How to get the non-overlapping area ??

Hi all,

My question is; How to compute [(sum of circles' areas)-(area of overlaps of the circles)]?

Any suggestions for code to do that??

[Walter Roberson]

Note that this does not calculate "non-overlapping area". You have places where the same area is overlapping between three circles. That overlapping area will be included three times, once for each circle it is inside, but the overlap will be subtracted only once.

Question: is this a numeric question or a theoretical question? Are you expected to give the symbolic formula for the result, or are you expected to give a floating point number? If you are expected to give a floating point number, then is there any guidance as to how many digits are required? I ask because there are techniques for calculating this that approximate it by small squares instead of by calculating exactly where all of the intersections are.

[Eman S]

Thanks so much for your reply. I expected to give a floating point number. 5 digits are required.

[Walter Roberson]

Unfortunately the efficient but dumb way of solving this to 5 decimal places would require somewhere around 20 gigabytes of memory.

[Eman S]

Thanks so much for your help. I really appreciate that. But, why does it need this large memory. Is there any possible way to do that if it would be solved for less number of digits; e.g. 2 digits?

[Image Analyst]

Original question by Erman (in case he deletes it):

Hi all, I have the following code to draw 10 circles;

%%center of circles
 X = [0.02  0.44; 0.08 0.58;...
      0.22 0.57; 0.3 0.9;...
      0.44 0.79; 0.39 0.24;...
      0.65  0.19; 0.95 0.11;...
      0.93 0.7; 0.82 0.93];

   theta = linspace(0,2*pi) ; 
   C_M=[X(:,1),X(:,2)]; 
   r=[0.18,0.18,0.18,0.18,0.18,0.18,0.13,0.13,0.13,0.13];

      for index = 1:length(C_M)
          if ~isinf(C_M(index))
             x(:,index) =r(index)*cos(theta) ; 
             y(:,index) =r(index)*sin(theta) ;
             xx(:,index)= x(:,index)+C_M(index,1);
             yy(:,index)= y(:,index)+C_M(index,2);
            %%plot circles
              plot(xx,yy,'-b','linewidth',1.5)

          end
       end 

%%x and y axis limits
  xlim([-0.2,1.2])
  ylim([-0.2,1.2])
grid on
grid minor

Six of these circles have radius of r=0.18 and the rest have radius of r=0.13. So, there are overlapping areas as the following image;

My question is; How to compute [(sum of circles' areas)-(area of overlaps of the circles)], i.e.

Any suggestions for code to do that??

Answer 1

[Walter Roberson]

You can calculate lower and upper bounds of the data based upon the starting positions and the radii. Then you can create x and y grids like

N = 500;
xv = linspace(min_x, max_x, N);
dx = xv(2) - xv(1);
yv = linspace(min_y, max_y, N);
dy = yv(2) - yv(1);
[X, Y] = ndgrid(xv, yv);
counts = zeros(N, N, 'uint8');
unscaled_areas = zeros(number_of_circles, 1);

After that you can loop,

mask = (X-x_center(K)).^2 + (Y-y_center(K)).^2 <= radii(K).^2;
unscaled_areas(K) = nnz(mask);
counts(mask) = counts(mask) + 1;

and do that for all of the circles.

Then

scaled_areas = unscaled_areas * (dx * dy);
total_scaled_area = sum(scaled_areas);   %which is the first summation in your equation

Now examine the counts matrix. Anywhere location that is 0 had no circles present. Anywhere that is 1 had exactly 1 circle present, but you do not need to count those because size of the circles without overlap was already calculated in scaled_areas. Anywhere that is more than 1 had more than 1 circle present. Count how many locations have value greater than 1, multiply by dx * dy to get the total area occupied by overlaps, which is the second summation from your equations. Subtract appropriate variables to get the overall result of your calculations.

The memory requirements of this go up as you go for finer resolution of boxes, so the more decimal places you need, the more memory it uses, corresponding to increasing N.

[Eman S]

Thanks so much for your help.