Space Curves and T, N, B vectors

The unit tangent vector T, the unit normal vector N and the unit binormal vector B are three mutually perpendicular vectors used to describe a curve in two or three dimensions. This moving coordinate system is attached to the curve and describes the shape of the curve independent of any parameterization.

If the curve is given parametrically by

x = x\|t/|,  y = y|\t/|,   z = z|\t/|
the position, unit tangent, unit normal, and unit binormal vectors are
r|\t/| = x|\t/|i+ y|\t/|j+ z|\t/|k
         
T|\t/| =-r|\t/|
      |r |\t/||
        T |\t/|
N\|t/| = -- --
       |T |\t/||
B\|t/| = T|\t/| N|\t/|

Below is the graph of

x = 2 cos|\2p t/|,  y = sin|\2p t/|,  z = t, - 1 < t < 1.