Archimedean spiral points with Equal arc length discretization method

%% Archimedean spiral points with Equal angle discretization method

clear all
clc
% Parameters
R2 = 30; %outer radius (between this radius and R1 is constant arc increment)


b = 2; %incerement per rev, equivalent to feed
a = 0;    %inner radius

n = round((R2 - a)./(b));  %number  of revolutions and number of 

th = 2*n*pi;      %angle obtained for n number of revolution, for one revoultion 2*pi

%% parameters defined for constant angle like incremental x value and theta value
dtheta= 1;   %% incremental angle (degree)

eqangle= dtheta* pi/(180);   %% equal angle obtianed for each spiral revolution 

npt= 2*pi/(eqangle); %% number of points in each spiral

tpoints= (npt*n)+1;   %% total number of points in n number of revolutions

theta = linspace(0,th,tpoints); 


r= a + b.*theta./(2*pi);

% Convert polar coordinates to Cartesian coordinates
xr = r.* cos(theta);
yr = r.* sin(theta);

plot(xr, yr, '-o'); 

NOTE:

The above code works well for equal angle discretization method. The code consists the information about equal angle i.e, 1 degree, the total number of points, number of points in each spiral, total angle obtained, etc.

Simialry the code for equal arc length discretization method should cosnsists the following information such as for incremental arc length i.e., 1 mm, what is number of points around the edge, spiral length, total number of points etc

Please help to developed the code based on the above information. The screenshot has been attached for your reference.

Looking forward for the reply of mail.

Answer 1

[Subhajyoti]

Hi @KUNDAN PRASAD,

It is my understanding that you are trying to generate the Archimedean spiral with equal arc-length discretization method.

In equal arc length discretization, we aim to ensure the distance between points along the spiral curve is constant (1 mm in your case). The relationship between the radius "r" and the angle "θ" in an Archimedean spiral is defined as: where:

a is the inner radius,

b is the distance between successive spiral turns,

θ is the angle in radians.

To ensure equal arc length, we need to iteratively compute θ based on the desired arc length between points. The arc length between two nearby points can be computed using the following formula:

Here, in the following implementation, I have generated an Archimedean spiral with equal arc length discretization method

% Define your layers in the dlnetwork format
% Parameters
R2 = 30; % Outer radius (mm)
b = 2;   % Increment per revolution (equivalent to feed) (mm)
a = 0;   % Inner radius (mm)
arcLengthIncrement = 1; % Desired arc length increment (mm) between points

% Calculate the total number of revolutions
n = round((R2 - a) / b);  % Number of revolutions
theta_max = 2 * n * pi;   % Total angle (radians) for n revolutions

% Initialize variables
theta = 0;   % Start angle
r = a;       % Start radius
tpoints = [0, 0]; % To store points

% Arc length calculation and point generation
while r <= R2
    % Calculate the next theta and r based on equal arc length
    % Arc length formula: dS = sqrt((r*dtheta)^2 + (dr)^2)
    % First, compute the current radius and its rate of change (dr/dtheta)
    dr_dtheta = b / (2 * pi);   % Derivative of r with respect to theta
    
    % Calculate dtheta for the current step to keep arc length constant
    dtheta = arcLengthIncrement / sqrt(r^2 + (dr_dtheta)^2);
    
    % Update theta and r
    theta = theta + dtheta;
    r = a + b * theta / (2 * pi);
    
    % Convert polar coordinates to Cartesian
    xr = r * cos(theta);
    yr = r * sin(theta);
    
    % Append new point
    tpoints = [tpoints; xr, yr];
    
    % Stop if we've reached or exceeded the outer radius
    if r > R2
        break;
    end
end

% Plot the spiral
figure;
plot(tpoints(:,1), tpoints(:,2), '-o');
xlabel('X (mm)');
ylabel('Y (mm)');
title('Archimedean Spiral with Equal Arc Length Discretization');
axis equal;
grid on;

[KUNDAN PRASAD]

Thank you

[Subhajyoti]

You are Welcome 😄

You can accept the answer if the answer helped you and you feel it might help others also.