### Space curves

Given a curve \(f:\, I\rightarrow\mathbb{R}^{3}\) parametrized by arc length, the vector \(T=\left(T_{1},T_{2},T_{3}\right)\) given by \(f'(t)\) is the **tangent vector** of \(f\) at \(t\) while \(N=\left(N_{1},N_{2},N_{3}\right)\) given by \[\frac{f''(t)}{\left|f''(t)\right|}\] is the **normal vector**.

In the 3D case, there is a third important vector: the **binormal**, defined as follows:
\[\begin{array}{ccl}
B & = & \left(B_{1},B_{2},B_{3}\right)=T\times N=\\
& = & "\mbox{det}"\left(\begin{array}{ccc}
e_{1} & e_{2} & e_{3}\\
T_{1} & T_{2} & T_{3}\\
N_{1} & N_{2} & N_{3}
\end{array}\right)=\\
& = & \left(T_{2}N_{3}-T_{3}N_{2},T_{3}N_{1}-T_{1}N_{3},T_{1}N_{2}-T_{2}N_{1}\right)
\end{array}\]

It may be proved that also
\(\left|B\right|=1\). These three unitary vectors
(\(T\)),
(\(N\)) and (\(B\)) form the so called
**Frenet-Serret frame**. They are
orthogonal to each other; \(T\)
indicates the direction at which the curve moves; a
\(N\) indicates the direction at which the curve is turning while
\(B\) is the vector orthogonal to
\(T\)
and \(N\) such that the three vectors form an orthonormal basis with positive orientation.

The plane defined by
tangent vector \(T\)
and normal vector \(N\) is called the
**osculating plane**. The osculating plane of the curve at a given point is the plane that better approximates the curve in that point. Note also that the
binormal vector \(B\)
is orthogonal to the osculating plane.

A few examples: