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Curvatures, Principal Directions and Shape Operator Matrix of a Surface

1. The Curvature of a Surface at Any Given Point on the Surface in Any Given Direction is given by the Curvature of the Curve on the Surface which has Tangent in the Same Line Direction at that Point and whose Principal Unit Normal at that Point is in the Same Line Direction as that of the Normal to the Tangent Plane of the Surface at that Point.
2. Values of Curvatures present at Any Given Point on the Surface define the Type of the Point and locally define the Shape of Surface at/around that Point.
   
   The Curvatures on a Surface at a Point can have following values
   1. If the Values of Curvatures at a Point the Surface in All Directions are Zero then locally the Surface has the Shape of a Plane at/around that Point. Such Point is called a Planar Point of the Surface.
   2. If the Values of Curvatures at a Point the Surface in All Directions are Same and Non Zero then locally the Surface has the Shape of a Sphere at/around that Point. Such Point is called a Spherical Point or Umbilical Point of the Surface.
   3. If the Value of Curvature at a Point the Surface in Atleast One Direction is Zero (and in Other Directions are Either All Positive or All Negative) then locally the Surface has the Shape of a Cylinder at/around that Point. Such Point is called a Cylindrical Point of the Surface.
   4. If the Values of Curvatures at a Point the Surface in All Directions are Non Zero and have Same Sign (Either All Positive or All Negative) then locally the Surface has the Shape of a Elliptic Paraboloid/Ellipsoid at/around that Point. Such Point is called an Elliptic Point of the Surface.
   5. If the Values of Curvatures at a Point the Surface in Some Directions are Positive and in Some Directions are Negative then locally the Surface has the Shape of a Hyperbolic Paraboloid at/around that Point. Such Point is called a Hyperbolic Point of the Surface.
3. For Any Cylindrical, Elliptical or Hyperbolic Point on the Surface there exist 2 Mutually Perpendicular Line Directions, one of which has the Maximum Value of Curvature, and the oher has the Minimum Value of Curvature. These 2 Directions are called the Principal Directions of the Point on the Surface and the Curvatures along them are called the Principal Curvatures.
4. The Shape Operator Matrix (denoted by \(S\)) is calculated for Any Point on the Surface by using the First Fundamental Form Matrix \(\mathrm{I}\) and Second Fundamental Form Matrix \(\mathrm{II}\) of that Point as follows
   
   \(S = {\mathrm{I}}^{-1}\mathrm{II} = \begin{bmatrix}E & F \\ F & G\end{bmatrix}^{-1}\begin{bmatrix}L & M \\ M & N\end{bmatrix} =\begin{bmatrix}{\Large \frac{GL-FM}{EG-F^2}} & {\Large \frac{GM-FN}{EG-F^2}} \\ {\Large \frac{EM-FL}{EG-F^2}} & {\Large \frac{EN-FM}{EG-F^2}}\end{bmatrix}\)   ...(1)
   
   The Shape Operator Matrix for a Planar Point is a NULL Matrix.
5. For Any Cylindrical, Elliptical or Hyperbolic Point on the Surface, the 2 Eigen Values of the Shape Operator Matrix specify the 2 Principal Curvatures (denoted by \(\kappa_1\) and \(\kappa_2\)) at that Point. The corresponding Eigen Vectors give the Components of the 2 Principal Directions for that Point. The Actual Principal Directions can be found out by Scaling the Tangent Vectors of the Plane \(\vec{R_u}\) and \(\vec{R_v}\) with the Components of the Eigen Vectors.
   
   For example if \(\begin{bmatrix}a_1 \\ a_2\end{bmatrix}\) and \(\begin{bmatrix}b_1 \\ b_2\end{bmatrix}\) are the 2 Eigen Vectors corresponding to \(\kappa_1\) and \(\kappa_2\) respectively, then their Principal Directions are
   
   Principal Direction of \(\kappa_1 = a_1\vec{R_u} + a_2\vec{R_v}\)   ...(2)
   
   Principal Direction of \(\kappa_2 = b_1\vec{R_u} + b_2\vec{R_v}\)   ...(3)
   
   For Any Spherical/Umbilical Point on the Surface, the 2 Eigen Values are Same.
6. The Determinant of the Shape Operator Matrix (denoted by \(|S|\)), gives the Product of the 2 Principal Curvatures and is called the Gaussian Curvature or Total Curvature of the Surface and is denoted by \(K\)
   
   \(|S| = K = \kappa_1\kappa_2 = \begin{vmatrix}{\Large \frac{GL-FM}{EG-F^2}} & {\Large \frac{GM-FN}{EG-F^2}} \\ {\Large \frac{EM-FL}{EG-F^2}} & {\Large \frac{EN-FM}{EG-F^2}}\end{vmatrix} ={\Large \frac{LM-N^2}{EG-F^2}}\)   ...(4)
   
   
7. Half of the Value of Trace of the Shape Operator Matrix (denoted by \({\Large\frac{Trace(S)}{2}}\)), gives the Average Value of the 2 Principal Curvatures and is called the Mean Curvature of the Surface and is denoted by \(H\)
   
   \({\Large\frac{Trace(S)}{2}} = H = {\Large\frac{\kappa_1 + \kappa_2}{2}} ={\Large \frac{EN+GL-2FM}{2(EG-F^2)}}\)   ...(5)
   
   
8. Given the Gaussian Curvature \(K\) and Mean Curvature \(H\) of a Surface at a Point, the corresponding Principal Curvatures at that Point on Surface can be found out by solving the following Quadratic Equation
   
   \(\kappa^2 - 2H\kappa + K =0\)
   
   \(\Rightarrow \kappa^2 - 2{\Large \frac{EN+GL-2FM}{2(EG-F^2)}}\kappa + {\Large \frac{LM-N^2}{EG-F^2}} =0\)
   
   \(\Rightarrow(EG-F^2)\kappa^2 - (EN+GL-2FM)\kappa + (LM-N^2) =0\)   ...(6)
   
   The 2 Principal Curvatures \(\kappa_1\) and \(\kappa_2\) get calculated as
   
   \(\kappa_1= H + \sqrt{H^2 - K}= ,\hspace{6mm}\kappa_2= H - \sqrt{H^2 - K}\)   ...(7)
9. Given the Gaussian Curvature \(K\) and Mean Curvature \(H\) of a Surface at a Point, the Type of Point and Shape of Surface at/around that Point can be found as follows
   1. If Gaussian Curvature \(K=0\) and Mean Curvature \(H=0\) then locally the Surface has the Shape of a Plane at/around that Point. It is a Planar Point of the Surface.
   2. If Gaussian Curvature \(K=0\) and Mean Curvature \(H\neq0\) then locally the Surface has the Shape of a Cylinder at/around that Point. It is a Cylindrical Point of the Surface.
   3. If Gaussian Curvature \(K>0\) then locally the Surface has the Shape of a Elliptic Paraboloid/Ellipsoid at/around that Point. It is an Elliptic Point of the Surface.
   4. If Gaussian Curvature \(K<0\) then locally the Surface has the Shape of a Hyperbolic Paraboloid at/around that Point. It is a Hyperbolic Point of the Surface.