The curvature of an intersection curve I am the teaching assistant of a calculus class, and a student asked me this question. A smooth surface $z=S(x,y)$ intersects with a plane, say $z=0$. For any point $P$ on intersection curve $L$, you can calculate three curvatures, two principal curvatures $k_1$ and $k_2$ of the surface, and the curvature of the curve $k_L$. Is there a relationship for the three curvatures? 
In fact the student asked whether the mean curvature $H=(k_1+k_2)/2$ is greater or smaller than $k_L$.
 A: This seems pretty sophisticated for a calculus class. :)
The simple answer is no. For example, on a plane ($k_1=k_2=0$), you can have curves with arbitrarily large curvature. The same is true on a sphere ($k_1=k_2=1$) or any other surface. The answer to your student's question is, yet again, that there's no relation with the mean curvature. You can make $k_L$ as small as the smaller principal curvature and as large as you want.
There are, however, two relevant formulas. Meusnier's formula says that the normal curvature $\kappa_n$ of the curve satisfies the equation $\kappa_n = \kappa\cos\phi$, where $\phi$ is the angle between the surface normal and the principal normal of the curve. And Euler's formula says that $\kappa_n = k_1\cos^2\theta + k_2\sin^2\theta$, where $\theta$ is the angle between the tangent vector of the curve and the first principal direction.
A: It can perhaps be introduced for students after an exposure to stress/strain in mechanics of materials. What is stated as a concept for principal stresses can be extended to principal curvatures as well, only the symbols change.
Euler's relation expresses intermediate normal curvature in terms of major and minor (principal) curvatures $ k_1, k_2,$  as a function of rotation of reference plane of osculation.
Diagrammatically it can be represented on a a Mohr's circle of curvatures, where it is necessary to bring in geodesic torsion $\tau_g$, similar to shear for case of stress.
It is an excellent diagram catering to such normal curvature depiction requirements. Useful for positive and negative Gauss curvatures K. In case of K<0 it can also locate the asymptotic directions.
The Mohr's  circle is nothing but  a representation of varying curvatures given by Euler relation in a parametric form, beautifully integrating it with the definition of geodesic torsion.
$$ k_n = k_1 \cos^2 \alpha + k_2 \sin^2 \alpha $$
$$ \tau_g = ( k_1-k_2)/2* \sin 2 \alpha$$
The diagram uses rotation $ 2 \alpha $.
The diagram also fixes positions of mean curvature and even  K ( as tangents to Apollonian circles in Bipolar coordinates). 
On the x-axis of the Mohr's diagram we can have any sign for $k_1,k_2$, just by shifting y-axis. We have $k_2< H < k_1$ in ascending order by convention.  For  minimal surfaces ( soap films)  mean curvature H ( represented by center of circle) is zero. 
$ \alpha $ denotes which direction in the tangent plane we are referring to in between  $k_1,k_2$ extreme values. At reference direction $ 2 \alpha= \pi/2 $ or $ \alpha= \pi/4 $ the mean curvature is the average of principal curvatures. ( In differential geometry one simply says "lines of curvature".)
BTW what is required to be clarified is about normal curvature of any curve on a surface. To add to the above, there is also a geodesic curvature in the tangent plane, which vanishes for geodesic shortest lines.

EDIT1:
Diagram updated
A: $k_L$ can be smaller or greater than $\frac{k_1+k_2}{2}$ as depending to the position of the plane and of the surface, you can have $k_L=k_1$ or $k_L=k_2$ if the plane is such that it is normal to the surface at a given point and intersect the tangent plane in one of the principal curvature directions.
