MATLAB: Radius of curvature calculation

principle curvatures

I have a set of lines/curves ( discontinuous lines) from which i revolve them to get a surface. I need to find the principle curvatures on this surface. As i understand, to calculate the principle curvatures you would require the analytical equation. But i have only a numerical representation of the base curve. How do i go about in doing this?

Best Answer

While gradient() replies the slope, gradient(gradient()) determines 2nd derivative. The signed curvature is defined as:

k = y'' / (1 + y'^2)^1.5

with primes mean the derivative to x here.

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Try not to use global variables if its possible

function [x0, y0, lambda] = mycurvature(xinput, yinput)
alpha=atan2(yinput(2)-yinput(1),xinput(2)-xinput(1))-atan2(yinput(3)-yinput(2),xinput(3)-xinput(2));
if abs(alpha)<0.002;                       %set the curvature to 0 if the points are aligned
    R=inf; lambda=0;
    %plot(xinput,yinput,'-r',xinput,yinput,'*b','linewidth',2,'markersize',10);
else
initialguess = [1; 1; 1];                % Make a starting guess at the solution
options=optimset('Display','none');      % Option to display output
res = fsolve(@(vars)userfun(vars,xinput,yinput),initialguess,options);    % Call optimizer
x0=res(1);y0=res(2);R=res(3);     %x-coord of center; y-coord of center; radius of curvature
lambda=1/R;           %curvature
%t=0:0.01:2*pi;xx=x0+R*cos(t);yy=y0+R*sin(t);
%plot(xx,yy,'-r',xinput,yinput,'*b','linewidth',2,'markersize',10);
end
%str=sprintf('radius=%6.3f, curvature=%6.3f',R,lambda);
%title(str, 'fontsize', 14);xlabel('x'),ylabel('y');
%axis equal;
function F = userfun(vars,xinput,yinput)
F = [...
(xinput(1)-vars(1))^2+(yinput(1)-vars(2))^2-(vars(3))^2;  ...
(xinput(2)-vars(1))^2+(yinput(2)-vars(2))^2-(vars(3))^2;  ...
(xinput(3)-vars(1))^2+(yinput(3)-vars(2))^2-(vars(3))^2;  ...  
];

Main code:

clc , clear
a = 2;
b = 2;
t = linspace(0,pi/2,30);
x = a*cos(t);
y = b*sin(t);
plot(x,y,'.-r')
hold on
for i = 2:length(x)-1
    xinput = [x(i-1) x(i) x(i+1)];
    yinput = [y(i-1) y(i) y(i+1)];
    [xc, yc, lam] = mycurvature(xinput,yinput);
    plot([xc x(i)],[yc y(i)],'.-b')
end
hold off
grid on

Best Answer

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