Curvature and the Shape Operator

line, curve, tangent and normal

Cutting Along The Normal Plane

Just to make sure that the idea behind what we mean when we say cut the surface doesn't get lost in explanation, let's look at a quick example.

Say we have the surface we have been looking at this time, and we take the normal plane and rotate it 90 degrees, as in the diagram.

If we now increase the width of this plane so that it extends the entire length of our surface, we may use it to cut the surface.

To cut the surface means to remove the entire surface on either side of the normal plane, and examine only the portion of the surface that lies at the intersection with the plane--this portion is the 2-dimensional curve that we began this tutorial by discussing!

cutting along the normal plane