An example of a normal section to a surface
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The picture below shows the surface \(S\) defined as the graph of the function \(f : \mathbb{R}^2 \rightarrow \mathbb{R}^3\) given by \(f(x,y) = x^2 - y^2\). At the origin, a normal vector is given by \(\xi = (0,0,1)\). Given any tangent vector \(X \in T_{(0,0,0)} S\), we can construct the plane through the origin that contains the vectors \(\xi\) and \(X\), and then define the normal section determined by \(X\) to be the curve given by the intersection of this plane with \(X\).
You can see this in the picture below. The tangent vector \(X\) is given by \((\cos(\theta), \sin(\theta))\) and you can vary the value of \(\theta\) by moving the slider on the left. The plane and the normal section will move automatically.
The normal curvature is the dot product of the curvature vector of this curve with the normal vector \(\xi\), and you can also see that the maximum and minimum values of the normal curvature will occur when \(X\) is parallel to the \(x\)-axis or the \(y\)-axis, which matches the eigenvectors of the shape operator that we computed in the example of the paraboloid.
