In the paper Surface sampling and the intrinsic Voronoi diagram (2008), Ramsay Dyer defines the Gaussian curvature radius at a point $x$ of a surface $S$ to be $\rho_K(x) = 1/\sqrt{K(x)}$ where $K(x)=\kappa_1(x) \kappa_2(x)$ is the Gaussian curvature at $x$.
Trying to track back the notion in Berger's A panoramic view of Riemannian geometry, and in Lee's Riemannian manifolds and in Chavel's Riemannian Geometry yielded nothing.
My question is two-folded:
- Where can I find more information about this notion?
- Is there a reason not to define it as $\rho_K(x) = 1/|K(x)|$? Otherwise, this definition is only valid for non-negatively curved surfaces.
EDIT As pointed out by Deane Yang, there is no sense in the definition I suggested. Nevertheless, if one wants to relate the Gaussian curvature to a radius (for either negatively or positively curved surfaces) how about this alternative: $\rho_{K}(x)=1/\sqrt{|K(x)|}$?
Best Answer
As for question #2, why does your definition make sense for a negatively curved surface? For a positively curved surface it does not give the right answer for spheres, since presumably you would want a sphere of radius $r$ to have a Gauss curvature radius of $r$. In particular, the word "radius" reflects a linear measurement and therefore should scale linearly if you rescale the surface.
The "radius of curvature" at a point on a curve is the radius of an osculating circle and turns out to be the reciprocal of the geodesic curvature.
On a point of a surface in $R^3$, you get a radius of curvature for each tangent direction, corresponding to the osculating circle in that direction. In particular, there are the two principal radii of curvature corresponding to the principal directions. The Gauss curvature radius, as defined above, is the geometric average. Since it can be defined in terms of Gauss curvature only, it has the advantage of being intrinsic. You could also define the "mean radius" by taking the arithmetic average. I don't recall seeing this before, but it also seems reasonable to study.
I recommend working out the example of $z = f(x,y)$ at the origin, where $f(0, 0) = \partial_xf(0,0) = \partial_yf(0,0) = 0$.