![]() \Īppendix B: Why Are Principal Directions Orthogonal?Įarlier we stated that the unit principal directions \(X_1, X_2\) are orthogonal with respect to the metric \(g\) induced by the immersion \(f\), i.e., We can therefore get the normal curvature along \(X\) by extracting the tangential part of \(dN\): Remember the Frenet-Serret formulas (Theorem 1.1)? They tell us that the change in the normal along a curve is given by \(dN = \kappa T \tau B\). This plane intersects the surface in a curve, and the curvature \(\kappa_n\) of this curve is called the normal curvature in the direction \(X\): In particular, let \(X\) be a unit tangent direction at some distinguished point on the surface, and consider a plane containing both \(df(X)\) and the corresponding normal \(N\). This way of looking at curvature - in terms of curves traveling along the surface - is often how we treat curvature in general. But what about something like a beer bottle? Along one direction the bottle quickly curves around in a circle along another direction it’s completely flat and travels along a straight line: The word “curvature” really corresponds to our everyday understanding of what it means for something to be curved: eggshells, donuts, and cavatappi pasta have a lot of curvature floors, ceilings, and cardboard boxes do not. Let’s take a more in-depth look at the curvature of surfaces. ![]()
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