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I am trying to draw a cone, connected to the sphere in Matlab. I have the point [x1,y1,z1] outside of the sphere [x2,y2,z2] with R radius and I want it to be the top of the cone, created out of tangents.
On this pictures you can see what I have in mind:
Below you can see what I have already done. I am using it in order to mark the part of the Earth's surface, visible from the satellite position in orbit. Unfortunately, the cone in this picture is approximate, I need to create accurate one, connected with surface. For now, it is not only inaccurate, but also goes under it.
I am creating the sphere with this simple code (I am skipping the part of putting the map on it, it's just an image):
r = 6371.0087714;
[X,Y,Z] = sphere(50);
X2 = X * r;
Y2 = Y * r;
Z2 = Z * r;
surf(X2,Y2,Z2)
props.FaceColor= 'texture';
props.EdgeColor = 'none';
props.FaceLighting = 'phong';
figure();
globe = surface(X2,Y2,Z2,props);
Let's assume that I have the single point in 3D:
plot3(0,0,7000,'o');
How can I create such a cone?
There are two different questions here:
How to calculate cone dimensions?
How to plot lateral faces of a 3D cone?
Calculating Cone Dimensions
Assuming that center of sphere is located on [0 0 0]:
d = sqrt(Ax^2+Ay^2+Az^2);
l = sqrt(d^2-rs^2);
t = asin(rs/d);
h = l * cos(t);
rc = l * sin(t);
Plotting the Cone
The following function returns coordinates of lateral faces of cone with give apex point, axis direction, base radius and height, and the number of lateral faces.
function [X, Y, Z] = cone3(A, V, r, h, n)
% A: apex, [x y z]
% V: axis direction, [x y z]
% r: radius, scalar
% h: height, scalar
% n: number of lateral surfaces, integer
% X, Y, Z: coordinates of lateral points of the cone, all (n+1) by 2. You draw the sphere with surf(X,Y,Z) or mesh(X,Y,Z)
v1 = V./norm(V);
B = h*v1+A;
v23 = null(v1);
th = linspace(0, 2*pi, n+1);
P = r*(v23(:,1)*cos(th)+v23(:,2)*sin(th));
P = bsxfun(#plus, P', B);
zr = zeros(n+1, 1);
X = [A(1)+zr P(:, 1)];
Y = [A(2)+zr P(:, 2)];
Z = [A(3)+zr P(:, 3)];
end
The Results
rs = 6371.0087714; % globe radius
A = rs * 2 * [1 1 1]; % sattelite location
V = -A; % vector from sat to the globe center
% calculating cone dimensions
d = norm(A); % distance from cone apex to sphere center
l = (d^2-rs^2)^.5; % length of generating line of cone
sint = rs/d; % sine of half of apperture
cost = l/d; % cosine of half of apperture
h = l * cost; % cone height
rc = l * sint; % cone radius
% globe surface points
[XS,YS,ZS] = sphere(32);
% cone surface points
[XC, YC, ZC] = cone3(A, V, rc, h, 32);
% plotting results
hold on
surf(XS*rs,YS*rs,ZS*rs, 'facecolor', 'b', 'facealpha', 0.5, 'edgealpha', 0.5)
surf(XC, YC, ZC, 'facecolor', 'r', 'facealpha', 0.5, 'edgealpha', 0.5);
axis equal
grid on
Animating the satellite
The simplest way to animate objects is to clear the whole figure by clf and plot objects again in new positions. But a way more efficient method is to plot all objects once and in each frame, only update positioning data of moving objects:
clc; close all; clc
rs = 6371.0087714; % globe radius
r = rs * 1.2;
n = 121;
t = linspace(0, 2*pi, n)';
% point on orbit
Ai = [r.*cos(t) r.*sin(t) zeros(n, 1)];
[XS,YS,ZS] = sphere(32);
surf(XS*rs,YS*rs,ZS*rs, 'facecolor', 'b', 'facealpha', 0.5, 'edgealpha', 0.5)
hold on
[XC, YC, ZC] = cone3(Ai(1, :), Ai(1, :), 1, 1, 32);
% plot a cone and store handel of surf object
hS = surf(XC, YC, ZC, 'facecolor', 'r', 'facealpha', 0.5, 'edgealpha', 0.5);
for i=1:n
% calculating new point coordinates of cone
A = Ai(i, :);
V = -A;
d = norm(A);
l = (d^2-rs^2)^.5;
sint = rs/d;
cost = l/d;
h = l * cost;
rc = l * sint;
[XC, YC, ZC] = cone3(A, V, rc, h, 32);
% updating surf object
set(hS, 'xdata', XC, 'ydata', YC, 'zdata', ZC);
pause(0.01); % wait 0.01 seconds
drawnow(); % repaint figure
end
Another sample with 3 orbiting satellites:
I modified a code based on the shown in https://www.mathworks.com/matlabcentral/answers/377838-please-how-can-i-find-the-center-and-the-radius-of-the-inscribed-circle-of-a-set-of-points in order to find the inscribe circle but I do not understand why the image is rotated. Why and how can I solve it?
Code:
url='https://i.pcmag.com/imagery/reviews/00uaCVfzQ4Gsuhmh85WvT3x-4.fit_scale.size_1028x578.v_1569481686.jpg';
Image = rgb2gray(imread(url));
Image = imcomplement(Image);
fontSize = 10;
% determine contours
BW = imbinarize(Image);
BW = imfill(BW,'holes');
[B,L] = bwboundaries(BW,'noholes');
k = 1;
b = B{k};
y = b(:,2);
x = b(:,1);
subplot(2, 2, 1);
plot(x, y, 'b.-', 'MarkerSize', 3);
grid on;
title('Original Points', 'FontSize', fontSize);
% Enlarge figure to full screen.
set(gcf, 'Units', 'Normalized', 'OuterPosition', [0, 0.04, 1, 0.96]);
% Make data into a 1000x1000 image.
xMin = min(x)
xMax = max(x)
yMin = min(y)
yMax = max(y)
scalingFactor = 1000 / min([xMax-xMin, yMax-yMin])
x2 = (x - xMin) * scalingFactor + 1;
y2 = (y - yMin) * scalingFactor + 1;
mask = poly2mask(x2, y2, ceil(max(y2)), ceil(max(x2)));
% Display the image.
p2 = subplot(2, 2, 2);
imshow(mask);
axis(p2, 'on', 'xy');
title('Mask Image', 'FontSize', fontSize);
% Compute the Euclidean Distance Transform
edtImage = bwdist(~mask);
% Display the image.
p3 = subplot(2, 2, 3);
imshow(edtImage, []);
axis(p3, 'on', 'xy');
% Find the max
radius = max(edtImage(:))
% Find the center
[yCenter, xCenter] = find(edtImage == radius)
% Display circles over edt image.
viscircles(p3, [xCenter, yCenter], radius,'Color','g');
% Display polygon over image also.
hold on;
plot(x2, y2, 'r.-', 'MarkerSize', 3, 'LineWidth', 2);
title('Euclidean Distance Transform with Circle on it', 'FontSize', fontSize);
% Display the plot again.
subplot(2, 2, 4);
plot(x, y, 'b.-', 'MarkerSize', 3);
grid on;
% Show the circle on it.
hold on;
% Scale and shift the center back to the original coordinates.
xCenter = (xCenter - 1)/ scalingFactor + xMin
yCenter = (yCenter - 1)/ scalingFactor + yMin
radius = radius / scalingFactor
rectangle('Position',[xCenter-radius, yCenter-radius, 2*radius, 2*radius],'Curvature',[1,1]);
title('Original Points with Inscribed Circle', 'FontSize', fontSize);
Original image:
Output image
[B,L] = bwboundaries(BW,...) returns in B the row and column values (documentation). That is, the first column of B{k} is y, the second one is x.
After changing this bit of code as follows:
y = b(:,1);
x = b(:,2);
you will notice that the image is upside down! That is because in an image the y-axis increases down (y is the row number in the matrix), whereas in the plot the y-axis increases up (mathematical coordinate system).
The axes where you use imshow in are automatically set to the right coordinate system, but then you do axis(p3, 'on', 'xy');, turning it upside down again. Instead, use
axis(p1, 'on', 'image');
on the axes where you don't use imshow (i.e. the top-left and bottom-right ones).
I am plotting a phase map and using a circular colormap like hsv. The issue I am having is at the pi/-pi angle interfaces, the interpolation (shading interp command in MATLAB) gives me 0, which leads to strange lines in the resulting figure (see below, the below figure is the 4 tangent arctan function). Is there a way I can get rid of these artifacts? I like the smoothness of interpolated shading as opposed to flat shading, but flat shading avoids these artifacts.
Here is the code that generates the above image:
[x_grid,y_grid] = meshgrid(-31:32,-31:32);
phase = atan2(y_grid,x_grid);
surf(x_grid,y_grid,phase);
view(0,90);
shading interp
colorbar
axis([-31 32 -31 32])
colormap hsv
The problem is that using an N-by-N set of grid points is always going to have a discontinuity at the pi/-pi interface which it will try to interpolate across. You can instead create a 2-by-N strip of coordinates that wrap around the origin and are disconnected at the pi/-pi interface. The following illustrates how the discontinuity looks for the 2 approaches:
% N-by-N grid:
[x_grid, y_grid] = meshgrid(-31:32, -31:32);
phase = atan2(y_grid, x_grid);
subplot(1, 2, 1);
surf(x_grid, y_grid, phase);
title('N-by-N grid');
% 2-by-N strip:
X = [-31.*ones(1, 33) -30:31 32.*ones(1, 64) 31:-1:-30 -31.*ones(1, 32); ...
zeros(1, 253)];
Y = [0:32 32.*ones(1, 62) 32:-1:-31 -31.*ones(1, 62) -31:0; ...
zeros(1, 253)];
phase = atan2(Y([1 1], :), X([1 1], :));
phase(:, end) = -pi;
subplot(1, 2, 2);
surf(X, Y, phase);
title('2-by-N strip');
And here's how the final 2-D view would look (with a higher-resolution colormap):
X = [-31.*ones(1, 33) -30:31 32.*ones(1, 64) 31:-1:-30 -31.*ones(1, 32); ...
zeros(1, 253)];
Y = [0:32 32.*ones(1, 62) 32:-1:-31 -31.*ones(1, 62) -31:0; ...
zeros(1, 253)];
phase = atan2(Y([1 1], :), X([1 1], :));
phase(:, end) = -pi;
surf(X, Y, phase);
view(0, 90);
shading interp;
colorbar;
axis([-31 32 -31 32]);
colormap(hsv(256));
I am trying to program scatterplot with specific errorbars. The only build in function i found is
errorbar()
but this only enables me to make a 2d plot with errorbars in y direction. What i am asking for is a method to plot this with errorbars in x and y direction.
At the end my goal is to make a 3D-scatter-plot with 3 errorbars.
Perfect would be if the resulting image would be a 3d-plot with 3d geometric shapes (coordinate x,y,z with expansion in the dimension proportional to the errorbars) as 'marker'.
I found this page while searching the internet: http://code.izzid.com/2007/08/19/How-to-make-a-3D-plot-with-errorbars-in-matlab.html
But unfortunately they use only one errorbar.
My data is set of 6 arrays each containing either the x,y or z coordinate or the specific standard derivation i want to show as errorbar.
The code you posted looks very easy to adapt to draw all three error bars. Try this (note that I adapted it also so that you can change the shape and colour etc of the plots as you normally would by using varargin, e.g. you can call plot3d_errorbars(...., '.r'):
function [h]=plot3d_errorbars(x, y, z, ex, ey, ez, varargin)
% create the standard 3d scatterplot
hold off;
h=plot3(x, y, z, varargin{:});
% looks better with large points
set(h, 'MarkerSize', 25);
hold on
% now draw the vertical errorbar for each point
for i=1:length(x)
xV = [x(i); x(i)];
yV = [y(i); y(i)];
zV = [z(i); z(i)];
xMin = x(i) + ex(i);
xMax = x(i) - ex(i);
yMin = y(i) + ey(i);
yMax = y(i) - ey(i);
zMin = z(i) + ez(i);
zMax = z(i) - ez(i);
xB = [xMin, xMax];
yB = [yMin, yMax];
zB = [zMin, zMax];
% draw error bars
h=plot3(xV, yV, zB, '-k');
set(h, 'LineWidth', 2);
h=plot3(xB, yV, zV, '-k');
set(h, 'LineWidth', 2);
h=plot3(xV, yB, zV, '-k');
set(h, 'LineWidth', 2);
end
Example of use:
x = [1, 2];
y = [1, 2];
z = [1, 2];
ex = [0.1, 0.1];
ey = [0.1, 0.5];
ez = [0.1, 0.3];
plot3d_errorbars(x, y, z, ex, ey, ez, 'or')
How could I create a 3D binary matrix/image from a surface mesh in Matlab?
For instance, when I create ellipsoid using:
[x, y, z] = ellipsoid(0,0,0,5.9,3.25,3.25,30);
X, Y and X are all 2D matrix with size 31 x 31.
Edited based on suggestion of #Magla:
function Create_Mask_Basedon_Ellapsoid3()
close all
SurroundingVol = [50, 50, 20];
%DATA
[MatX,MatY,MatZ] = meshgrid(-24:1:25, -24:1:25, -9:1:10);
[mask1, x, y, z] = DrawEllipsoid([0, -10, 0], [6, 3, 3], MatX,MatY,MatZ);
[mask2, x2, y2, z2] = DrawEllipsoid([15, 14, 6], [6, 3, 3], MatX,MatY,MatZ);
mask = mask1 + mask2;
%Surface PLOT
figure('Color', 'w');
subplot(1,2,1);
%help: Ideally I would like to generate surf plot directly from combined mask= mask1 + mask2;
s = surf(x,y,z); hold on;
s2 = surf(x2,y2,z2); hold off;
title('SURFACE', 'FontSize', 16);
view(-78,22)
subplot(1,2,2);
xslice = median(MatX(:));
yslice = median(MatY(:));
zslice = median(MatZ(:));
%help: Also how do I decide correct "slice" and angles to 3D visualization.
h = slice(MatX, MatY, MatZ, double(mask), xslice, yslice, zslice)
title('BINARY MASK - SLICE VOLUME', 'FontSize', 16);
set(h, 'EdgeColor','none');
view(-78, 22)
%az = 0; el = 90;
%view(az, el);
end
function [mask, Ellipsoid_x, Ellipsoid_y, Ellipsoid_z] = DrawEllipsoid(CenterEllipsoid, SizeEllipsoid, MatX, MatY, MatZ)
[Ellipsoid_x, Ellipsoid_y, Ellipsoid_z] = ellipsoid(CenterEllipsoid(1), CenterEllipsoid(2), CenterEllipsoid(3), SizeEllipsoid(1)/2 , SizeEllipsoid(2)/2 , SizeEllipsoid(3)/2 ,30);
v = [Ellipsoid_x(:), Ellipsoid_y(:), Ellipsoid_z(:)]; %3D points
%v = [x(:), y(:), z(:)]; %3D points
tri = DelaunayTri(v); %triangulation
SI = pointLocation(tri,MatX(:),MatY(:),MatZ(:)); %index of simplex (returns NaN for all points outside the convex hull)
mask = ~isnan(SI); %binary
mask = reshape(mask,size(MatX)); %reshape the mask
end
There you go:
%// Points you want to test. Define as you need. This example uses a grid of 1e6
%// points on a cube of sides [-10,10]:
[x y z] = meshgrid(linspace(-10,10,100));
x = x(:);
y = y(:);
z = z(:); %// linearize
%// Ellipsoid data
center = [0 0 0]; %// center
semiaxes = [5 4 3]; %// semiaxes
%// Actual computation:
inner = (x-center(1)).^2/semiaxes(1).^2 ...
+ (y-center(2)).^2/semiaxes(2).^2 ...
+ (z-center(3)).^2/semiaxes(3).^2 <= 1;
For the n-th point of the grid, whose coordinates are x(n), y(n), z(n), inner(n) is 1 if the point lies in the interior of the ellipsoid and 0 otherwise.
For example: draw the interior points:
plot3(x(inner), y(inner), z(inner), '.' , 'markersize', .5)
Here is a method for creating a binary mask from an ellipsoid. It creates a corresponding volume and sets to NaN the points outside the ellipsoid (ones inside).
It doesn't take into consideration the formula of the ellipsoid, but uses a convex hull. Actually, it works for any volume that can be correctly described by a 3D convex hull. Here, the convexhulln step is bypassed since the ellipsoid is already a convex hull.
All credits go to Converting convex hull to binary mask
The following plot
is produced by
%DATA
[x, y, z] = ellipsoid(0,0,0,5.9,3.25,3.25,30);
%METHOD
v = [x(:), y(:), z(:)]; %3D points
[X,Y,Z] = meshgrid(min(v(:)):0.1:max(v(:))); %volume mesh
tri = DelaunayTri(v); %triangulation
SI = pointLocation(tri,X(:),Y(:),Z(:)); %index of simplex (returns NaN for all points outside the convex hull)
mask = ~isnan(SI); %binary
mask = reshape(mask,size(X)); %reshape the mask
%PLOT
figure('Color', 'w');
subplot(1,2,1);
s = surf(x,y,z);
title('SURFACE', 'FontSize', 16);
view(-78,22)
subplot(1,2,2);
xslice = median(X(:));
yslice = median(Y(:));
zslice = median(Z(:));
h = slice(X, Y, Z, double(mask), xslice, yslice, zslice)
title('BINARY MASK - SLICE VOLUME', 'FontSize', 16);
set(h, 'EdgeColor','none');
view(-78,22)
Several ellipsoids
If you have more than one ellipsoid, one may use this masking method for each of them, and then combine the resulting masks with &.
Choice of slices and angle
"Correct" is a matter of personal choice. You can either
create the unrotated mask and rotate it after (Rotate a 3D array in matlab).
create a mask on already rotated ellipsoid.
create a mask on a slightly rotated ellipsoid (that gives you the choice of a "correct" slicing), and rotate it further to its final position.