What are the principal directions of a cylinder?

Question

I'm trying to understand why, for $S=\left\{(x, y, z) \in \mathbb{R}^{3}: x^{2}+y^{2}=1\right\}$ the unit cylinder in $\mathbb{R}^{3}$, the vertical and horizontal directions are the principal directions of $S$.

How can I see this? What is the intuition? Of course, if I define charts, and do explicit calculations, I will get the above result, but is there another way of seeing this immediately? (I'm afraid that I do not have an geometric intuition as to what principal directions are)

Answer 1

The principal directions of curvature are the eigenspaces of the Shape operator $S = \nabla \nu$ where $\nu$ is the outward unit normal.

In the case of the cylinder $C = \{ (x,y,z)\mid x^2 + y^2 = 1 \}$, the outward unit vector field is $\nu(x_0,y_0,z_0) = (x_0,y_0,0)$.

One overkill solution is this:

• we know $S$ is a symmetric operator, hence has orthogonal eigenspaces,
• it is clear by its expression that $S v = 0$ if $v$ is vertical (i.e if $v$ is colinear to the $z$ axis): the vertical direction is thus an eigenspace of $S$,
• the only direction that is orthogonal to the vertical direction in the tangent space is the horizontal one, hence by the symmetry of $S$, it has to be an eigenspace of $S$, and thus, a principal direction of curvature.

Another way to prove this is simply by computing $S$ and proving that $S$ is zero on the vertical direction, $1$ on the horizontal.

Answer 2

Regarding intuition: the principal directions are the directions in which the normal curvature attains it’s minimum and maximum values. A theorem of Euler tells us that these directions are always at right angles to each other.

So, take a point on a cylinder, and construct a line through this point that’s normal to the cylinder, and a plane containing this line. This plane intersects the cylinder in a curve, which will be an ellipse, in general. Now gradually rotate the plane around the normal line. In one rotated position, the intersection curve will be a circle, and 90 degrees away from this, the intersection curve will be a straight line. Among all the (generally elliptical) intersection curves we obtain as the plane rotates, the circle has the largest curvature, and the straight line has the smallest curvature. The tangent directions of the circle and the straight line are the principal directions.

Useful pictures here.