Curvature and the Shape Operator

line, curve, tangent and normal

What's in a straight line?

The problem, however, with measuring curvature using this principal is that we have to first define a reference line that is straight relative to the curve.

But where would we place this line and how would we orient it to the curve?

We could draw it so that it bisects the curve (a line along the x-axis would operate in this manner); however, we would then have to subtract the sum of the distances on one side of this line from the distances on the other.

We could also draw a line above or below the entire curve, and measure our distances from that reference line to the curve. However, where would we draw this line? Which side of the curve should it be on? How far away from the curve should we place it?

You can probably already see that anchoring the curvature of a curve to an arbitrary reference line is probably not the way to go. But how else can we measure it?