= Principal curvature
{title2=$k_1,k_2$}
{wiki}
At a point of a <smooth surface>, a <normal section of a surface> can bend by different amounts depending on its <tangent vector>. The <principal curvatures> are the largest and smallest signed <normal curvatures> among unit <tangent vectors>. Equivalently they are the two <eigenvalues> $k_1,k_2$ of the <shape operator>, with an <orthonormal basis> $e_1,e_2$ of corresponding <principal directions>. Here the labels follow the selected directions, without requiring $k_1\leq k_2$. https://ghomi.math.gatech.edu/Classes/Math497C/LectureNotes10.pdf[Ghomi, Lecture Notes 10, §2.3.3: normal curvature and principal curvatures]{ref}
For a unit <tangent vector> making <angle> $\theta$ with $e_1$, <Euler formula for normal curvature> gives
$$
v=\cos\theta\,e_1+\sin\theta\,e_2,\qquad k_n(\theta)=II(v,v)=k_1\cos^2\theta+k_2\sin^2\theta.
$$
The <normal curvature> is a weighted average of the <principal curvatures>; the extremes occur in the <principal directions>. If the <principal curvatures> coincide, every direction in the <tangent plane> is a <principal direction>: this is an <umbilical point>. The <Gaussian curvature> and <mean curvature> summarize the pair by
$$
K=k_1k_2,\qquad H=\frac{k_1+k_2}{2}.
$$
For an outward-oriented <sphere> of <radius> $R$, both <principal curvatures> are $-1/R$, so the <normal curvature> is $-1/R$ in every direction. For an outward-oriented <circular cylinder>, choose $e_1$ around the circumference and $e_2$ along the axis. The <principal curvatures> are $-1/R,0$, giving $k_n(\theta)=-\cos^2\theta/R$; halfway between the <principal directions>, $k_n(\pi/4)=-1/(2R)$.
At the origin of the upward-oriented <hyperbolic paraboloid> $z=(x^2-y^2)/2$, the <principal directions> are the coordinate axes and the <principal curvatures> are $1,-1$ in the chosen length unit. Thus $k_n(\theta)=\cos(2\theta)$, with zero <normal curvature> at $\theta=\pi/4,3\pi/4$. Zero <normal curvature> in these directions coexists with <Gaussian curvature> $-1$: the <shape operator> is not zero. The <mean curvature> is zero at this point; this does not say that the whole <hyperbolic paraboloid> is a <minimal surface>.
\Image[/principal-normal-curvature-directions.png]
{title=Directional normal curvature on a sphere, cylinder, and saddle with the convention S=-dN}
Each graph uses <angle> $\theta$ in the <tangent plane>, measured from the indicated first <principal direction>. The <sphere> and <circular cylinder> have $R=1$; the saddle graph refers only to the origin. Reversing the <unit normal> reflects each <normal curvature> graph through zero, negates both <principal curvatures> and the <mean curvature>, and preserves their product, the <Gaussian curvature>.
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