conformal mapping of circle

how can I transform a circle to ellipse or airfoil in complex plane using conformal mapping and w = z + 1/z transform function?

Answer 1

[Gautam]

To transform a circle using conformal mapping, you can directly apply the transformation to the equation of the circle.

Here’s the code that performs the mapping to produce the corresponding output

% Define the parameters
theta = linspace(0, 2*pi, 1000); % Parameter for the circle

% Define the circle in the complex plane
r = 0.5; 
z = r * exp(1i * theta);

% Apply the conformal mapping: w = z + 1/z
w = z + 1 ./ z;

% Plot the original circle
figure;
subplot(1, 2, 1);
plot(real(z), imag(z), 'b', 'LineWidth', 2);
axis equal;
title('Original Circle');
grid on;

% Plot the transformed shape (ellipse)
subplot(1, 2, 2);
plot(real(w), imag(w), 'r', 'LineWidth', 2);
axis equal;
title('Transformed Shape (Ellipse)');
grid on;