So I have a 3 dimensional matrix of points that (presumably) define a surface. For my purposes, X and Y can be random values but when plotted along with their Z coordinates, they will define some undulating surface. I'd like to measure the local curvatures of said surface, and in order to do that, I need to be able to find the gradient of said surface, at which point calculating the curvature is trivial.
I have not yet found an implementation of how to measure this curvature that doesn't make use of Matlab's gradient function. The problem with Matlab's gradient function is that it assumes that the points are in some sort of order, similar to diff(X). This would suffice if my points were spaced along a grid, which is not necessarily the case.
One possible solution to measuring the gradient is to give in and assign each point to a discrete coordinate in a grid in the XY plane, thus overcoming this issue. However, this solution seems somewhat inelegant and was curious to see if anyone had suggestions. Thanks!
You can use griddata to interpolate from your scattered data points to grid spaced points and then calculate the gradient.
Related
I have a surface by the code below and another surface which is created by the exact same code. I want to see the height differences in another figure. How am I able to do that? Already operated with the Minus-operator but this won't work.
Furthermore the matrices have NOT the same size!
Appreciate your help!
x1 = Cx1;
y1 = Cy1;
z1 = Cz1;
tri1 = delaunay(x1,y1);
fig1 = figure%('units','normalized','outerposition',[0 0 1 1]);
trisurf(tri1,x2,y2,z2)
xlabel('x [mm] ','FontSize',30)
ylabel('y [mm] ','FontSize',30)
zlabel('z [mm] ','FontSize',30)
The simplest way to solve this problem is to interpolate from one mesh onto the other one. Such an approach works well when one is more highly resolved than the other, or when you're not as concerned with results at individual nodes, but rather the overall pattern across elements. If that's not the case, then you have a very complicated problem because you need to create a polygonal surface that fully captures all nodes and edges of both triangulations. Consider the following pair of triangular patterns:
A surface that captured all the variations would need to have all the vertices and edges that make up both of them, which is not a purely triangular surface. So, let us instead assume the easier case. To map results from one triangulation to the other, you simply need to formulate functions that define how the values vary along the triangles, which are more broadly called basis functions. It is often assumed that values betweeen the nodes (i.e. vertices) of the triangles vary linearly along the surfaces of the triangles. You can do it differently if you want, it just requires defining new basis functions. If we go for linear functions, then the equations in 2D are pretty simple. Let's say you make an array trimap that has which triangle each of the vertices of the other triangulation is inside of. This can be accomplished using the info here. Then, we set the coordinates of the vertices of the current triangle to (x1,y1), (x2,y2), and (x3,y3), and then do the math:
for cnt1=1,npoints
x1=x(tri1(trimap(cnt1),1));
x2=x(tri1(trimap(cnt1),2));
x3=x(tri1(trimap(cnt1),3));
y1=y(tri1(trimap(cnt1),1));
y2=y(tri1(trimap(cnt1),2));
y3=y(tri1(trimap(cnt1),3));
delta=x2*y3+x1*y2+x3*y1-x2*y1-x1*y3-x3*y2;
delta1=(x2*y3-x3*y2+xstat(cnt1)*(y2-y3)+ystat(cnt1)*(x3-x2));
delta2=(x3*y1-x1*y3+xstat(cnt1)*(y3-y1)+ystat(cnt1)*(x1-x3));
delta3=(x1*y2-x2*y1+xstat(cnt1)*(y1-y2)+ystat(cnt1)*(x2-x1));
weights(cnt1,1)=delta1/delta;
weights(cnt1,2)=delta2/delta;
weights(cnt1,3)=delta3/delta;
z1=z(tri1(trimap(cnt1),1));
z2=z(tri1(trimap(cnt1),2));
z3=z(tri1(trimap(cnt1),3));
valinterp(cnt1)=sum(weights(cnt1,:).*[z1,z2,z3]);
end
valinterp is the interpolated value for each point. Here and here are some nice slides explaining the mathematics behind all this. Note that I've not tested any of this code. Note also that you will need to do something to assign to values outside of the triangulation. Perhaps a null value, or an inverse-distance weighted value.
I would like to find residuals of my solutions. However, each solution uses a different grid, thus I would like to interpolate the solutions in 2D to a mesh grid for each solution such that I can easily find the residuals.
I used griddata which seems to work, but it doesn't work very well in regions of discontinuities and near edges. I would like to interpolate data of flow over a sphere for example. The area of the sphere should be confined to zero since that is outside of the grid boundaries; however, after using griddata, there's some non zero values in the sphere region. I also get a warning saying there were duplicate x and y points.
Does anyone know how to preserve the integrity of the sphere after using griddata? I've double checked the original x,y data, and the area of the sphere all have values of 0.
Thank you. I hope that made sense.
Best,
Yuki
I have a convex polygon in 3D. For simplicity, let it be a square with vertices, (0,0,0),(1,1,0),(1,1,1),(0,0,1).. I need to arrange these vertices in counter clockwise order. I found a solution here. It is suggested to determine the angle at the center of the polygon and sort them. I am not clear how is that going to work. Does anyone have a solution? I need a solution which is robust and even works when the vertices get very close.
A sample MATLAB code would be much appreciated!
This is actually quite a tedious problem so instead of actually doing it I am just going to explain how I would do it. First find the equation of the plane (you only need to use 3 points for this) and then find your rotation matrix. Then find your vectors in your new rotated space. After that is all said and done find which quadrant your point is in and if n > 1 in a particular quadrant then you must find the angle of each point (theta = arctan(y/x)). Then simply sort each quadrant by their angle (arguably you can just do separation by pi instead of quadrants (sort the points into when the y-component (post-rotation) is greater than zero).
Sorry I don't have time to actually test this but give it a go and feel free to post your code and I can help debug it if you like.
Luckily you have a convex polygon, so you can use the angle trick: find a point in the interior (e.g., find the midpoint of two non-adjacent points), and draw vectors to all the vertices. Choose one vector as a base, calculate the angles to the other vectors and order them. You can calculate the angles using the dot product: A · B = A B cos θ = |A||B| cos θ.
Below are the steps I followed.
The 3D planar polygon can be rotated to 2D plane using the known formulas. Use the one under the section Rotation matrix from axis and angle.
Then as indicated by #Glenn, an internal points needs to be calculated to find the angles. I take that internal point as the mean of the vertex locations.
Using the x-axis as the reference axis, the angle, on a 0 to 2pi scale, for each vertex can be calculated using atan2 function as explained here.
The non-negative angle measured counterclockwise from vector a to vector b, in the range [0,2pi], if a = [x1,y1] and b = [x2,y2], is given by:
angle = mod(atan2(y2-y1,x2-x1),2*pi);
Finally, sort the angles, [~,XI] = sort(angle);.
It's a long time since I used this, so I might be wrong, but I believe the command convhull does what you need - it returns the convex hull of a set of points (which, since you say your points are a convex set, should be the set of points themselves), arranged in counter-clockwise order.
Note that MathWorks recently delivered a new class DelaunayTri which is intended to superseded the functionality of convhull and other older computational geometry stuff. I believe it's more accurate, especially when the points get very close together. However I haven't tried it.
Hope that helps!
So here's another answer if you want to use convhull. Easily project your polygon into an axes plane by setting one coordinate zero. For example, in (0,0,0),(1,1,0),(1,1,1),(0,0,1) set y=0 to get (0,0),(1,0),(1,1),(0,1). Now your problem is 2D.
You might have to do some work to pick the right coordinate if your polygon's plane is orthogonal to some axis, if it is, pick that axis. The criterion is to make sure that your projected points don't end up on a line.
I have a 3D data set of a surface that is not a function graph. The data is just a bunch of points in 3D, and the only thing I could think of was to try scatter3 in Matlab. Surf will not work since the surface is not a function graph.
Using scatter3 gave a not so ideal result since there is no perspective/shading of any sort.
Any thoughts? It does not have to be Matlab, but that is my go-to source for plotting.
To get an idea of the type of surface I have, consider the four images:
The first is a 3D contour plot, the second is a slice in a plane {z = 1.8} of the contour. My goal is to pick up all the red areas. I have a method to do this for each slice {z = k}. This is the 3rd plot, and I like what I see here a lot.
Iterating this over z give will give a surface, which is the 4th plot, which is a bit noisy (though I have ideas to reduce the noise...). If I plot just the black surface using scatter3 without the contour all I get is a black indistinguishable blob, but for every slice I get a smooth curve, and I have noticed that the curves vary pretty smoothly when I adjust z.
Some fine-tuning will give a much better 4th plot, but still, even if I get the 4th plot to have no noise at all, the result using scatter3 will be a black incomprehensible blob when plotted alone and not on top of the 3D contour. I would like to get a nice picture of the full surface that is not plotted on top of the 3D contour plot
In fact, just to compare and show how bad scatter3 is for surfaces, even if you had exact points on a sphere and used scatter3 the result would be a black blob, and wouldn't even look like a sphere
Can POV-Ray handle this? I've never used it...
If you have a triangulation of your points, you could consider using the trisurf function. I have used that before to generate closed surfaces that have no boundary (such as polyhedra and spheres). The downside is that you have to generate a triangulation of your points. This may not be ideal to your needs but it definitely an option.
EDIT: As #High Performance Mark suggests, you could try using delaunay to generate a triangulation in Matlab
just wanted to follow up on this question. A quick nice way to do this in Matlab is the following:
Consider the function d(x, y, z) defined as the minimum distance from (x, y, z) to your data set. Make sure d(x, y, z) is defined on some grid that contains the data set you're trying to plot.
Then use isosurface to plot a (some) countour(s) of d(x, y, z). For me plotting the contour 0.1 of d(x, y ,z) was enough: Matlab will plot a nice looking surface of all points within a distance 0.1 of the data set with good lighting and all.
In povray, a blob object could be used to display a very dense collection of points, if you make them centers of spheres.
http://www.povray.org/documentation/view/3.6.1/71/
If you want to be able to make slices of "space" and have them colored as per your data, then maybe the object pattern (based on a #declared blob object) might do the trick.
Povray also has a way to work with df3 files, which I've never worked with, but this user appears to have done something similar to your visualization.
http://paulbourke.net/miscellaneous/df3/
I have a formula that depends on theta and phi (spherical coordinates 0<=theta<=2*pi and 0<=phi<=pi). By inserting each engle, I obtained a quantity. Now I have a set of data for different angles and I need to plot the surface. My data is a 180*360 matrix, so I am not sure if I can use SURF or MESH or PLOT3. The figure should be a surface that include all data and the axes should be in terms of the quantity, not the quantity versus the angles. How can I plot such a surface?
I see no reason why you cannot use mesh or surf to plot such data. Another option I tend to use is that of density plots. You basically display the dependent variable (quantity) as an image and include the independent variables (angles) along the axis, much like you would with the aforementioned 3D plotting functions. This can be done with imagesc.
Typically you would want your axes to be the dependent variables. Could you elaborate more on this point?
If I understand you correctly you have calculated a function f(theta,phi) and now you want to plot the surface containing all the points with the polar coordinated (r,theta,phi) where r=f(theta,phi).
If this is what you want to do, the 2D version of such a plot is included in MATLAB under the name polar. Unfortunately, as you pointed out, polar3 on MatlabCentral is not the generalization you are looking for.
I have been able to plot a sphere with the following code, using constant r=1. You can give it a try with your function:
phi1=0:1/(3*pi):pi; %# this would be your 180 points
theta1=-pi:1/(3*pi):pi; % your 360 points
r=ones(numel(theta1),numel(phi1));
[phi,theta]=meshgrid(phi1,theta1);
x=r.*sin(theta).*cos(phi);
y=r.*sin(theta).*sin(phi);
z=r.*cos(theta);
tri=delaunay(x(:),y(:),z(:));
trisurf(tri,x,y,z);
From my tests it seems that delaunay also includes a lot of triangles which go through the volume of my sphere, so it seems this is not optimal. So maybe you can have a look at fill3 and construct the triangles it draws itself: as a first approximation, you could have the points [x(n,m) x(n+1,m) x(n,m+1)] combined into one triangle, and [x(n+1,m) x(n+1,m+1) x(n+1,m+1)] into another...?