The derivatives of the vectors t, p, and b can be expressed as a linear combination of these vectors. The formulae for these expressions are called the Frenet-Serret Formulae. This is natural because t, p, and b form an orthogonal basis for a three-dimensional vector space.
Of course, we know already that and so it remains to find . First, we differentiate to obtain so it takes on the form . We take the dot product of this with t to obtain . Taking the derivative of , we get or . Also, taking the dot product of with b, we obtain . Taking the derivative of , we get . Thus, we arrive at the following expression for :
This formula, combined with the previous two formulae, are together called the Frenet-Serret Formulae and they can be represented by a skew-symmetric matrix.