How to find points inside a region defined by a rotated ellipse

Hello. I have a rotated ellipse defined by a center [x0,y0], the major and minor axes r1 and r2 as well as the angle of rotation, theta. My goal is to determine whether a given point [x y] lie within the regoin defined by this ellipse. Thanks

Answer 1

[Wan Ji]

Here is an example of telling how to calculate whether a given point [x y] lie within the regoin defined by this ellipse.

function main
x = rand(1000,1); % the x coordinates of 1000 points
y = rand(1000,1); % the y coordinates of 1000 points
xc = 0.4; % centre x coordinate of the ellipse
yc = 0.5; % centre y coordinate of the ellipse
a = 0.3; % major semi-axis
b = 0.1; % minor semi-axis
theta = 54 * pi / 180; % the orientation of ellipse 54 degree
p = inellipse(x, y, xc, yc, a, b, theta) ;

% plot the ellipse boundary
plot_ellipse(xc, yc, a, b, theta); hold on
% scatter the points in the ellipse, markered in red
scatter(x(p), y(p), 10, 'r', 'filled');
% scatter the points out of the ellipse, markered in blue
scatter(x(~p), y(~p), 10, 'b', 'filled');
legend('Ellipse boundary','In ellipse', 'Out ellipse')
axis equal
end

function p = inellipse(x, y, xc, yc, a, b, theta) 
% return a logical array, indices of the points in the ellipse
xr = x - xc;
yr = y - yc;
x0 = cos(theta)*xr + sin(theta)*yr;
y0 = -sin(theta)*xr + cos(theta)*yr;
p = x0.^2 / a^2 + y0.^2 / b^2 < 1;
end

function plot_ellipse(xc, yc, a, b, theta)
% plot an ellipse
alpha = linspace(0,2*pi,101);
x0 = a*cos(alpha);
y0 = b*sin(alpha);
x = cos(theta)*x0 - sin(theta)*y0 + xc; % rotate and translate
y = sin(theta)*x0 + cos(theta)*y0 + yc;
plot(x,y,'k--')
end

To this example, you will obtain a figure which validates this method.

[Yakubu Galadima]

Thank you so much Wan. Your suggestion did solved my problem. Once again, many thanks