## Local parabolic curvature

fit.tar.gz (including plane fit, bond curvature and local parabolic curvature calculations.)

#### Parabolic fitting

After we found out the tangent plane of a local surface structure, we can use the normal vector of the tangent plane as the z-axis of the fitting parabolic surface nz and find out the two principle axis and their corresponding curvatures.
1. Take arbitrary nx and ny orthogonal to each other and nz.

2. Map the coordiantes of all particles to this new coordinate system n = (nx, ny, nz):

xi = (qi - q0) * n,

where qi is the old coordinate of ith particle, xi = (xi, yi, zi ) is the new coordinate, q0 is the reference point which can be either the desiganted central particle or the center of mass of this local surface structure.

3. Define parabolic fit function

f(x, y) = axxx2 + 2axyxy + ayyy2 + a0

and minimize

S = Sigmai(zi-f(xi, yi)) 2

with respect to axx, axy, ayy, a0.

4. Once we have axx, axy, ayy, we diagonize the matrix

( axx axy )
( axy ayy )
and get two principle axis and the correponding local curvatures (with a factor of 2). Note the parabolic curvatures now have signs to indicate directions ( positive as bending toward center of mass ).

Here is an illustration of the principle axis for a near-flat facet. Here is the local curvature distributions for the icosahedral gold cluster with 2624 atoms.