i have two concentric circles and a set of random nodes generated inside both the circles, How to index these nodes which lie inside the smaller circle and which lie between the two circles

after indexing i want to find the distance of every nodefrom the center of the two circles

Answer 1

[Image Analyst]

Try this:

clc;    % Clear the command window.
clear all;
workspace;  % Make sure the workspace panel is showing.
format long g;
format compact;

x0=3; % Center of the circle in the x direction.
y0=2; % Center of the circle in the y direction.
r=1;
r1=0.65;
teta=-pi:0.01:pi;
x=r*cos(teta)+x0;
y=r*sin(teta)+y0
x1=r1*cos(teta)+x0
y1=r1*sin(teta)+y0
plot(x,y,x1,y1)
hold on
scatter(x0,y0,'or')
axis square
%----------------------------------------
% divide your circle to n sectors
n=8
tet=linspace(-pi,pi,n+1)
xi=r*cos(tet)+x0
yi=r*sin(tet)+y0;
for k=1:numel(xi)
  plot([x0 xi(k)],[y0 yi(k)], 'r*', 'MarkerSize', 10, 'LineWidth', 2)
  hold on
end
% distribution of nodes
n=30;
angle1 = 0;
angle2 = 2*pi;
t = (angle2 - angle1) * rand(n,1) + angle1;
R = r*sqrt(rand(n,1));
xr = x0 + R.*cos(t);
yr = y0 + R.*sin(t);
% Now display our random set of nodes in a figure.
scatter(xr,yr,'g','MarkerFaceColor','c')
axis square;
grid on;
% Enlarge figure to full screen.
set(gcf, 'units','normalized','outerposition',[0 0 1 1]);
fontSize = 30;
xlabel('X', 'FontSize', fontSize);
ylabel('Y', 'FontSize', fontSize);
title('Random Locations Within a Circle', 'FontSize', fontSize)
% Find out which (xr, yr) are in between the circles.
radii = sqrt((xr-x0).^2 + (yr-y0).^2);
inBetween = radii > r1;
plot(xr(inBetween), yr(inBetween), 'bo', 'MarkerSize', 15);

[Image Analyst]

To also get those points within a sector, just calculate the angles and use the same kind of logic:

clc;    % Clear the command window.
clear all;
close all;
workspace;  % Make sure the workspace panel is showing.
format long g;
format compact;

x0=3; % Center of the circle in the x direction.
y0=2; % Center of the circle in the y direction.
r=1;
r1=0.65;
teta=-pi:0.01:pi;
x=r*cos(teta)+x0;
y=r*sin(teta)+y0
x1=r1*cos(teta)+x0
y1=r1*sin(teta)+y0
plot(x,y,x1,y1)
hold on
scatter(x0,y0,'or')
axis square
%----------------------------------------
% divide your circle to n sectors
numberOfPoints=8
tet=linspace(-pi,pi,numberOfPoints+1)
xi=r*cos(tet)+x0
yi=r*sin(tet)+y0;
for k=1:numel(xi)
  plot([x0 xi(k)],[y0 yi(k)], 'r*', 'MarkerSize', 10, 'LineWidth', 2)
  hold on
end
% distribution of nodes
numberOfPoints = 150;
angle1 = 0;
angle2 = 2*pi;
t = (angle2 - angle1) * rand(numberOfPoints,1) + angle1;
R = r*sqrt(rand(numberOfPoints,1));
xr = x0 + R.*cos(t);
yr = y0 + R.*sin(t);
% Now display our random set of nodes in a figure.
scatter(xr,yr,'g','MarkerFaceColor','c')
axis square;
grid on;
% Enlarge figure to full screen.
set(gcf, 'units','normalized','outerposition',[0 0 1 1]);
fontSize = 30;
xlabel('X', 'FontSize', fontSize);
ylabel('Y', 'FontSize', fontSize);
title('Random Locations Within a Circle', 'FontSize', fontSize)
% Find out which (xr, yr) are in between the circles.
radii = sqrt((xr-x0).^2 + (yr-y0).^2);
inBetween = radii > r1;
plot(xr(inBetween), yr(inBetween), 'bo', 'MarkerSize', 15);

% Find angles of all points from the center of the circles.
angles = atan2((yr-y0),(xr-x0))
% Plot a blue diamond over angles between 3*pi/4 and pi/2
% First find indexes of those points in that angle sector.
selectedAngles = angles > 2*pi/4 & angles < 3*pi/4;
% Also make sure it's in the "in between" annular region
selectedAngles = selectedAngles & inBetween;
plot(xr(selectedAngles), yr(selectedAngles), 'bd', 'MarkerSize', 25, 'LineWidth', 2);

See the points outlined in a blue diamond in the upper left of the figure below: