To generate constant arc points on Archimedean Spiral

Dear Sir/ Mam,

The code for constant angle has been attahced. In this code i want to modify and need to generate constant arc points on Archimedean Spiral instead of constant angle

clear all

clc

% Parameters

R = 0; %outer radius

b = 2; %incerement per rev, equivalent to feed

a = -20; %inner radius

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

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

desiredAngle = 10;% Desired constant angle between points

npt= 360/desiredAngle; %% number of points in each spiral

t= ((-a/(2))*npt);

theta = 0:th/t:th;

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

% Interpolate points along the Archimedean spiral with constant angle

interpPoints = interp1(theta, [r', theta'], 0:th/t:th);

% Extract interpolated r and theta values

interp_r = interpPoints(:, 1);

interp_theta = interpPoints(:, 2);

% Convert polar coordinates to Cartesian coordinates

x =(-interp_r.* cos(interp_theta))';

y = (-interp_r.* sin(interp_theta))';

% Plot the Archimedean spiral and equidistant points in terms of angle

figure;

plot((-r.* cos(theta)), (-r .* sin(theta)), 'LineWidth', 2); % Original Archimedean Spiral

hold on;

plot(x, y, '-o'); % Equidistant Points in terms of Angle

axis equal;

xlabel('X-axis');

ylabel('Y-axis');

title('Spiral Toolpath with Constant Angle');

legend('Original Archimedean Spiral', 'Equidistant Points in terms of Angle');

Answer 1

[Mathieu NOE]

hello

try this

also I prefer to have

R = 20; %outer radius

a = 0; %inner radius

instead of

R = 0; %outer radius

a = -20; %inner radius

as for me inner radius (or start point) is at the center and not the most outer point - same comment for the other point

code

% Parameters
R = 20; %outer radius
b = 2; %incerement per rev, equivalent to feed
a = 0;    %inner radius
n = round((R - a)./(b));  %number  of revolutions and number of 
th = 2*n*pi;      %angle obtained for n number of revolution, for one revoultion 2*pi
theta = linspace(0,th,n*360);
r= a + b.*theta./(2*pi);
% Convert polar coordinates to Cartesian coordinates
xr = r.* cos(theta);
yr = r.* sin(theta);

%% constant arc code
% curvilinear abscissa : s = integral of a*sqrt(1+theta²)
y = b*sqrt(1+theta.^2);
s = cumtrapz(theta,y);

desiredArcsPoints = 3*360; % n*360;% Desired arcs points
sd = linspace(0,s(end),desiredArcsPoints);
% Interpolate points along the Archimedean spiral with constant arc
ti = interp1(s, theta, sd);  % angle values with constant arc spacing
r= a + b.*ti./(2*pi);
% Convert polar coordinates to Cartesian coordinates
x = r.* cos(ti);
y = r.* sin(ti);
% Plot the Archimedean spiral and equidistant points in terms of angle
figure;
 plot(xr, yr, 'LineWidth', 2);  % Original Archimedean Spiral
 hold on;
plot(x, y, '-o');  % Equidistant Points in terms of Angle
axis equal;
xlabel('X-axis');
ylabel('Y-axis');
title('Spiral Toolpath with Constant Arc');
legend('Original Archimedean Spiral', 'Equidistant Points in terms of Arc');

[KUNDAN PRASAD]

Thank you for informative details provide on the code.

Still i have one question?

How we can make the constant angle in the inner portion of the spiral toolpath with constant arc.

i.e., if total diameter is 40; then 10 mm is made with constant angle and rest is 30 mm diameter with constant arc.

Looking for the reply of question.

Once again thank you

[Mathieu NOE]

try this

clear all
clc
% Parameters
R2 = 40; %outer radius (between this radius and R1 is constant arc increment)
R1 = 10; %intermediate radius (below this radius is constant angle 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
theta = linspace(0,th,n*360);
r= a + b.*theta./(2*pi);
% Convert polar coordinates to Cartesian coordinates
xr = r.* cos(theta);
yr = r.* sin(theta);

%% constant angular increment section
ind1 = (r<=R1);
x1 = xr(ind1);
y1 = yr(ind1);


%% constant arc increment section
ind2 = (r>R1);
theta2 = theta(ind2);

% curvilinear abscissa : s = integral of a*sqrt(1+theta²)
y = b*sqrt(1+theta2.^2);
s = cumtrapz(theta2,y);

desiredArcsPoints = 3*360; % n*360;% Desired arcs points
sd = linspace(0,s(end),desiredArcsPoints);
% Interpolate points along the Archimedean spiral with constant arc
ti = interp1(s, theta2, sd);  % angle values with constant arc spacing
r= a + b.*ti./(2*pi);
% Convert polar coordinates to Cartesian coordinates
x = r.* cos(ti);
y = r.* sin(ti);
% combine both data sets (below and above R1)
x = [x1 x];
y = [y1 y];
% Plot the Archimedean spiral 
figure;
 plot(xr, yr, 'LineWidth', 2);  % Original Archimedean Spiral
 hold on;
plot(x, y, '-o');  % Equidistant Points in terms of Angle then Arc
axis equal;
xlabel('X-axis');
ylabel('Y-axis');
title('Spiral Toolpath with Constant Arc');
legend('Original Archimedean Spiral', 'Equidistant Points in terms of Angle / Arc');