I solved the temperature profile of a sphere being cooled in a gas stream with pdepe, but the assignment was to calculate at which time the average temperature of the sphere is 16C. Now I used the function mean, but this isn't correct for a sphere. How would one calculate the mean temperature of a sphere?
PDE toolbox functions area() and volume() might help with your case. For a 2D mesh, area() gives the area of each mesh element; one then needs to find the centroid coordinates for the element and interpolate the solution at these coordinates. Similarly, for a 3D mesh and the element volumes. The answer to this question on Mathworks' site also showcases some code that can be used for axisymmetric geometries.
Mean in spherical geometry is usually weighted by radius, the final formula for <f> is as follows: f_mean = 3*trapz(r, r.^2.*f)
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TL;DR - How do I only analyse the surface data points for surface estimation?
I have a 3D object, and I would like to estimate the surface shape.
The problem is: MATLAB curve fitting toolbox takes into account all the data points of the object. See the example of a cylinder and it's approximated polynomial. MATLAB is taking into account all the data points for surface estimation, what can I do to over come this?
Assuming that the outer and inner surface have identical shape, you can first do an imfill to make the object solid and then use morphological skeleton, bwmorph(BW,'skel',Inf) to turn it into a single line, which you then can approximate the shape from.
Basically, I have a many irregular circle on the ground in the form of x,y,z coordinates (of 200*3 matrix). but I want to fix a best circle in to the data of x,y,z coordinates (of 200*3 matrix).
Any help will be greatly appreciated.
I would try using the RANSAC algorithm which finds the parameters of your model (in your case a circle) given noisy data. The algorithm is quite easy to understand and robust against outliers.
The wikipedia article has a Matlab example for fitting a line but it shouldn't be too hard to adapt it to fit a circle.
These slides give a good introduction to the RANSAC algorithm (starting from page 42). They even show examples for fitting a circle.
Though this answer is late, I hope this helps others
To fit a circle to 3d points
Find the centroid of the 3d points (nx3 matrix)
Subtract the centroid from the 3D points.
Using RANSAC, fit a plane to the 3D points. You can refer here for the function to fit plane using RANSAC
Apply SVD to the 3d points (nx3 matrix) and get the v matrix
Generate the axes along the RANSAC plane using the axes from SVD. For example, if the plane norm is along the z-direction, then cross product between the 1st column of v matrix and the plane norm will generate the vector along the y-direction, then the cross product between the generated y-vector and plane norm will generate a vector along the x-direction. Using the generated vectors, form a Rotation matrix [x_vector y_vector z_vector]
Multiply the Rotation matrix with the centroid subtracted 3d points so that the points will be parallel to the XY plane.
Project the points to XY plane by simply removing the Z-axes from the 3d points
fit a circle using Least squares circle fit
Rotate the center of the circle using the inverse of the rotation matrix obtained from step 5
Translate back the center to the original location using the centroid
The circle in 3D will have the center, the radius will be the same as the 2D circle we obtained from step 8, the circle plane will be the RANSAC plane we obtained from step 3
Given a matrix nx3 that represents n points in 3D space. All points lie on a plane. The plane is given by its normal and a point lying on it. Is there a Matlab function or any Matlabby way to find the area directly from the matrix?
What i was trying to do is write a function that first computes the centroid,c, of the n-gon. Then form triangles : (1,2,c),(2,3,c),...,(n,1,c). Compute their area and sum up. But then i had to organise the polygon points in a cyclic order as they were unordered which i figured was hard. Is there a easy way to do so?
Is there a easier way in Matlab to just call some function on the matrix?
Here is perhaps an easier method.
First suppose that your plane is not parallel to the z-axis.
Then project the polygon down to the xy-plane simply by removing the 3rd coordinate.
Now compute the area A' in the xy-plane by the usual techniques.
If your plane makes an angle θ with the xy-plane, then your 3D
area A = A' / cos θ.
If your plane is parallel to the z-axis, do the same computation
w.r.t. the y-axis instead, projecting to the xz-plane.
To project from 3D to the plane normal to N, take some non-parallel vector A and compute the cross products U = N x A and V = N x U. After normalizing U and V, the dot products P.U and P.V give you 2D coordinates in the plane.
Joseph's solution is even easier (I'd recommend to drop the coordinate with the smallest absolute cosine).
You said the points all lie on a plane and you have the normal. You should then be able to reproject the 3-D points into 2-D coordinates in a new 2-D basis. I am not aware of a canned function in Matlab to do this , but coding it should not be difficult, this answer from Math.SE and this Matlab Central post should help you.
If you already solved the problem of finding the coordinates of the points in the 2-D plane they are in, you could use the Matlab boundary or convex hull function to compute the area of the boundary or convex hull enclosing the points.
[k,v]= boundary(x,y)
or
[k,v] =convhull(x,y)
where k is the vector of indices into points x,y, that define the boundary or convex hull, v is the area enclosed, and x, y are vectors of the x and y coordinates of your points.
What you were describing with trying to find triangles with the points sounds like a first attempt toward Delaunay triangulation. I think more recent versions of Matlab have functions to do Delaunay triangulation as well.
I have N 3D observations taken from an optical motion capture system in XYZ form.
The motion that was captured was just a simple circle arc, derived from a rigid body with fixed axis of rotation.
I used the princomp function in matlab to get all marker points on the same plane i.e. the plane on which the motion has been done.
(See a pic representing 3D data on the plane that was found, below)
What i want to do after the previous step is to look the fitted data on the plane that was found and get the curve of the captured motion in 2D.
In the princomp how to, it is said that
The first two coordinates of the principal component scores give the
projection of each point onto the plane, in the coordinate system of
the plane.
(from "Fitting an Orthogonal Regression Using Principal Components Analysis" article on mathworks help site)
So i thought that if i just plot those pc scores -plot(score(:,1),score(:,2))- i'll get the motion curve. Instead what i got is this.
(See a pic representing curve data in 2D derived from pc scores, below)
The 2d curve seems stretched and nonlinear (different y values for same x values) when it shouldn't be. The curve that i am looking for, should be interpolated by just using simple polynomial (polyfit) or circle fit in matlab.
Is this happening because the plane that was found looks like rhombus relative to the original coordinate system and the pc axes are rotated with respect to the basis of plane in such way that produce this stretch?
Then i thought that, this is happening because of the different coordinate systems of optical system and Matlab. Optical system's (ie cameras) co.sys. is XZY oriented and Matlab's default (i think) co.sys is XYZ oriented. I transformed my data to correspond to Matlab's co.sys through a rotation matrix, run again princomp but i got the same stretch in the 2D curve (the new curve just had different orientation now).
Somewhere else i read that
Principal Components Analysis chooses the first PCA axis as that line
that goes through the centroid, but also minimizes the square of the
distance of each point to that line. Thus, in some sense, the line is
as close to all of the data as possible. Equivalently, the line goes
through the maximum variation in the data. The second PCA axis also
must go through the centroid, and also goes through the maximum
variation in the data, but with a certain constraint: It must be
completely uncorrelated (i.e. at right angles, or "orthogonal") to PCA
axis 1.
I know that i am missing something but i have a problem understanding why i get a stretched curve. What i have to do so i can get the curve right?
Thanks in advance.
EDIT: Here is a sample data file (3 columns XYZ coords for 2 markers)
w w w.sendspace.com/file/2hiezc
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.