Center and Radius of Osculating Sphere of a Curve

1. An Osculating Sphere at any given Point on a Curve is a Sphere that has a \(3\)-rd Order Contact with the Curve.
2. The equation of the Osculating Sphere is given as
   
   \(|\vec{R}-\vec{C}|=R\)
   
   \(\Rightarrow \sqrt{(\vec{R}-\vec{C}) \cdot (\vec{R}-\vec{C})} =R\)   ...(1)
   
   \(\Rightarrow (\vec{R}-\vec{C}) \cdot (\vec{R}-\vec{C}) =R^2\)
   
   \(\Rightarrow (\vec{R}-\vec{C})^2 - R^2= 0\)   ...(2)
   
   where
   
   \(\vec{R}=\) Position Vector of the Point of Contact of the Curve and the Osculating Sphere
   
   \(\vec{C}=\) Position Vector of Center of the Osculating Sphere
   
   \(R=\) Radius of the Osculating Sphere
   
   Taking the Derivative of equation (2) with respect to Arc Length \(s\) we get
   
   \(2(\vec{R}-\vec{C})\cdot{\Large \frac{d\vec{R}}{ds}}=0\)
   
   \(\Rightarrow (\vec{R}-\vec{C})\cdot\hat{\mathbf{T}}=0\)   (\(\because {\Large \frac{d\vec{R}}{ds}}=\hat{\mathbf{T}}\)) ...(3)
   
   The equation (3) above gives the Projection of Vector \((\vec{R}-\vec{C})\) on Unit Vector \(\hat{\mathbf{T}}\)
   
   Now, taking the Derivative of equation (3) with respect to Arc Length \(s\) we get
   
   \({\Large \frac{d\vec{R}}{ds}} \cdot \hat{\mathbf{T}} + (\vec{R}-\vec{C})\cdot {\Large \frac{d\hat{\mathbf{T}}}{ds}}=0\)
   
   \(\Rightarrow \hat{\mathbf{T}} \cdot \hat{\mathbf{T}} + (\vec{R}-\vec{C})\cdot \kappa\hat{\mathbf{N}}=0\)   (\(\because {\Large \frac{d\hat{\mathbf{T}}}{ds}}=\kappa\hat{\mathbf{N}}\))
   
   \(\Rightarrow (\vec{R}-\vec{C})\cdot \kappa\hat{\mathbf{N}}=-1\)   (\(\because \hat{\mathbf{T}}\cdot \hat{\mathbf{T}}=1\))
   
   \(\Rightarrow (\vec{R}-\vec{C})\cdot \hat{\mathbf{N}}={\Large\frac{-1}{\kappa}}\)
   
   \(\Rightarrow (\vec{R}-\vec{C})\cdot \hat{\mathbf{N}}=-\rho\)   (\(\because {\Large \frac{1}{\kappa}}=\rho\)) ...(4)
   
   The equation (4) above gives the Projection of Vector \((\vec{R}-\vec{C})\) on Unit Vector \(\hat{\mathbf{N}}\)
   
   Now, taking the Derivative of equation (4) with respect to Arc Length \(s\) we get
   
   \({\Large \frac{d\vec{R}}{ds}} \cdot \hat{\mathbf{N}} + (\vec{R}-\vec{C})\cdot {\Large \frac{d\hat{\mathbf{N}}}{ds}}=-{\Large \frac{d\rho}{ds}}\)
   
   \(\Rightarrow \hat{\mathbf{T}} \cdot \hat{\mathbf{N}} + (\vec{R}-\vec{C})\cdot (-\kappa \hat{\mathbf{T}} + \tau \hat{\mathbf{B}})=-{\Large \frac{d\rho}{ds}}\)   (\(\because {\Large \frac{d\hat{\mathbf{N}}}{ds}}=-\kappa\hat{\mathbf{T}} + \tau\hat{\mathbf{B}}\))
   
   \(\Rightarrow -\kappa (\vec{R}-\vec{C})\cdot \hat{\mathbf{T}} + \tau (\vec{R}-\vec{C})\cdot \hat{\mathbf{B}}=-{\Large \frac{d\rho}{ds}}\)   (\(\because \hat{\mathbf{T}}\cdot \hat{\mathbf{N}}=0\))
   
   \(\Rightarrow (\vec{R}-\vec{C})\cdot \hat{\mathbf{B}}=-{\Large \frac{1}{\tau}}{\Large \frac{d\rho}{ds}}\)   (\(\because\) from equation (1) \((\vec{R}-\vec{C})\cdot \hat{\mathbf{T}}=0\))
   
   \(\Rightarrow (\vec{R}-\vec{C})\cdot \hat{\mathbf{B}}=-\sigma{\Large \frac{d\rho}{ds}}\)   (\(\because {\Large \frac{1}{\tau}}=\sigma\)) ...(5)
   
   The equation (5) above gives the Projection of Vector \((\vec{R}-\vec{C})\) on Unit Vector \(\hat{\mathbf{B}}\)
   
   Now, the Position Vector of the Center of Osculating Sphere \(\vec{C}\) can be found out by adding the Projection of the Vector \((\vec{R}-\vec{C})\) on the Unit Vectors \(\hat{\mathbf{T}}\), \(\hat{\mathbf{N}}\) and \(\hat{\mathbf{B}}\) given in equations (3), (4) and (5) In the Direction of Unit Vectors \(\hat{\mathbf{T}}\), \(\hat{\mathbf{N}}\) and \(\hat{\mathbf{B}}\) as follows
   
   \((\vec{R}-\vec{C})=0\hat{\mathbf{T}} -\rho \hat{\mathbf{N}} - \sigma{\Large \frac{d\rho}{ds}} \hat{\mathbf{B}}= -\rho \hat{\mathbf{N}} - \sigma{\Large \frac{d\rho}{ds}} \hat{\mathbf{B}}\)   ...(6)
   
   \(\therefore \vec{C}= \vec{R} + \rho \hat{\mathbf{N}} + \sigma{\Large \frac{d\rho}{ds}} \hat{\mathbf{B}}\)   ...(7)
   
   Now, using equations (1) and (6) we can find the Radius \(R\) of the Osculating Sphere as follows
   
   \(R=\sqrt{(\vec{R}-\vec{C}) \cdot (\vec{R}-\vec{C})}=\sqrt{(-\rho \hat{\mathbf{N}} - \sigma{\Large \frac{d\rho}{ds}} \hat{\mathbf{B}})\cdot (-\rho \hat{\mathbf{N}} - \sigma{\Large \frac{d\rho}{ds}} \hat{\mathbf{B}})} =\sqrt{\rho^2 + {(\sigma{\Large \frac{d\rho}{ds}})}^2 }\)   ...(8)