A Question of Curvature: When is a sphere the best approximation to a surface at  a point?
To be more specific: 
Let $S$ be a smooth surface in $R^3$. $P$ a point on $S$. $N$ normal to $S$ at $P$. $\Pi$ a plane through $N$. $C$ the intersection of $\Pi$ with $S$. $R$ the radius of curvature of $C$ at $P$. 
Under what conditions would $R$ not depend on $\Pi$?  
That is, all the normal planes that intersect the surface at $P$ leave a trace with the same curvature?
 A: The best approximation to a surface at a point is the surface itself. Therefore, a sphere is the best approximation to a surface at a point if that surface forms part of a sphere itself. 
An umbilic point is a point where the oscullating sphere approximates the surface better than usual. There is no notion of a best approximation, except for the surface itself. At such points the principal curvatures are equal, and the shape operator (equiv. the second fundamental form) is a multiple of the identity. There are also so-called flat-umbilics. These are points where the principal curvatures are zero and the centres of the oscullating spheres have receded to infinity (the oscullating spheres have tended to planes in the limit).
One way to measure the contact between surfaces is to consider the contact function. If one surface is given by the zero-level set of an equation, say $f : (\mathbb{R}^3,0) \to (\mathbb{R},0)$, and the other is given by a parametrisation, say $g : (\mathbb{R}^2,0) \to (\mathbb{R}^3,0)$, then the contact function is the composite $f \circ g : (\mathbb{R}^2,0) \to (\mathbb{R},0)$. The singularity type of $f \circ g$ is used to classify the contact between the two surfaces, and is invariant under $\mathscr{K}$-equivalence. 
