Rotate line to arbitary position around unit circle-CodePudding

I have a unit circle with n roots of unity marked. I would like to be able to rotate, translate, and scale a line resting on the x-axis (between -1 and 1) to connect any pair of marked roots. Currently my code can do this in some cases, but doesn't work in general. I want to avoid hard-coding how the line should move for each possible pair. Here's what I have so far:

clear

%% Roots of unity
n = 10;
roots = zeros(1, n);
for k = 1 : n
    roots(k) = exp(2 * k* pi * 1i / n);
end

%% Move line
% Pair of roots the line should connect
point_1 = roots(2);
point_2 = roots(6);

% Coordinates of pair of roots
x1 = real(point_1);
x2 = real(point_2);
y1 = imag(point_1);
y2 = imag(point_2);

d = sqrt((x1-x2)^2 (y1-y2)^2); % Euclidean distance between pair of roots
m = (y1 - y2) / (x1 - x2); % Gradient of line connecting pair of roots
c = y1 - m * x1; % y-intercept of line

int = -c / m; % x-coordinate that the rotation should occur at
shift = [int; 0];

x = linspace(-1, 1, 10); % Initial line lying purely on x-axis
y = 0 * x;
v = [x; y];

theta = atan((y2-shift(2))/(x2-shift(1))); % Angle by which to rotate

rot = [cos(theta), -sin(theta); sin(theta), cos(theta)]; % Rotation matrix

u = v * (d / 2); % Scale initial line

if m < 1e-3  % Horizontal case
    shift = [0; 0];
end

w = (rot * (u - shift))   shift; % Apply rotation

% Another shift that seems necessary
% This is definitely a problematic section
shift_x = w(1, 1) - x2;
shift_y = w(2, 1) - y2;
shift_2 = [shift_x; shift_y];
w = w - shift_2;


%% Plot
fig = figure;
fig.Units = 'inches';
fig.Position = [1, 1, 9, 9];
ax = gca;
tt = title(ax, 'Title');
tt.FontWeight = 'bold';
tt.FontSize = 20;
st = subtitle(ax, sprintf('More text here'));
st.FontAngle = 'italic';
st.FontSize = 15;

hold on
hCircle = viscircles([0, 0], 1, 'Color', 'k');
for i = 1 : n
   x_point = real(roots(i));
   y_point = imag(roots(i));
   hPin = plot(x_point, y_point, 'Marker', 'o', 'MarkerSize', 20, 'MarkerfaceColor', 'red', ...
       'MarkerEdgeColor', 'black');
end

% Plot original and shifted line, split into colours so direction is easier to see
plot(v(1,1:4), v(2,1:4), 'b');
plot(v(1,4:7), v(2,4:7), 'r');
plot(v(1,7:end), v(2,7:end), 'g');

plot(w(1,1:4), w(2,1:4), 'b');
plot(w(1,4:7), w(2,4:7), 'r');
plot(w(1,7:end), w(2,7:end), 'g');

For example, keeping point_1 = roots(2); and changing only point_2 = roots(p); works as intended for only p=3, 4, 6, 7, 8.

Any guidance on how to get this working would be greatly appreciated, thanks!

Edit: To give some more details, basically I have an array of numbers between 0 and 1 (rather than just a line) which I want to plot on the line that would connect two roots. E.g. if my array is x=[0.2, 0.5, 0.9], then I want three points between point_1 and point_2, the first being 0.2d down the connecting line away from point_1, the second 0.5d (i.e. halfway), and the final being 0.9d away.

CodePudding user response：

First of all, since the points you want to connect are complex numbers, it is easier to work with complex coordinate directly.

Roots of unity
I have simplified your code a bit.
Some may raise a flag on naming a variable roots since roots is a built-in matlab function. I am fine with it, as long as the usage does not cause any confusion, namely, don't use roots as a variable and as a function in the same context.
As matlab provides so many built-in functions, it is impossible to avoid name collision unless one knows them all by heart or searches before naming every single variable.

n = 10;
k = 1:n;
roots = exp(2 * k * pi * 1i / n);

Scaling, rotating, and translating

% Pair of roots the line should connect
point_1 = roots(2);
point_2 = roots(6);

d = abs(point_2 - point_1);         % distance between pair of roots
theta = angle(point_2 - point_1);   % rotation angle

p_on_line = linspace(-1, 1, 10);                    % Initial line lying on x-axis
p_on_line = p_on_line * d/2;                        % scale
p_on_line = p_on_line * exp(1i*theta);              % rotate
p_on_line = p_on_line   (point_1 - p_on_line(1));   % translate

Plot
I added some scatter points and removed irrevelant parts (e.g. title, fonts).

fig = figure;
fig.Units = 'inches';
fig.Position = [1, 1, 9, 9];

hold on
hCircle = viscircles([0, 0], 1, 'Color', 'k');
hPin = plot(roots, 'o', 'MarkerSize', 20, 'MarkerfaceColor', 'red', ...
            'MarkerEdgeColor', 'black');

% Plot line connecting roots
plot(p_on_line(1:4), 'b.-', 'MarkerSize', 20);
plot(p_on_line(4:7), 'r.-', 'MarkerSize', 20);
plot(p_on_line(7:end), 'g.-', 'MarkerSize', 20);

% Plot original line
original_x = linspace(-1, 1, 10);
original_y = zeros(1, 10);
plot(original_x(1:4), original_y(1:4), 'b.-', 'MarkerSize', 20);
plot(original_x(4:7), original_y(4:7), 'r.-', 'MarkerSize', 20);
plot(original_x(7:end), original_y(7:end), 'g.-', 'MarkerSize', 20);
hold off

This should work for all combinations of root pairs.