# Serret-Frenet equations in ${\mathbb{R}}^{2}$

Given a plane curve^{}, we may associate to each point on the curve an
orthonormal basis consisting of the unit normal^{} tangent vector^{} and
the unit normal. In general, different points will have different
bases associated to them, so we may ask how the basis depends upon
the choice of point. The Serret-Frenet equations answer this
question by relating the rte of change of the basis vectors to
the curvature^{} of the curve.

Suppose $I$ is an open interval^{} and $c:I\to {\mathbb{R}}^{2}$ is a twice
continuously differentiable curve such that $\parallel {c}^{\prime}\parallel =1$.
Let us then
define the *tangent vector* and *normal vector ^{}* as

$\mathbf{T}$ | $=$ | ${c}^{\prime},$ | ||

$\mathbf{N}$ | $=$ | $J\cdot \mathbf{T},$ |

where $J=\left(\begin{array}{cc}\hfill 0\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$ is the rotational matrix that rotates the plane $90$ degrees counterclockwise.

## Curvature

Differentiating $\u27e8{c}^{\prime},{c}^{\prime}\u27e9=1$ yields
$\u27e8{\mathbf{T}}^{\prime},\mathbf{T}\u27e9=0$,
so ${\mathbf{T}}^{\prime}$ is in the orthogonal complement^{} of $\mathbf{T}$,
which is $1$-dimensional. Since $J\cdot \mathbf{T}$ is also in
the orthogonal complement,
it follows that there exists a function^{} $\kappa :I\to \mathbb{R}$ such that

$${\mathbf{T}}^{\prime}=\kappa J\cdot \mathbf{T}.$$ |

Furthermore, $\kappa $ is uniquely determined by this equation.
We define this unique $\kappa $ to
be the *curvature* of $c$. Explicitly,

$$\kappa =\u27e8{\mathbf{T}}^{\prime},J\cdot \mathbf{T}\u27e9.$$ |

## Serret-Frenet equations

By the definition of curvature

${\mathbf{T}}^{\prime}$ | $=$ | $\kappa J\cdot \mathbf{T}=\kappa \mathbf{N}$ |

and so

${\mathbf{N}}^{\prime}$ | $=$ | $J\cdot {\mathbf{T}}^{\prime}=\kappa J\mathbf{N}=-\kappa \mathbf{T}$ |

since ${J}^{2}=-\mathrm{I}$. These are the *Serret-Frenet*
equations

${\left(\begin{array}{c}\hfill \mathbf{T}\hfill \\ \hfill \mathbf{N}\hfill \end{array}\right)}^{\prime}=\left(\begin{array}{cc}\hfill 0\hfill & \hfill \kappa \hfill \\ \hfill -\kappa \hfill & \hfill 0\hfill \end{array}\right)\left(\begin{array}{c}\hfill \mathbf{T}\hfill \\ \hfill \mathbf{N}\hfill \end{array}\right).$ |

Title | Serret-Frenet equations in ${\mathbb{R}}^{2}$ |
---|---|

Canonical name | SerretFrenetEquationsInmathbbR2 |

Date of creation | 2013-03-22 15:16:57 |

Last modified on | 2013-03-22 15:16:57 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 8 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 53A04 |

Related topic | SerretFrenetFormulas |