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: