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:
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;
