# Differential Geometry/Frenet-Serret Formulae

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 :

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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.