Curvature and the Shape Operator

line, curve, tangent and normal

The Normal Plane

The importance of the normal plane is that if we give it an infinite height and width, it would essentially cut through the entire surface perpendicularly.

If we now remove half of this cut surface (in other words, if we take a cross section of the surface where the normal plane intersects it), we can see that what we have done by dissecting the surface is to form a plane curve that runs through our point of interest.

We can now analyze this curve just as we did earlier!

A curve, however, tells us very little about the surface in the area immediately surrounding the point of interest. In order to obtain a better understanding of this entire surface, we need to be able to analyze all of the curves that travel through this point.

To see how this is done, imagine that we now rotate this normal plane about the line that is normal to our point of interest.

This normal line would then show us all of the curves that run through this point!