I have two equations:
ellipseOne = '((x-1)^2)/6^2 + y^2/3^2 = 1';
and
ellipseTwo = '((x+2)^2)/2^2 + ((y-5)^2)/4^2 = 1';
and I plotted them:
ezplot(ellipseOne, [-10, 10, -10, 10])
hold on
ezplot(ellipseTwo, [-10, 10, -10, 10])
title('Ellipses')
hold off
Now I'm trying to find the intersection of the two ellipses. I tried:
intersection = solve(ellipseOne, ellipseTwo)
intersection.x
intersection.y
to find the points where they intersect, but MATLAB is giving me a matrix and an equation as an answer which I don't understand. Could anyone point me in the right direction to get the coordinates of intersection?
The solution is in symbolic form. The last step you need to take is to transform it into numeric. Simply convert the result using double. However, because this is a pair of quadratic equations, there are 4 possible solutions due to the sign ambiguity (i.e. +/-) and can possibly give imaginary roots. Therefore, isolate out the real solutions and what is left should be your answer.
Therefore:
% Your code
ellipseOne = '((x-1)^2)/6^2 + y^2/3^2 = 1';
ellipseTwo = '((x+2)^2)/2^2 + ((y-5)^2)/4^2 = 1';
intersection = solve(ellipseOne, ellipseTwo);
% Find the points of intersection
X = double(intersection.x);
Y = double(intersection.y);
mask = ~any(imag(X), 2) | ~any(imag(Y), 2);
X = X(mask); Y = Y(mask);
The first three lines of code are what you did. The next four lines extract out the roots in numeric form, then I create a logical mask that isolates out only the points that are real. This looks at the imaginary portion of both the X and Y coordinate of all of the roots and if there is such a component for any of the roots, we want to eliminate those. We finally remove the roots that are imaginary and we should be left with two real roots.
We thus get for our intersection points:
>> disp([X Y])
-3.3574 2.0623
-0.2886 2.9300
The first column is the X coordinate and the second column is the Y coordinate. This also seems to agree with the plot. I'll take your ellipse plotting code and also insert the intersection points as blue dots using the above solution:
ezplot(ellipseOne, [-10, 10, -10, 10])
hold on
ezplot(ellipseTwo, [-10, 10, -10, 10])
% New - place intersection points
plot(X, Y, 'b.');
title('Ellipses');
hold off;
Related
I am generating a scatter plot containing data from multiple sources, as displayed below.
I would like to be able to generate a curve surrounding an arbitrary query point and passing through points on scatter plot. Final goal is to calculate the area between the lines on the plot.
I have implemented solution using finding points with knnsearch in a circular fashion and then applying hampel filter to eliminate noise. In the example below, I have selected a point right about in the middle of the blue-shaded area. As you can see, the result is far from perfect, and I need more precision.
I am looking for something similar to boundary function, but to work from the inside of the point cloud, not from the outside.
Final goal is to calculate the area between the lines on the plot.
I would do it differently. Just take any two lines of the plot, calculate the area under the curves with some kind of numerical approximation (for example trapezoidal numerical integration), then subtract the areas and obtain the area between the lines.
Thank to idea in Trilarion's answer, I was able to come up with the better solution.
Note that I use notation for YZ plane instead of XY (to keep consistent with robot coordinate system).
Solution
Generate curves for each set of scatter data
% Scatter data is in iy and iz vectors.
curve = fit(iy, iz, 'smoothingspline', 'SmoothingParam', 0.5);
% Remove outliers.
fdata = feval(curve, iy);
I = abs(fdata - iz) > 0.5 * std(iz);
outliers = excludedata(iy, iz, 'indices', I);
% Final curve without outliers.
curve = fit(iy, iz, 'smoothingspline', 'Exclude', outliers, 'SmoothingParam', 0.5);
Plot curves and scatter data
% Color maps generated by MATLAB's colormap function.
h_curve = plot(curve);
set(h_curve, 'Color', color_map_light(i,:));
scatter(iy, iz, '.', 'MarkerFaceColor', color_map(i,:))
Let user provide an input by selecting points
User selects one point as a query point and two points for limits along Y axis. This is because some curves come close, but never intersect.
[cs_position.y, cs_position.z] = ginput(1);
[cs_area_limits, ~] = ginput(2);
if cs_area_limits(1) > cs_area_limits(2)
cs_area_limits = flipud(cs_area_limits);
end
plot_cross_section(cs_position);
Finally calculate and plot surface area
This section uses fantastic answer by Doresoom.
function [ ] = plot_cross_section(query_point)
%PLOT_CROSS_SECTION Calculates and plots cross-section area.
% query_point Query point.
% Find values on query point's Y on each of the curves.
z_values = cellfun(#(x, y) feval(x, y),...
curves, num2cell(ones(size(curves)) * query_point.y))
% Find which curves are right above and below the query point.
id_top = find(z_values >= query_point.z, 1, 'first')
id_bottom = find(z_values < query_point.z, 1, 'last')
if isempty(id_top) || isempty(id_bottom)
return
end
% Generate points along curves on the range over Y.
y_range = cs_area_limits(1):0.1:cs_area_limits(2);
z_top = feval(curves{id_top}, y_range).';
z_bottom = feval(curves{id_bottom}, y_range).';
% Plot area.
Y = [ y_range, fliplr(y_range) ];
Z = [ z_top, fliplr(z_bottom) ];
fill(Y, Z, 'b', 'LineStyle', 'none')
alpha 0.5
hold on
% Calculate area and show to user.
cs_area = polyarea(Y, Z);
area_string = sprintf('%.2f mm^2', cs_area);
text(0, -3, area_string, 'HorizontalAlignment', 'center')
end
Result
I want to plot the area above and below a particular value in x axis.
The problem i am facing is with discrete values. The code below for instance has an explicit X=10 so i have written it in such a way that i can find the index and calculate the values above and below that particular value but if i want to find the area under the curve above and below 4 this program will now work.
Though in the plot matlab does a spline fitting(or some sort of fitting for connecting discrete values) there is a value for y corresponding to x=4 that matlab computes i cant seem to store or access it.
%Example for Area under the curve and partial area under the curve using Trapezoidal rule of integration
clc;
close all;
clear all;
x=[0,5,10,15,20];% domain
y=[0,25,50,25,0];% Values
LP=log2(y);
plot(x,y);
full = trapz(x,y);% plot of the total area
I=find(x==10);% in our case will be the distance value up to which we want
half = trapz(x(1:I),y(1:I));%Plot of the partial area
How can we find the area under the curve for a value of ie x = 2 or 3 or 4 or 6 or 7 or ...
This is an elaboration of patrik's comment, "first interpolate and then integrate".
For the purpose of this answer I'll assume that the area in question is the area that can be seen in the plot, and since plot connects points by straight lines I assume that linear interpolation is adequate. Moreover, since the trapezoidal rule itself is based on linear interpolation, we only need interpolated values at the beginning and end of the interval.
Starting from the given points
x = [0, 5, 10, 15, 20];
y = [0, 25, 50, 25, 0];
and the integration interval limits, say
xa = 4;
xb = 20;
we first select the data points within the limits
ind = (x > xa) & (x < xb);
xw = x(ind);
yw = y(ind);
and then complete them by interpolation values at the edges:
ya = interp1(x, y, xa);
yb = interp1(x, y, xb);
xw = [xa, xw, xb];
yw = [ya, yw, yb];
Now we can simply apply trapezoidal integration:
area = trapz(xw, yw);
I think that you either need more samples, or to interpolate the data. Another alternative is to use a function handle. Then you need to know the function though. Example using linear interpolation follows.
x0 = [0;5;10;15;20];
y0 = [0,25,50,25,0];
x1 = 0:20;
y1 = interp1(x0,y0,x1,'linear');
xMax = 4;
partInt = trapz(x1(x1<=xMax),y1(x1<=xMax));
Some other kind of interpolation may be suitable, but that is hard to say without more information. Also, this interpolates from the beginning to x. However, I guess figuring out how to change the limits should be easy from here. This solution is different than the former, since it is less depending on the pyramid shape of the data. So to say, it is more general.
I am writing a code to find the intersection point value of two independent lines , but I am confused that how to obtain the value of the intersection point , till now I have coded :
y = [2.63 8.12 13.01 21.87 35.19 58.49];
x = [200 400 500 600 800 1000];
plot(x,y)
hold on
plot([200, 1000], [10, 10]) % this [10, 10] is a straight line
hold off
I want to find the meeting point of plot(x,y) and Straight line , Can anyone give me hint for this Thanks :)
The x-coordinate at which the monotonically increasing piecewise linear curve plot(x,y) crosses v is given by:
interp1q(y,x,v);
Ok, this is the formula JakubT was assuming for:
yIntersect = 10;
dy = diff(y);
dx = diff(x);
i=find(diff(y > yIntersect));
xIntersect = x(i)+dx(i)*(yIntersect-y(i))/dy(i);
-->
xIntersect = 438.45
Of course this is not production code.
Not a very elegant solution, but you could, for each pair of consecutive elements of x and y (e.g., [8.12 13.01] and [400 500] being the second such pair), take the equation of the line passing through these two points, compute intersection with your crossing line (I assume that you have/can get the analytical formula for that one?) - and for each such pair of points, you check if the intersection actually happens between these two boundary points - if so, you have both the equation of the line passing through these two points as well as the equation of the crossing line, which yields the point of intersection.
i just started with my master thesis and i already am in trouble with my capability/understanding of matlab.
The thing is, i have a trajectory on a surface of a planet/moon whatever (a .mat with the time, and the coordinates. Then i have some .mat with time and the measurement at that time.
I am able to plot this as a color coded trajectory (using the measurement and the coordinates) in scatter(). This works awesomely nice.
However my problem is that i need something more sophisticated.
I now need to take the trajectory and instead of color-coding it, i am supposed to add the graph (value) of the measurement (which is given for each point) to the trajectory (which is not always a straight line). I will added a little sketch to explain what i want. The red arrow shows what i want to add to my plot and the green shows what i have.
You can always transform your data yourself: (using the same notation as #Shai)
x = 0:0.1:10;
y = x;
m = 10*sin(x);
So what you need is the vector normal to the curve at each datapoint:
dx = diff(x); % backward finite differences for 2:end points
dx = [dx(1) dx]; % forward finite difference for 1th point
dy = diff(y);
dy = [dy(1) dy];
curve_tang = [dx ; dy];
% rotate tangential vectors 90° counterclockwise
curve_norm = [-dy; dx];
% normalize the vectors:
nrm_cn = sqrt(sum(abs(curve_norm).^2,1));
curve_norm = curve_norm ./ repmat(sqrt(sum(abs(curve_norm).^2,1)),2,1);
Multiply that vector with the measurement (m), offset it with the datapoint coordinates and you're done:
mx = x + curve_norm(1,:).*m;
my = y + curve_norm(2,:).*m;
plot it with:
figure; hold on
axis equal;
scatter(x,y,[],m);
plot(mx,my)
which is imo exactly what you want. This example has just a straight line as coordinates, but this code can handle any curve just fine:
x=0:0.1:10;y=x.^2;m=sin(x);
t=0:pi/50:2*pi;x=5*cos(t);y=5*sin(t);m=sin(5*t);
If I understand your question correctly, what you need is to rotate your actual data around an origin point at a certain angle. This is pretty simple, as you only need to multiply the coordinates by a rotation matrix. You can then use hold on and plot to overlay your plot with the rotated points, as suggested in the comments.
Example
First, let's generate some data that resembles yours and create a scatter plot:
% # Generate some data
t = -20:0.1:20;
idx = (t ~= 0);
y = ones(size(t));
y(idx) = abs(sin(t(idx)) ./ t(idx)) .^ 0.25;
% # Create a scatter plot
x = 1:numel(y);
figure
scatter(x, x, 10, y, 'filled')
Now let's rotate the points (specified by the values of x and y) around (0, 0) at a 45° angle:
P = [x(:) * sqrt(2), y(:) * 100] * [1, 1; -1, 1] / sqrt(2);
and then plot them on top of the scatter plot:
hold on
axis square
plot(P(:, 1), P(:, 2))
Note the additional things have been done here for visualization purposes:
The final x-coordinates have been stretched (by sqrt(2)) to the appropriate length.
The final y-coordinates have been magnified (by 100) so that the rotated plot stands out.
The axes have been squared to avoid distortion.
This is what you should get:
It seems like you are interested in 3D plotting.
If I understand your question correctly, you have a 2D curve represented as [x(t), y(t)].
Additionally, you have some value m(t) for each point.
Thus we are looking at the plot of a 3D curve [x(t) y(t) m(t)].
you can easily achieve this using
plot3( x, y, m ); % assuming x,y, and m are sorted w.r.t t
alternatively, you can use the 3D version of scatter
scatter3( x, y, m );
pick your choice.
Nice plot BTW.
Good luck with your thesis.
Using MatLab, I know how to create a line segment connecting two points using this code:
line([0 1],[0 1])
This draws a straight line segment from the point (0,0) to the point (1,1).
What I am trying to do is continue that line to the edge of the plot. Rather than just drawing a line between these two points I want to draw a line through those two points that spans the entire figure for any set of two points.
For this particular line and a x=-10:10, y=-10:10 plot I could write:
line([-10 10], [-10 10]);
But I would need to generalize this for any set of points.
Solve the line equation going through those two points:
y = a*x + b;
for a and b:
a = (yp(2)-yp(1)) / (xp(2)-xp(1));
b = yp(1)-a*xp(1);
Find the edges of the plotting window
xlims = xlim(gca);
ylims = ylim(gca);
or take a edges far away, so you can still zoomout, later reset the x/y limits.
or if there is no plot at the moment, define your desired edges:
xlims = [-10 10];
ylims = [-10 10];
Fill in those edges into the line equation and plot the corresponding points:
y = xlims*a+b;
line( xlims, y );
And reset the edges
xlim(xlims);
ylim(ylims);
There is one special case, the vertical line, which you'll have to take care of separately.
What about
function = long_line(X,Y,sym_len)
dir = (Y-X)/norm(Y-X);
Yp = X + dir*sym_len;
Xp = X - dir*sym_len;
line(Xp,Yp);
end
being sym_len one half of the expected length of the plotted line around X?