Let’s z be a 2D point in the space as \(z = x + jy\), if we want to rotate this point a given angle \(\theta\), we get the following expressions:
\[e^{j\theta} \cdot z = \left(\cos{\theta} + j \sin{\theta}\right)\left(x+jy\right) \\ = x\cos{\theta}-y\sin{\theta} + j \left(y \cos{\theta} + x \sin{\theta} \right) \\ = x’ + j y’ \]
Then, for a generic point, the rotation can be expressed as an equation system, where \(x’\) and \(y’\) are the new coordinates, \(\theta\) is the rotation angle and \(x\) and \(y\) are the original coordinates:
\[\begin{bmatrix}
x’\\
y’
\end{bmatrix}=
\begin{bmatrix}
\cos{\theta} & -\sin{\theta}\\
\sin{\theta} & \cos{\theta}
\end{bmatrix}\begin{bmatrix}
x\\
y
\end{bmatrix} \]
This rotation can be coded in MATLAB as:
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%% Function to rotate vector function v = rotate(P, theta) rot = [cos(theta) -sin(theta); sin(theta) cos(theta)]; v = rot*P; end |
A possible implementation of the cordic algorithm could be:
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%% Clear all previous values clear all; %% Define vectors A = [2;-3]; O = [0;0]; %% Define accuracy error_limit = 0.01; %% Initialize variables start_angle = pi/2; % 90º current_angle = start_angle; acc_angle = 0; A_rotated = A; steps = 0; % Second quadrant (90º - 180º) if(A_rotated(1) < 0 && A_rotated(2) > 0) A_rotated = rotate(A_rotated, -pi/2); acc_angle = pi/2; % Third quadrant (180º - 270º) elseif(A_rotated(1) < 0 && A_rotated(2) < 0) A_rotated = rotate(A_rotated, -pi); acc_angle = pi; % Forth quadrant (270º - 360º) elseif(A_rotated(1) > 0 && A_rotated(2) < 0) A_rotated = rotate(A_rotated, -3*pi/2); acc_angle = 3*pi/2; end %% Compute angle % Keep rotating while error is too high while(abs(A_rotated(2)) > error_limit) % Represent current vector quiver(0, 0,A_rotated(1), A_rotated(2)); % Keep previous vectors hold on; % Decrease angle rotation current_angle = current_angle/2; % (For debugging purposes) current_angle_deg = current_angle*180/pi; % Save current error error = A_rotated(2); % If y coordinate is still positive if(error > 0) % Rotate again conterclockwise A_rotated = rotate(A_rotated, -1*current_angle); % Accumulate rotated angle acc_angle = acc_angle + current_angle; % If y coordinate is negative else % Rotate vector clockwise A_rotated = rotate(A_rotated, current_angle); % Substract current angle to the accumulator because we have % overcome the actual angle value acc_angle = acc_angle - current_angle; end % (For debugging purposes) acc_angle_deg = acc_angle*180/pi; % Increase step counter steps = steps + 1; end % Print angle, hypotenuse length and number of steps needed to compute fprintf('Angle = %f\nHypotenuse = %f\nNumber of steps: %d\n', acc_angle*180/pi, A_rotated(1), steps); %% Function to rotate vector function v = rotate(P, theta) rot = [cos(theta) -sin(theta); sin(theta) cos(theta)]; v = rot*P; end |
I have coded an interactive applet to illustrate the algorithm. It has been done using the p5.js library. The error limit has been set to \(0.5\).
