# Surface Curvature Matlab equivalent in Python

### Question

I was trying to calculate the curvature of a surface given by array of points (x,y,z). Initially I was trying to fit a polynomial equation z=a + bx + cx^2 + dy + exy + fy^2) and then calculate the gaussian curvature

$K = \frac{F_{xx}\cdot F_{yy}-{F_{xy}}^2}{(1+{F_x}^2+{F_y}^2)^2}$

However the problem is fitting if the surface is complex. I found this Matlab code to numerically calculate curvature. I wonder how to do the same in Python.

function [K,H,Pmax,Pmin] = surfature(X,Y,Z),
% SURFATURE -  COMPUTE GAUSSIAN AND MEAN CURVATURES OF A SURFACE
%   [K,H] = SURFATURE(X,Y,Z), WHERE X,Y,Z ARE 2D ARRAYS OF POINTS ON THE
%   SURFACE.  K AND H ARE THE GAUSSIAN AND MEAN CURVATURES, RESPECTIVELY.
%   SURFATURE RETURNS 2 ADDITIONAL ARGUEMENTS,
%   [K,H,Pmax,Pmin] = SURFATURE(...), WHERE Pmax AND Pmin ARE THE MINIMUM
%   AND MAXIMUM CURVATURES AT EACH POINT, RESPECTIVELY.

% First Derivatives

% Second Derivatives

% Reshape 2D Arrays into Vectors
Xu = Xu(:);   Yu = Yu(:);   Zu = Zu(:);
Xv = Xv(:);   Yv = Yv(:);   Zv = Zv(:);
Xuu = Xuu(:); Yuu = Yuu(:); Zuu = Zuu(:);
Xuv = Xuv(:); Yuv = Yuv(:); Zuv = Zuv(:);
Xvv = Xvv(:); Yvv = Yvv(:); Zvv = Zvv(:);

Xu          =   [Xu Yu Zu];
Xv          =   [Xv Yv Zv];
Xuu         =   [Xuu Yuu Zuu];
Xuv         =   [Xuv Yuv Zuv];
Xvv         =   [Xvv Yvv Zvv];

% First fundamental Coeffecients of the surface (E,F,G)
E           =   dot(Xu,Xu,2);
F           =   dot(Xu,Xv,2);
G           =   dot(Xv,Xv,2);

m           =   cross(Xu,Xv,2);
p           =   sqrt(dot(m,m,2));
n           =   m./[p p p];

% Second fundamental Coeffecients of the surface (L,M,N)
L           =   dot(Xuu,n,2);
M           =   dot(Xuv,n,2);
N           =   dot(Xvv,n,2);

[s,t] = size(Z);

% Gaussian Curvature
K = (L.*N - M.^2)./(E.*G - F.^2);
K = reshape(K,s,t);

% Mean Curvature
H = (E.*N + G.*L - 2.*F.*M)./(2*(E.*G - F.^2));
H = reshape(H,s,t);

% Principal Curvatures
Pmax = H + sqrt(H.^2 - K);
Pmin = H - sqrt(H.^2 - K);

1
4
3/24/2013 3:29:31 PM

I hope I'm not too late here. I work with exactely the same problem (a product for the company I work to).

The first thing you must consider is that the points must represent a rectangular mesh. X is a 2D array, Y is a 2D array, and Z is a 2D array. If you have an unstructured cloudpoint, with a single matrix shaped Nx3 (the first column being X, the second being Y and the third being Z) then you can't apply this matlab function.

I have developed a Python equivalent of this Matlab script, where I only calculate Mean curvature (I guess you can get inspired by the script and adapt it to get all your desired curvatures) for the Z matrix, ignoring the X and Y by assuming the grid is square. I think you can "grasp" what and how I am doing, and adapt it for your needs:

def mean_curvature(Z):