MATLAB: Curvature of a discrete function

curvature

Hello,

I need to compute a curvature of a simple 2D discrete function like this one:

x=1:0.5:20;
y=exp(x);

can anybody help how to do that? thanks

Best Answer

You function seems to be a 1D function.

Are you looking for the 2nd derivative? While diff calculates the one-sided differential quotient, gradient uses the two-sided inside the interval:

gradient(gradient(y))

If you mean the curvature as reciprocal radius of the local fitting circle:

dx  = gradient(x);
ddx = gradient(dx);
dy  = gradient(y);
ddy = gradient(dy);
num   = dx .* ddy - ddx .* dy;
denom = dx .* dx + dy .* dy;
denom = sqrt(denom);
denom = denom * denom * denom;
curvatur = num ./ denom;
curvature(denom < 0) = NaN;

Please test this, because I'm not sure if I remember the formulas correctly.

Related Solutions

MATLAB: 1-d gradient

Your question contains the answer already. If the data is a vector, conv is sufficient instead of conv2:

x    = rand(1, 10)
dx_1 = gradient(x)
dx_2 = conv(x, [0.5, 0, -0.5], 'same')

The first and last element of dx_2 differ from dx_1, but you can adjust it with the one-sided quotient of differences easily.

MATLAB: Calling gradient multiple times

It's because of calculation first and last point. gradient does it this way:

first_grad = (y2-y1)/dx;
second_grad = 1/2*(y4-y3)/dx + 1/2*(y3-y2)/dx;
third_grad = 1/2*(y5-y4)/dx + 1/2*(y4-y3)/dx;
%% ...
last_grad = (ylast-ylast_1)/dx;

BLack curve and black points is your original data. Red points are points where gradient/derivative is calculated. Pay attention that first and last points are not at the beginning and end

Second time you use gradient you use the same step (again). But for the first and last points step/2 should be used.

Interesting fact

If you have not regular step (h is not equal) gradient can't handle it. Better use diff

    % generate some data
x = sort(10*rand(100,1));
y = sin(x);
dy1 = gradient(y,x);
dy2 = diff(y)./diff(x);
x2 = x(2:end)-diff(x)/2;        % actual x for diff/derivative
plot(x,y)                       % original data
hold on
plot(x,dy1,'g')                 % derivative obtained with gradient
plot(x2,dy2,'.m')               % derivative obtained with diff
plot(x,cos(x),'k')              % exact derivative
hold off
legend('orignal data','gradient','diff','exact')

Best Answer

Related Solutions

MATLAB: 1-d gradient

MATLAB: Calling gradient multiple times

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