Principal Curvatures & Normal Curvature

Euler's formula, shape-operator eigenvalues, and point classification.

The idea

The normal curvature measures how much a surface bends in a chosen tangent direction.

Definition (Normal and principal curvatures).

At a point $p$ of a surface with unit normal $N$, the normal curvature $k_n$ in a unit tangent direction is the signed curvature at $p$ of the plane curve cut out by the plane spanned by that direction and $N$. The maximum and minimum of $k_n$ over all directions are the principal curvatures $k_1$ and $k_2$, attained along the principal directions.

As the direction rotates around the tangent plane, $k_n$ varies continuously, and its two extremes occur in perpendicular directions: $k_1$ and $k_2$ are the eigenvalues of the shape operator $S = -dN$, and the principal directions are its eigenvectors. Every other normal curvature is determined by these two.

Theorem (Euler's formula).

Let $k_1$ and $k_2$ be the principal curvatures at $p$, and let $\theta$ be the angle between a unit tangent direction and the first principal direction. The normal curvature in that direction is $k_n(\theta) = k_1\cos^{2}\theta + k_2\sin^{2}\theta.$

These quantities are extrinsic: they record how the surface sits in space. Rolling a flat sheet into a cylinder changes no distance within the sheet, yet it gives the sheet a nonzero principal curvature. They also depend on the choice of unit normal, since reversing it reverses both signs. The product $K = k_1 k_2$ survives the reversal, and its sign classifies the point: elliptic when $K > 0$, a dome bending the same way in every direction; hyperbolic when $K < 0$, a saddle; parabolic when $K = 0$, flat along one principal direction, like a cylinder.

Ways to work on it

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