Frenet-Serret Frame of a Curve and Derivation of Frenet-Serret Equations

1. Each Point on a Curve can be said to be associated with 3 Mutually Perpendicular Unit Vectors (and 3 Mutually Perpendicular Planes formed by these 3 Unit Vectors), which can be used to study and understand the Nature / Properties of the Curve at that Point.
   
   The 3 Mutually Perpendicular Unit Vectors are the Unit Tangent Vector (denoted by \(\hat{\mathbf{T}}\)) , the Unit Prinicipal Normal Vector (denoted by \(\hat{\mathbf{N}}\)), and the Unit Binormal Vector (denoted by \(\hat{\mathbf{B}}\)) of the Curve at a Point.
   
   The 3 Mutually Perpendicular Planes are
   1. Osculating Plane: The Plane containg the Unit Tangent Vector \(\hat{\mathbf{T}}\) and the Unit Prinicipal Normal Vector \(\hat{\mathbf{N}}\) perpendicular to the Unit Binormal Vector \(\hat{\mathbf{B}}\).
   2. Normal Plane: The Plane containg the Unit Prinicipal Normal Vector \(\hat{\mathbf{N}}\) and the Unit Binormal Vector \(\hat{\mathbf{B}}\) perpendicular to the Unit Tangent Vector \(\hat{\mathbf{T}}\).
   3. Rectifying Plane: The Plane containg the Unit Binormal Vector \(\hat{\mathbf{B}}\) and the Unit Tangent Vector \(\hat{\mathbf{T}}\) perpendicular to the Unit Prinicipal Normal Vector \(\hat{\mathbf{N}}\).
2. These 3 Unit Vectors and 3 Planes form a Local Coordinate System or Local Reference Frame for/at every Point on the Curve called the Frenet-Serret Frame. The 3 Unit Vectors are related to each other via the following equations
   
   \(\hat{\mathbf{B}}=\hat{\mathbf{T}} \times \hat{\mathbf{N}}\)   ...(1)
   
   \(\hat{\mathbf{T}}=\hat{\mathbf{N}} \times \hat{\mathbf{B}}\)   ...(2)
   
   \(\hat{\mathbf{N}}=\hat{\mathbf{B}} \times \hat{\mathbf{T}}\)   ...(3)
3. The Instantaneous Rate of Change of these 3 Unit Vectors are determined by Calculating the Derivative of the Function representing these Vectors with respect to the Arc Length of the Curve \(s\). These Derivatives are represented using a set of 3 equations called Frenet-Serret Equations given as follows
   
   \(\hat{\mathbf{T\hspace{.1cm}'}}={\Large \frac{d \hat{\mathbf{T}}}{ds}}=\kappa \hat{\mathbf{N}}\)   ...(4)
   
   \(\hat{\mathbf{B\hspace{.1cm}'}}={\Large \frac{d \hat{\mathbf{B}}}{ds}} =-\tau \hat{\mathbf{N}}\)   ...(5)
   
   \(\hat{\mathbf{N\hspace{.1cm}'}}={\Large \frac{d \hat{\mathbf{N}}}{ds}}=\tau \hat{\mathbf{B}} - \kappa \hat{\mathbf{T}}\)   ...(6)
   
   These equations can help in finding the Properties of Curve like the Curvature \(\kappa\) and Torsion \(\tau\) at any Point of Curve.
4. The Frenet-Serret Equations can be represented in form of a Matrix Equation as
   
   \(\begin{bmatrix} \hat{\mathbf{T\hspace{.1cm}'}} \\ \hat{\mathbf{N\hspace{.1cm}'}} \\ \hat{\mathbf{B\hspace{.1cm}'}}\end{bmatrix} =\begin{bmatrix} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{bmatrix}\begin{bmatrix} \hat{\mathbf{T}} \\ \hat{\mathbf{N}} \\ \hat{\mathbf{B}}\end{bmatrix}\)   ...(7)
   
   
5. The following gives the derivation of Frenet-Serret Frame Vectors, Frenet-Serret Equations and related formulae
   
   The Parametric Position Vector Function of any Curve can be given as follows
   
   \(\vec{R}=x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}}\)   ...(8)
   
   where Vector \(\vec{R}\) gives the Position Vector of the Curve. Now, if \(x\), \(y\) and \(z\) are Functions of Variable \(t\) given as follows
   
   \(x=f_x(t)\),   \(y=f_y(t)\),   \(z=f_z(t)\)   ...(9)
   
   then, using equation (8) and (9)
   
   \(\vec{R}=f_x(t)\hat{\mathbf{i}} + f_y(t)\hat{\mathbf{j}} + f_z(t)\hat{\mathbf{k}}\)   ...(10)
   
   The Tangent Vector to the Curve denoted by \(\vec{R\hspace{.1cm}'}\) is given by Derivative of Vector \(\vec{R}\) with respect to Variable \(t\) as follows
   
   \(\vec{R\hspace{.1cm}'}={\Large\frac{d \vec{R}}{dt}} = {\Large \frac{dx}{dt}}\hat{\mathbf{i}} + {\Large \frac{dy}{dt}}\hat{\mathbf{j}} + {\Large \frac{dz}{dt}}\hat{\mathbf{k}} = {\Large \frac{d (f_x(t))}{dt}}\hat{\mathbf{i}} + {\Large \frac{d (f_y(t))}{dt}}\hat{\mathbf{j}} + {\Large \frac{d (f_z(t))}{dt}}\hat{\mathbf{k}} \)   ...(11)
   
   The Magnitude of the Tangent Vector \(\vec{R\hspace{1mm}'}\), gives the Derivative of Arc Length of Curve \(s\) with repect to Parameter Variable \(t\) as follows
   
   \({\Large \frac{ds}{dt}} ={|\Large\frac{d \vec{R}}{dt}|} = | \vec{R\hspace{.1cm}'}|={\Large\sqrt{\frac{d \vec{R}}{dt} \cdot \frac{d \vec{R}}{dt}}} = \sqrt{\vec{R\hspace{.1cm}'} \cdot \vec{R\hspace{.1cm}'}} = \sqrt{({\Large \frac{dx}{dt}}\hat{\mathbf{i}} + {\Large \frac{dy}{dt}}\hat{\mathbf{j}} + {\Large \frac{dz}{dt}}\hat{\mathbf{k}}) \cdot ({\Large \frac{dx}{dt}}\hat{\mathbf{i}} + {\Large \frac{dy}{dt}}\hat{\mathbf{j}} + {\Large \frac{dz}{dt}}\hat{\mathbf{k}})} =\sqrt{{\Large (\frac{dx}{dt})}^2 + {\Large (\frac{dy}{dt})}^2 + {\Large (\frac{dz}{dt})}^2}\)   ...(12)
   
   Therefore, using equations (11) and (12), the Unit Tangent Vector to the Curve denoted by \(\hat{\mathbf{T}}\) is calculated as
   
   \(\hat{\mathbf{T}}={\Large\frac{\vec{R\hspace{.1cm}'}}{|\vec{R\hspace{.1cm}'}|}} ={\Large \frac{\frac{d \vec{R}}{dt}} {\frac{ds}{dt}}}= {\Large \frac{d \vec{R}}{ds}} \)   ...(13)
   
   The Rate of Change of Unit Tangent Vector to the Curve denoted by \(\vec{T}\) is given by Derivative of Vector \(\hat{\mathbf{T}}\) with respect to Variable \(t\) as follows
   
   \(\vec{T}={\Large\frac{d \hat{\mathbf{T}}}{dt}} ={\Large(\frac{\vec{R\hspace{.1cm}'}}{|\vec{R\hspace{.1cm}'}|})}' ={\Large(\frac{\vec{R\hspace{.1cm}'}}{\sqrt{\vec{R\hspace{.1cm}'} \cdot \vec{R\hspace{.1cm}'}}})}'\)
   
   \(\Rightarrow \vec{T}={\Large \frac{\sqrt{\vec{R\hspace{.1cm}'} \cdot \vec{R\hspace{.1cm}'}}\hspace{.2cm}\vec{R\hspace{.1cm}''} \hspace{.1cm}-\hspace{.1cm}\frac{2 (\vec{R\hspace{.1cm}'} \cdot \vec{R\hspace{.1cm}''}) \hspace{.2cm} \vec{R\hspace{.1cm}' }}{2 \sqrt{\vec{R\hspace{.1cm}'} \cdot \vec{R\hspace{.1cm}'}}} }{\vec{R\hspace{.1cm}'} \cdot \vec{R\hspace{.1cm}'}}} ={\Large \frac{(\vec{R\hspace{.1cm}'} \cdot \vec{R\hspace{.1cm}'})\hspace{.2cm}\vec{R\hspace{.1cm}''} \hspace{.1cm}-\hspace{.1cm}(\vec{R\hspace{.1cm}'} \cdot \vec{R\hspace{.1cm}''}) \hspace{.2cm} \vec{R\hspace{.1cm}'}} {\vec{R\hspace{.1cm}'} \cdot \vec{R\hspace{.1cm}'} \sqrt{\vec{R\hspace{.1cm}'} \cdot \vec{R\hspace{.1cm}'}} }}\)
   
   \(\Rightarrow \vec{T}={\Large \frac{\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})} {|\vec{R\hspace{.1cm}'}|^2 |\vec{R\hspace{.1cm}'}|}} ={\Large \frac{\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})} {|\vec{R\hspace{.1cm}'}|^3}} ={\Large \frac{(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'}} {|\vec{R\hspace{.1cm}'}|^3}}\)   ...(14)
   
   \(\Rightarrow |\vec{T}|={\Large \frac{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})|} {|\vec{R\hspace{.1cm}'}|^3}} ={\Large \frac{|(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'}|} {|\vec{R\hspace{.1cm}'}|^3}}\)   ...(15)
   
   Using equations (15) and (16), the Unit Principal Normal Vector to the Curve denoted by \(\hat{\mathbf{N}}\) is given as
   
   \(\hat{\mathbf{N}}={\Large \frac{\frac{d \hat{\mathbf{T}}}{dt}}{|\frac{d \hat{\mathbf{T}}}{dt}|}}={\Large \frac{\vec{T}}{|\vec{T}|}} ={\Large \frac{\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})} {|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})|}} ={\Large \frac{(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'}} {|(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'}|}}\)   ...(16)
   
   Please note that the Unit Principal Normal Vector \(\hat{\mathbf{N}}\) Always Points at the Direction of Concavity of the Curve.
   
   Also note that the Unit Principal Normal Vector \(\hat{\mathbf{N}}\) becomes 0 at Points of any Curve where the Direction of Concavity is Not Cleary Defined, for example at Cusps, Inflection Points or Places where the Curve Intersects itself.
   
   The Curvature Vector to the Curve denoted by \(\hat{\mathbf{T\hspace{.1cm}'}}\) or \(\vec{\kappa}\) is given as
   
   \(\hat{\mathbf{T\hspace{.1cm}'}}=\vec{\kappa}={\Large \frac{\frac{d \hat{\mathbf{T}}}{dt}} {\frac{ds}{dt}}}= {\Large \frac{d \hat{\mathbf{T}}}{ds}} = {\Large \frac{d^2 \vec{R}}{ds^2}} ={\Large\frac{\vec{T}}{|\vec{R\hspace{.1cm}'}|}} \)   ...(17)
   
   \(\Rightarrow \hat{\mathbf{T\hspace{.1cm}'}}=\vec{\kappa}={\Large\frac{\frac{\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})} {|\vec{R\hspace{.1cm}'}|^3}}{|\vec{R\hspace{.1cm}'}|}} ={\Large \frac{\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})} {|\vec{R\hspace{.1cm}'}|^4}} ={\Large \frac{(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'}} {|\vec{R\hspace{.1cm}'}|^4}}\)   ...(18)
   
   \(\Rightarrow |\hat{\mathbf{T\hspace{.1cm}'}}|=|\vec{\kappa}|=\kappa={\Large |\frac{d \hat{\mathbf{T}}}{ds}|}={\Large |\frac{d^2 \vec{R}}{ds^2}|}={\Large\frac{|\vec{T}|}{|\vec{R\hspace{.1cm}'}|}} ={\Large \frac{|(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'}|} {|\vec{R\hspace{.1cm}'}|^4}} ={\Large \frac{|\vec{R\hspace{.1cm}'}| |\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'}|} {|\vec{R\hspace{.1cm}'}|^4}} ={\Large \frac{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|} {|\vec{R\hspace{.1cm}'}|^3}}\)   ...(19)
   
   Also
   
   \(\hat{\mathbf{T\hspace{.1cm}'}}=\vec{\kappa}= {\Large \frac{d \hat{\mathbf{T}}}{ds}} ={\Large \frac{\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})} {|\vec{R\hspace{.1cm}'}|^4}} ={\Large \frac{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|} {|\vec{R\hspace{.1cm}'}|^3}} {\Large \frac{\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})} {|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})|}}=\kappa \hat{\mathbf{N}} \)   ...(20)
   
   As given in equation (1), the Unit Binormal Vector to the Curve denoted by \(\hat{\mathbf{B}}\) is given as
   
   \(\hat{\mathbf{B}}=\hat{\mathbf{T}} \times \hat{\mathbf{N}}\)
   
   Using values of \(\hat{\mathbf{T}}\) and \(\hat{\mathbf{N}}\) from equations (14) and (17) respectively we get
   
   \(\Rightarrow \hat{\mathbf{B}}={\Large\frac{\vec{R\hspace{.1cm}'}}{|\vec{R\hspace{.1cm}'}|}} \times {\Large \frac{\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})} {|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})|}} ={\Large \frac{\vec{R\hspace{.1cm}'}\times (\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'}))} {|\vec{R\hspace{.1cm}'}||\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})|}}\)
   
   \(\Rightarrow \hat{\mathbf{B}}={\Large \frac{\vec{R\hspace{.1cm}'}\cdot (\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})\hspace{.2cm}-\hspace{.2cm}(\vec{R\hspace{.1cm}'} \cdot \vec{R\hspace{.1cm}'}) (\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'}) } {|\vec{R\hspace{.1cm}'}||\vec{R\hspace{.1cm}'}||\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'}|}} ={\Large \frac{|\vec{R\hspace{.1cm}'}|^2 (\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}) } {|\vec{R\hspace{.1cm}'}|^2|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|}} ={\Large \frac{\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''} } {|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|}}\)   ...(21)
   
   The Rate of Change of Unit Binormal Vector to the Curve denoted by \(\vec{B}\) is given by Derivative of Vector \(\hat{\mathbf{B}}\) with respect to Variable \(t\) as follows
   
   \(\vec{B}={\Large \frac{d \hat{\mathbf{B}}}{dt}}= {\Large (\frac{\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''} } {|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|})}' = {\Large \frac{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}| (\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})' \hspace{.1cm}-\hspace{.1cm}|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|' (\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}) }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^2}} \)
   
   \(\Rightarrow \vec{B}={\Large \frac{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}| [(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}+\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'''})] \hspace{.1cm}-\hspace{.1cm} \frac{[2 (\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}) \hspace{.1cm}\cdot\hspace{.1cm}[(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}+\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'''})]](\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})}{2 |\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|} }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^2}} \)
   
   \(\Rightarrow \vec{B}={\Large \frac{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}| (\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'''}) \hspace{.1cm}-\hspace{.1cm} \frac{[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}) \hspace{.1cm}\cdot\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'''})](\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})}{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|} }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^2}} \)
   
   \(\Rightarrow \vec{B}={\Large \frac{[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\cdot\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})] (\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'''}) \hspace{.1cm}-\hspace{.1cm} [(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}) \hspace{.1cm}\cdot\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'''})](\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}) }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^3}} \)
   
   \(\Rightarrow \vec{B}={\Large \frac{(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\times\hspace{.1cm}[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'''})\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})] }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^3}} ={\Large \frac{[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'''})]\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}) }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^3}} \)
   
   \(\Rightarrow \vec{B}={\Large \frac{[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\cdot\hspace{.1cm}\hspace{.1cm}\vec{R\hspace{.1cm}'''}]\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}) }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^3}} \)
   
   \(\Rightarrow \vec{B}={\Large \frac{[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\cdot\hspace{.1cm}\vec{R\hspace{.1cm}'''}]\hspace{.1cm}[\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})] }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^3}} \)   ...(22)
   
   \(\Rightarrow |\vec{B}|={\Large \frac{[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\cdot\hspace{.1cm}\vec{R\hspace{.1cm}'''}]\hspace{.1cm}|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})| }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^3}} ={\Large \frac{[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\cdot\hspace{.1cm}\vec{R\hspace{.1cm}'''}]\hspace{.1cm}|\vec{R\hspace{.1cm}'}| |(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})| }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^3}} \)
   
   \(\Rightarrow |\vec{B}|={\Large \frac{[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\cdot\hspace{.1cm}\vec{R\hspace{.1cm}'''}]\hspace{.1cm}|\vec{R\hspace{.1cm}'}| }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^2}} \)   ...(23)
   
   The Torsion Vector to the Curve denoted by \(\hat{\mathbf{B\hspace{.1cm}'}}\) or \(\vec{\tau}\) is given as
   
   \(\hat{\mathbf{B\hspace{.1cm}'}}=\vec{\tau}={\Large \frac{\frac{d \hat{\mathbf{B}}}{dt}} {\frac{ds}{dt}}}= {\Large \frac{d \hat{\mathbf{B}}}{ds}} ={\Large\frac{\vec{B}}{|\vec{R\hspace{.1cm}'}|}} \)   ...(24)
   
   \(\Rightarrow \hat{\mathbf{B\hspace{.1cm}'}}=\vec{\tau}={\Large\frac{ \frac{[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\cdot\hspace{.1cm}\vec{R\hspace{.1cm}'''}]\hspace{.1cm}[\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})] }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^3}}{|\vec{R\hspace{.1cm}'}|}} ={\Large \frac{[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\cdot\hspace{.1cm}\vec{R\hspace{.1cm}'''}]\hspace{.1cm}[\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})] }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^3 |\vec{R\hspace{.1cm}'}|} }\)   ...(25)
   
   \(\Rightarrow |\hat{\mathbf{B\hspace{.1cm}'}}|=|\vec{\tau}|=\tau={\Large |\frac{d \hat{\mathbf{B}}}{ds}|}={\Large\frac{|\vec{B}|}{|\vec{R\hspace{.1cm}'}|}} ={\Large\frac{\frac{[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\cdot\hspace{.1cm}\vec{R\hspace{.1cm}'''}]\hspace{.1cm}|\vec{R\hspace{.1cm}'}| }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^2} }{|\vec{R\hspace{.1cm}'}|}} ={\Large \frac{[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\cdot\hspace{.1cm}\vec{R\hspace{.1cm}'''}]\hspace{.1cm}|\vec{R\hspace{.1cm}'}| }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^2|\vec{R\hspace{.1cm}'}|}} ={\Large \frac{(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\cdot\hspace{.1cm}\vec{R\hspace{.1cm}'''} }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^2}}\)   ...(26)
   
   Also
   
   \(\hat{\mathbf{B\hspace{.1cm}'}}=\vec{\tau}= {\Large \frac{d \hat{\mathbf{B}}}{ds}} ={\Large \frac{[(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\cdot\hspace{.1cm}\vec{R\hspace{.1cm}'''}]\hspace{.1cm}[\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})] }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^3 |\vec{R\hspace{.1cm}'}|} } ={\Large \frac{(\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''})\hspace{.1cm}\cdot\hspace{.1cm}\vec{R\hspace{.1cm}'''} }{|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}''}|^2}} {\Large \frac{-[\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})]} {|\vec{R\hspace{.1cm}'}\hspace{.1cm}\times\hspace{.1cm}(\vec{R\hspace{.1cm}''}\hspace{.1cm}\times\hspace{.1cm}\vec{R\hspace{.1cm}'})|}} =-\tau \hat{\mathbf{N}}\)   ...(27)
   
   Now, taking Derivative of equation (3) with respect to Arc Length of Curve \(s\) we get the Vector \(\hat{\mathbf{N\hspace{.1cm}'}}\) as
   
   \(\hat{\mathbf{N\hspace{.1cm}'}}={\Large \frac{d \hat{\mathbf{N}}}{ds}}={\Large \frac{d \hat{\mathbf{B}}}{ds}} \times \hat{\mathbf{T}} + \hat{\mathbf{B}} \times {\Large \frac{d \hat{\mathbf{T}}}{ds}} =\hat{\mathbf{B\hspace{.1cm}'}} \times \hat{\mathbf{T}} + \hat{\mathbf{B}} \times \hat{\mathbf{T\hspace{.1cm}'}}\)   ...(28)
   
   Putting values of \(\hat{\mathbf{T\hspace{.1cm}'}}\) from equation (20) and \(\hat{\mathbf{B\hspace{.1cm}'}}\) from equation (27) in equation (28) and substituting the values of Cross Products from equations (1) and (2) we get
   
   \(\hat{\mathbf{N\hspace{.1cm}'}}={\Large \frac{d \hat{\mathbf{N}}}{ds}}= -\tau \hat{\mathbf{N}} \times \hat{\mathbf{T}} + \hat{\mathbf{B}} \times \kappa \hat{\mathbf{N}} =\tau (\hat{\mathbf{T}} \times \hat{\mathbf{N}}) - \kappa (\hat{\mathbf{N}} \times \hat{\mathbf{B}}) =\tau \hat{\mathbf{B}} - \kappa \hat{\mathbf{T}}\)   ...(29)