---- Let $\alpha:(a,b) \to \mathbb{R} ^{3}$ be a [[regular curve|regular]] [[parameterized curve]] [[parameterization by arc length|by arc length]], $\|\alpha'(s)\|_{2}\equiv 1$. Further assume $\alpha''(s)$ is never $0$ (no [[osculating plane|1st-order singular points]]). ^b13cef > [!definition] Definition. ([[Frenet frame]]) > To each value of the parameter $s$ we associate a [[frame]] of three [[orthonormal]] vectors: > - The [[velocity vector of a parameterized curve|tangent vector]] $t(s)$; > - The [[unit normal vector to a parameterized curve|normal vector]] $n(s)$; > - The [[binormal vector to a parameterized curve|binormal vector]] $b(s)$. > The trihedron thus formed is called the **Frenet Trihedron** at $s$. > > Taking [[derivative|derivatives]] yields the following first-order [[ODE]], called the **Frenet evolution formulas**: $\begin{align} t' = & \kappa n \\ n' = & -\kappa t - \tau b \\ b' = & \tau n, \end{align}$ where $\kappa$ denotes [[curvature of parameterized curve|curvature]] and $\tau$ denotes [[torsion of a parameterized curve|torsion]]. In [[matrix]] notation: $\begin{bmatrix} t' \\ n' \\ b' \end{bmatrix} = \begin{bmatrix} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{bmatrix} \begin{bmatrix} t \\ n \\ b \end{bmatrix}$ Terminology: >- The plane [[submodule generated by a subset|spanned]] by $t,b$ is called the **rectifying plane**; >- The plane [[submodule generated by a subset|spanned]] by $n,b$ is called the **normal plane**; >- The [[line|lines]] which contain $n(s)$ and $b(s)$ and pass through $\alpha(s)$ are called the **principal normal** and the **binormal**, respectively; >- The inverse $R(s):=\frac{1}{\kappa(s)}$ of the [[curvature of parameterized curve|curvature]] is called the **radius of curvature** at $s$.<p><a href="https://commons.wikimedia.org/wiki/File:Frenetframehelix.gif#/media/File:Frenetframehelix.gif"><img src="https://upload.wikimedia.org/wikipedia/commons/8/87/Frenetframehelix.gif" alt="Frenetframehelix.gif" height="205" width="290"></a><br> <a href="https://commons.wikimedia.org/w/index.php?curid=7519084"></a></p> ><p><a href="https://commons.wikimedia.org/wiki/File:Torus-Knot_nebeneinander_animated.gif#/media/File:Torus-Knot_nebeneinander_animated.gif"><img src="https://upload.wikimedia.org/wikipedia/commons/7/70/Torus-Knot_nebeneinander_animated.gif" alt="Torus-Knot nebeneinander animated.gif" height="435" width="1024"></a><br> <a href="https://creativecommons.org/licenses/by-sa/3.0" title="Creative Commons Attribution-Share Alike 3.0"></a>, <a href="https://commons.wikimedia.org/w/index.php?curid=18546395"></a></p> ^63129c > [!justification] > The formulas are quick to derive. >- $t'(s)=\kappa(s)n(s)$ immediately via the definition of $n(s)$. >- Since $n=b \times t$ (see the [[cross product]]'s first property), using the [[derivative of the cross product]] we have $\begin{align} n'(s)= & b(s)\times t'(s)+b'(s) \times t(s) \\ = & b(s) \times \kappa (s) n(s) + \tau(s) n(s) \times t(s) \\ = & -\kappa \big( b \times n \big) -\tau \big( n \times t \big) \\ = & -\kappa t - \tau b. \end{align}$ > >- $b'(s)=\tau(s)n(s)$ by definition of $\tau(s)$. ^1bcf3c ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` "{\n \"Examples\": \"[[Frenet frame#^63129c|animations in the Frenet frame note showing helix and torus knots]]\",\n \"Nonexamples\": \"N/A\",\n \"Used in Constructing\": \"[[the local canonical form in R3]]\",\n \"Constructions Used\": \"[[velocity vector of a parameterized curve]], [[unit normal vector to a parameterized curve]], [[binormal vector to a parameterized curve]], [[derivative of the cross product]], [[connection on a manifold]]\",\n \"Specializations\": \"N/A\",\n \"Generalizations\": \"[[frame]]\",\n \"Justifications and Intuition\": \"[[Frenet frame#^1bcf3c|Justification for the derivatives and cross product relations in the Frenet frame]]\",\n \"Properties\": \"[[Frenet frame#^b13cef|Properties of the Frenet frame related to regular parameterized curves with natural length parameterization]]\",\n \"Sufficiencies\": \"None yet\",\n \"Equivalences\": \"[[Frenet frame#^fd36b9|Equivalence of definitions for parameterizations by arc length]]\"\n}" "{\n \"Examples\": \"[[Frenet frame#^63129c|Animations in the Frenet frame note showing helix and torus knots]]\",\n \"Nonexamples\": \"N/A\",\n \"Used for Constructing\": \"[[the local canonical form in R3]]\",\n \"Constructions Used\": \"[[velocity vector of a parameterized curve]], [[unit normal vector to a parameterized curve]], [[binormal vector to a parameterized curve]], [[derivative of the cross product]], [[connection on a manifold]]\",\n \"Specializations\": \"N/A\",\n \"Generalizations\": \"[[frame]]\",\n \"Justifications and Intuition\": \"[[Frenet frame#^1bcf3c|Justification for the derivatives and cross product relations in the Frenet frame]]\",\n \"Properties\": \"[[Frenet frame#^b13cef|Properties of the Frenet frame related to regular parameterized curves with natural length parameterization]]\",\n \"Sufficiencies\": \"None yet\",\n \"Equivalences\": \"[[Frenet frame#^fd36b9|Equivalence of definitions for parameterizations by arc length]]\"\n}"