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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.
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.
I've been working on a project for my engineering school. We have to create a mathematical framework allowing us to understand the transformation from two input angles to one output point. The robot we are working on is a 5 bar parallel kinematic robot.
We have found the equation and we managed to find the trajectory.
We have now the error equation which is really good because, for a given point in space, we can find the angles required to achieve minimum error.
This is the error equation. Now we are struggling to find a way to compute all the angles so that the equation equals zero.
I wanted to visualize the graph from 0 to 2*pi for both variables. But on Octave and Matlab, I always end up with an ill matrix making it impossible to use. And now we just don't know what we can do?
Does anyone have any advice for me?
PS : this is the code I was using :
clear
X = 0.5;
Y = 0.5;
L = 1.0;
E = 1.0;
discret = 0.06;
x = [0:discret:2*pi];
y = [0:discret:2*pi];
[xx, yy] = meshgrid (x, y);
Cx = cos(xx)
Sx = sin(xx)
Cy = cos(yy)
Sy = sin(yy)
D = (((-2*(L*Cx - E)+(2*(E - L*Cy))))/((-2*(L)*Sy)+(2*(L)*Sx)));
K = ((-E+L*Cx)^2 - (E-L*Cy)^2 - (L*Sy)^2 - (L*Sx)^2)/((-2*(L)*Sy)+(2*(L)*Sx));
z = abs(Y - X*D - K);
mesh(x, y, z)
meshc(xx,yy,z)
xlabel ("theta1");
ylabel ("theta2");
zlabel ("Er(theta1,theta2)");
grid on
The bit about precision is hinting that you are performing matrix division, when in fact judging from your equations, you only seem to need normal array division (i.e. an elementwise/vectorised operation, not a matrix operation).
Simply replace any / or ^ with ./, .^ etc as appropriate.
Also, your equations can be simplified significantly to make your code slightly clearer and avoid bugs due to legibility:
clear
X = 0.5; Y = 0.5; L = 1.0; E = 1.0; stepsize = 0.06;
x = [0 : stepsize : 2*pi]; y = [0 : stepsize : 2*pi];
[xx, yy] = meshgrid (x, y);
Cx = cos(xx); Sx = sin(xx); Cy = cos(yy); Sy = sin(yy);
D = ( Cy + Cx - 2 .* E/L ) ./ ( Sy - Sx );
K = 0.5 .* ( Sy.^2 + Sx.^2 + (Cy - E/L).^2 - (Cx - E/L).^2 ) ./ (Sy-Sx);
z = abs(Y - X .* D - K);
surf(x, y, z, 'edgecolor', 'none', 'facecolor', 'interp' )
set( gcf, 'colormap', hot(256), 'color', 'k' )
set( gca, 'gridcolormode', 'manual', 'gridlinestyle', '-', 'gridcolor', 'w', 'gridalpha', 0.1, 'fontsize', 16, 'color', [0.2, 0.2, 0.2], 'xcolor', [0.8,0.8,0.8], 'ycolor', [0.8,0.8,0.8], 'zcolor', [0.8,0.8,0.8] )
set( get( gca, 'xlabel' ), 'string', '\theta_1', 'fontsize', 36 );
set( get( gca, 'ylabel' ), 'string', '\theta_2', 'fontsize', 36 );
set( get( gca, 'zlabel' ), 'string', {'Er( \theta_1, \theta_2 )',''}, 'fontsize', 36 );
view( 15, 30 );
My goal is to fit a sinusoid to data goming from a datalogger using Octave.
The datalogger logs force which is produced using an excenter, so it theoretically should be a sine wave.
I could not find any hint on how to do this elsewhere.
Currently I'm using the function "splinefit" followd by "ppval" to fit my data but I don't realy get the results I hoped from it...
Has anybody an idea how I could fit a sinusoid to my data?
Here's my current code I use to fit the data and a scrennshot of the result:
## splinefit force left
spfFL = splinefit(XAxis,forceL,50);
fitForceL=ppval(spfFL,XAxis);
##middle force left
meanForceL=mean(fitForceL);
middleedForceL=fitForceL-meanForceL;
result spline fit
on the X-Axis I have the 30'000 measurepoints or logs
on the Y-Axis I have the actual measured force values
the data comes from the datalogger in a .csv-file like this
You can do a simple regression using the sine and cosine of your (time) input as your regression features.
Here's an example
% Let's generate a dataset from a known sinusoid as an example
N = 1000;
Range = 100;
w = 0.25; % known frequency (e.g. from specs or from fourier analysis)
Inputs = randi(Range, [N, 1]);
Targets = 0.5 * sin( w * Inputs + pi/3 ) + 0.05 * randn( size( Inputs ) );
% Y = A + B sin(wx) + C cos(wx); <-- this is your model
Features = [ ones(N, 1), sin(w * Inputs), cos(w * Inputs) ];
Coefs = pinv(Features) * Targets;
A = Coefs(1); % your solutions
B = Coefs(2);
C = Coefs(3);
% print your nice solution against the input dataset
figure('position', [0, 0, 800, 400])
ax1 = axes()
plot(Inputs, Targets, 'o', 'markersize', 10, ...
'markeredgecolor', [0, 0.25, 0.5], ...
'markerfacecolor', [0, 0.5, 1], ...
'linewidth', 1.5)
set(ax1, 'color', [0.9, 0.9, 0.9])
ax2 = axes()
X = 1:0.1:Range;
plot( X, A + B*sin(w * X) + C*cos(w * X), 'k-', 'linewidth', 5 ); hold on
plot( X, A + B*sin(w * X) + C*cos(w * X), 'g-', 'linewidth', 2 ); hold off
set(ax2, 'xlim', get(ax1, 'xlim'), 'ylim', get(ax1, 'ylim'), 'color', 'none')
You could do a least squares optimization, using fminsearch
% sine to fit (in your case your data)
x = 0:0.01:50;
y = 2.6*sin(1.2*x+3.1) + 7.3 + 0.2*rand(size(x)); % create some noisy sine with known parameters
% function with parameters
fun = #(x,p) p(1)*sin(p(2)*x+p(3)) + p(4); % sine wave with 4 parameters to estimate
fcn = #(p) sum((fun(x,p)-y).^2); % cost function to minimize the sum of the squares
% initial guess for parameters
p0 = [0 0 0 0];
% parameter optimization
par = fminsearch(fcn, p0);
% see if estimated parameters match measured data
yest = fun(x, par)
plot(x,y,x,yest)
Replace x and y with your data. The par variable contains the parameters of the sine, as defined in fun.
I have 20 numerical data points with x and y coordinates. I would like to plot them in a 2D plot. They will be concentrated around an x and y coordinate. To better visualise this behaviour, I would like to add histogram bars on top of the 2D scatter plot for the x axis, and histogram bars on the right of the 2D plot for the y axis this way, they do not interfere with the axis labels. Now, my 20 numerical points are in fact two sets of 10 and I would like to have both sets plotted in different colours. Something like this:
python plot
How can I do this?
Update:
FWHM11Avg = [3.88,3.43,3.16,3.22,3.73,2.43,2.88,3.01,3.59,2.17];
FWHM11Med = [4.4,3.1,3,3.15,3.9,2,3.00,2.85,3.85,2.2];
FWHM12Avg = [3.50,2.30,2.97,2.97,2.98,2.28,2.94,2.36,3.51,1.7];
FWHM12Med = [3.3,2.1,2.9,2.8,2.9,2.1,2.8,2.30,3.5,1.7];
minx = min([FWHM11Avg; FWHM11Med]);
maxx = max([FWHM11Avg; FWHM11Med]);
miny = min([FWHM12Avg; FWHM12Med]);
maxy = max([FWHM12Avg; FWHM12Med]);
% make figure
figure(1)
clf
% first subplot -- y-data histc
ah1 = subplot(2, 2, 1);
y_bins = 1.5:.25:4.5;
n = hist(FWHM12Avg, y_bins);
bar(y_bins, n, 'vertical', 'on')
hold on
hist(FWHM12Med, y_bins)
bar(y_bins, n, 'vertical', 'on')
% x-data histc
ah2 = subplot(2, 2, 4);
x_bins = 1.5:.25:4.5;
n = hist(FWHM11Avg, x_bins);
bar(x_bins, n, 'horizontal', 'on')
hold on
n = hist(FWHM11Med, x_bins);
bar(x_bins, n, 'horizontal', 'on')
% scatterplot
ah3 = subplot(2, 2, 2);
hold on
scatter(FWHM11Avg, FWHM11Med)
scatter(FWHM12Avg, FWHM12Med)
% link axes, adjust histc orientation
linkaxes([ah1, ah3], 'y')
linkaxes([ah3, ah2], 'x')
set(ah3,'XLim',[minx, maxx]);
set(ah3,'YLim',[miny, maxy]);
ah1.Box = 'off';
ah1.View = [180, -90];
ah1.Visible = 'off';
ah2.Visible = 'off';
ah2.Box = 'off';
ah2.View = [0, -90];
Also there seems not to be an option available for adding numerical axes to the histograms to see how many points there are in a bar - at least in the documentation I did not see any option. Is that so?
Second Update with applied suggestions to the above syntax:
FWHM11Avg = [3.88,3.43,3.16,3.22,3.73,2.43,2.88,3.01,3.59,2.17];
FWHM11Med = [4.4,3.1,3,3.15,3.9,2,3.00,2.85,3.85,2.2];
FWHM12Avg = [3.50,2.30,2.97,2.97,2.98,2.28,2.94,2.36,3.51,1.7];
FWHM12Med = [3.3,2.1,2.9,2.8,2.9,2.1,2.8,2.30,3.5,1.7];
minx = min([FWHM11Avg; FWHM11Med]);
maxx = max([FWHM11Avg; FWHM11Med]);
miny = min([FWHM12Avg; FWHM12Med]);
maxy = max([FWHM12Avg; FWHM12Med]);
% make figure
figure(1)
clf
% first subplot -- y-data histc
ah1 = subplot(2, 2, 1);
y_bins = 1.5:.25:4.5;
n = hist(FWHM12Avg, y_bins);
bar(y_bins, n, 'vertical', 'on')
hold on
hist(FWHM12Med, y_bins)
bar(y_bins, n, 'vertical', 'on')
% x-data histc
ah2 = subplot(2, 2, 4);
x_bins = 1.5:.25:4.5;
n = hist(FWHM11Avg, x_bins);
bar(x_bins, n, 'horizontal', 'on')
hold on
n = hist(FWHM11Med, x_bins);
bar(x_bins, n, 'horizontal', 'on')
% scatterplot
ah3 = subplot(2, 2, 2);
hold on
scatter(FWHM11Avg, FWHM11Med)
scatter(FWHM12Avg, FWHM12Med)
% link axes, adjust histc orientation
linkaxes([ah1, ah3], 'y')
linkaxes([ah3, ah2], 'x')
set(ah3,'XLim',[minx, maxx]);
set(ah3,'YLim',[miny, maxy]);
set(ah1,'Box','off');
set(ah1,'View',[180, -90]);
set(ah1,'Visible','off');
set(ah2,'Visible','off');
set(ah2,'Box','off');
set(ah2,'View',[0, -90]);
Please research before asking. There is a function in Matlab scatterhist which does this
x0 = 6.1;
y0 = 3.2;
n = 50;
r = rand(n ,1 );
theta = 2*pi*rand(n, 1);
x = x0 + r.*cos(theta);
y = y0 + r.*sin(theta);
scatterhist(x,y, 'Direction','out', 'Location', 'NorthEast')
Edit: Using the data you provided. Is this what you want?
FWHM11Avg = [3.88,3.43,3.16,3.22,3.73,2.43,2.88,3.01,3.59,2.17];
FWHM11Med = [4.4,3.1,3,3.15,3.9,2,3.00,2.85,3.85,2.2];
FWHM12Avg = [3.50,2.30,2.97,2.97,2.98,2.28,2.94,2.36,3.51,1.7];
FWHM12Med = [3.3,2.1,2.9,2.8,2.9,2.1,2.8,2.30,3.5,1.7];
% make figure
figure(1)
clf
FWHM11Avg = FWHM11Avg(:);
FWHM11Med = FWHM11Med(:);
FWHM12Avg = FWHM12Avg(:);
FWHM12Med = FWHM12Med(:);
minX = min([FWHM11Avg; FWHM12Avg]);
maxX = max([FWHM11Avg; FWHM12Avg]);
minY = min([FWHM11Med; FWHM12Med]);
maxY = max([FWHM11Med; FWHM12Med]);
resX = 0.25;
resY = 0.25;
nBinsX = ceil((maxX - minX) / resX);
nBinsY = ceil((maxY - minY) / resY);
label = vertcat( ...
num2cell(repmat('FWHM11', size(FWHM11Avg)),2), ...
num2cell(repmat('FWHM12', size(FWHM11Avg)),2));
Avg = vertcat(FWHM11Avg, FWHM12Avg);
Med = vertcat(FWHM11Med, FWHM12Med);
% scatterplot
scatterhist(Avg, Med, 'Group', label, 'Direction','out', ...
'Location', 'NorthEast', 'NBins', [nBinsX, nBinsY])
This is something I've been using lately:
% generate some random data
mu = [1 2];
sigma = [1 0.5; 0.5 2];
R = chol(sigma);
my_data1 = repmat(mu,100,1) + randn(100,2)*R;
mu = [2 1];
sigma = [3 -0.5; -0.5 2];
R = chol(sigma);
my_data2 = repmat(mu,100,1) + randn(100,2)*R;
% find limits
minx = min([my_data1(:, 1); my_data2(:, 1)]);
maxx = max([my_data1(:, 1); my_data2(:, 1)]);
miny = min([my_data1(:, 2); my_data2(:, 2)]);
maxy = max([my_data1(:, 2); my_data2(:, 2)]);
% make figure
figure(1)
clf
% first subplot -- y-data histogram
ah1 = subplot(2, 2, 1);
histogram(my_data1(:, 2), 'Orientation','horizontal', 'Normalization', 'probability', 'BinWidth', 0.5)
hold on
histogram(my_data2(:, 2), 'Orientation','horizontal', 'Normalization', 'probability', 'BinWidth', 0.5)
% x-data histogram
ah2 = subplot(2, 2, 4);
histogram(my_data1(:, 1), 'Normalization', 'probability', 'BinWidth', 0.5)
hold on
histogram(my_data2(:, 1), 'Normalization', 'probability', 'BinWidth', 0.5)
% scatterplot
ah3 = subplot(2, 2, 2);
hold on
scatter(my_data1(:, 1), my_data1(:, 2))
scatter(my_data2(:, 1), my_data2(:, 2))
% link axes, adjust histogram orientation
linkaxes([ah1, ah3], 'y')
linkaxes([ah3, ah2], 'x')
ah3.XLim = [minx, maxx];
ah3.YLim = [miny, maxy];
ah1.Box = 'off';
ah1.View = [180, -90];
ah1.Visible = 'off';
ah2.Visible = 'off';
ah2.Box = 'off';
ah2.View = [0, -90];
producing this plot
This code assumes a recent version of MATLAB (I use 2014b), but can be easily adapted using the old histogram functions (hist, histc) and the set(..) syntax for graphical objects.
I'm working on a small simulation in matlab. For this purposes I want to create an animation of the simulated object (inverted pendulum). Unfortunatly the axes keep rescaling.
I tried everything. There are similar questions, but I just can't get it to work. The best i got is the code below. Where I get both axes at the same time, the scaled from -5 to 5 and those scaled by matlab.
%init visualisation
visualisation = figure();
axis([-5 5 -5 5]);
xlim([-5 5]);
ylim([-5 5]);
ax_hand = axes;
for i = 1:N
k1 = h * feval ( 'RHS', t0, x0, u );
k2 = h * feval ( 'RHS', t0 + (h/2), x0 + (k1/2), u);
k3 = h * feval ( 'RHS', t0 + h/2, x0 + k2/2, u);
k4 = h * feval ( 'RHS', t0 + h, x0 + k3, u);
x0 = x0 + ( k1 + 2*k2 + 2*k3 + k4 ) / 6;
t0 = t0 + h;
% model output
wi(1:neqn,i+1) = x0';
% model visualisation
%plotting cart
figure(visualisation);
plot(x0(3), 0, 'ro', 'LineWidth', 3);
%plotting pendulum
l = 2;
%plot(sin(x0(1))*l, cos(x0(1))*l, 'b*' , 'LineWidth', 2);
% regulator
end;
Here's one approach:
Do the first plot (possibly empty, doesn't matter). Get a handle to it, say h.
Set axis limits and include the statement axis manual fo freeze them.
Update plot (in a loop) via the 'XData' and 'YData' poperties of h.
Example:
h = plot(NaN, NaN, 'o'); %// empty plot
axis([0 10 0 5])
axis manual %// this line freezes the axes
for n = 1:10
x = 1:n;
y = sqrt(x);
set(h, 'XData', x, 'YData', y)
pause(.2)
end
Example with two plots:
h1 = plot(NaN, NaN, 'bo'); %// empty plot
hold on
h2 = plot(NaN, NaN, 'r*'); %// empty plot
axis([0 10 0 5])
axis manual %// this line freezes the axes
for n = 1:10
x1 = 1:n;
y1 = sqrt(x1);
set(h1, 'XData', x1, 'YData', y1)
x2 = 4.5;
y2 = n/2-.5;
set(h2, 'XData', x2, 'YData', y2)
pause(.2)
end