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?

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Unless you specify more about how $S$ is given, it is not clear that any given condition would be easier to apply than your definition itself. – Henning Makholm Sep 13 '12 at 21:17

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.