.

The tangent vector or velocity of \(\gamma\) is its derivative (provided that it exists):

\[\dot{\gamma}\]

Definition: Speed

The Euclidean norm of \(\gamma\)'s velocity is known as \(\gamma\)'s speed.

Definition: Unit Tangent Vector

The normalization of \(\gamma\)'s tangent vector is known as \(\gamma\)'s unit tangent vector:

\[\frac{1}{||\dot{\gamma}||}\dot{\gamma}\]

Notation

\[\mathbf{T}\]

Definition: Normal Vector

Let \(\gamma: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}^n\) be a vector-valued function.

The normal vector of \(\gamma\) is the derivative of its tangent vector (provided that it exists):

\[\ddot{\gamma}\]

Definition: Unit Normal Vector

The normalization of \(\gamma\)'s normal vector is known as \(\gamma\)'s unit normal vector:

\[\frac{1}{||\ddot{\gamma}||} \ddot{\gamma}\]

Notation

\[\mathbf{N}\]

Definition: Binormal Vector

Let \(\gamma: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}^3\) be a vector-valued function.

The binormal vector of \(\gamma\) is the cross product of its unit tangent vector and its unit normal vector (provided that these exist):

\[\mathbf{T} \times \mathbf{N}\]

Notation

\[\mathbf{B}\]