can anyone please explain me this code ..exactly whats happening here while computing curvature ?

function k = curvature_central(u)
% compute curvature
[ux,uy] = gradient(u);
normDu = sqrt(ux.^2+uy.^2+1e-10);
Nx = ux./normDu;
Ny = uy./normDu;
[nxx,junk] = gradient(Nx);
[junk,nyy] = gradient(Ny);
k = nxx+nyy;

Answer 1

[SAI SRUJAN]

Hi Nivedita,

This function calculates the curvature of a 2D field by first determining the direction of steepest ascent (the gradient), normalizing it, and then evaluating how this direction changes across the field. The result gives you an idea of how the surface bends or curves at each point.

Please go through the following updated code sample with comments to proceed further,

function k = curvature_central(u)
% Compute curvature of a 2D scalar field u

    % Calculate the gradient of u
    % ux and uy are the partial derivatives of u with respect to x and y
    [ux, uy] = gradient(u);
    
    % Compute the magnitude of the gradient vector
    % Adding 1e-10 to avoid division by zero and ensure numerical stability
    normDu = sqrt(ux.^2 + uy.^2 + 1e-10);
    
    % Normalize the gradient to obtain the unit normal vector components
    Nx = ux ./ normDu; % Normalized x-component
    Ny = uy ./ normDu; % Normalized y-component
    
    % Compute the second derivatives of the normalized components
    % nxx is the derivative of Nx with respect to x
    % nyy is the derivative of Ny with respect to y
    [nxx, junk] = gradient(Nx); % junk is used as a placeholder
    [junk, nyy] = gradient(Ny); % junk is used as a placeholder
    
    % Calculate curvature as the sum of these second derivatives
    % This represents the divergence of the unit normal vector
    k = nxx + nyy;
end

I hope this helps!