Osculating Circle, Radius and Center of Curvature of a Curve

1. An Osculating Circle at any given Point on a Curve is a Circle that has a \(2\)-nd Order Contact with the Curve, i.e. it has the Same Curvature as that of the Curve at that Point.
2. The Osculating Circle lies on the Plane spanned by the Unit Tangent Vector \(\hat{\mathbf{T}}\) and the Unit Principal Normal Vector \(\hat{\mathbf{N}}\). Hence, the Plane spanned by these 2 Vectors is called the Osculating Plane. The Osculating Circle is formed by the Intersection of the Osculating Sphere and the Osculating Plane.
3. The Radius of Curvature \(\rho\) of a Curve at any Point is the Radius of the Osculating Circle at that Point. It is the Inverse of the Curvature of the Curve \(\kappa\) at that Point (i.e. \(\rho=\Large{\frac{1}{\kappa}}\)).
4. The Center of Curvature of a Curve at any Point is the Center of the Osculating Circle at that Point. The Position Vector of the Center of Curvature \(\vec{C}\) is calculated as follows
   
   \(\vec{C}=\vec{R} + \rho\hat{\mathbf{N}}=\vec{R} + \Large{\frac{\hat{\mathbf{N}}}{\kappa}}\)
   
   where
   
   \(\vec{R}=\) Position Vector of a Point on the Curve
   
   \(\rho=\) Radius of Curvature Corresponding to the Point given by \(\vec{R}\)
   
   \(\kappa=\) Curvature Corresponding to the Point given by \(\vec{R}\)