I am doing some interpolation given 5000 points (t,x,y,z) where t is time. Some example points I have would be:
Sample data
I am using discrete velocity, change(Position)/change(time) for each consecutive point. I want to find a good way to interpolate the velocity at a given time. I know that I can use linear interpolation, but is there a better way to interpolate? I am using Python for this home project.
Related
As the title states, I need to make a plane above a 3D dataset in Python. There should be no data points above such plane, and the distances between the plane and the dataset should be optimized such that the plane somehow generalizes the whole data.
this is the 3d surface plot
(this is another example) the plane should look like this
I've been stuck for months on how to start/approach this problem. Should I start looking for the maximums in the data? Should I start on finding the peak in the middle and tamper around the possible slopes of the plane? Or are there other appropriate mathematical methods for this?
Thanks ahead.
I'm using numpy to generate a spline based on b-spline parameters. These parameters have been read from a STEP file exported from CAD with the end goal being to write a 2D unstructured mesh generator. I need to generate evenly spaced points on the spline to act as vertexes for the mesh elements.
Annoyingly, however, numpy is generating highly unevenly distributed points along the spline and I'm not sure why.
Here is an example of an aerofoil b-spline - the control points are relatively evenly placed around the spline and there is sufficient resolution in the spline coordinates.
But when I evaluate the same spline with 10x less points, you can see the distribution is heavily weighted to the upper line.
This is the code I'm using to generate the points (the bspline parameters have already been defined according to the STEP file):
N=int(ctrl_pts.shape[0]*100) # number of points to evaluate
spline_range=np.linspace(
knot_vector[0],
knot_vector[-1],
N
)
bspline=numpy.interpolate.BSpline(knot_vector,ctrl_pts,degree)
points=bspline(spline_range)
Is there a way to 'redistribute' the points on the spline?
I have data points in x,y,z format. They form a point cloud of a closed manifold. How can I interpolate them using R-Project or Python? (Like polynomial splines)
It depends on what the points originally represented. Just having an array of points is generally not enough to derive the original manifold from. You need to know which points go together.
The most common low-level boundary representation ("brep") is a bunch of triangles. This is e.g. what OpenGL and Directx get as input. I've written a Python software that can convert triangular meshes in STL format to e.g. a PDF image. Maybe you can adapt that to for your purpose. Interpolating a triangle is usually not necessary, but rather trivail to do. Create three new points each halfway between two original point. These three points form an inner triangle, and the rest of the surface forms three triangles. So with this you have transformed one triangle into four triangles.
If the points are control points for spline surface patches (like NURBS, or Bézier surfaces), you have to know which points together form a patch. Since these are parametric surfaces, once you know the control points, all the points on the surface can be determined. Below is the function for a Bézier surface. The parameters u and v are the the parametric coordinates of the surface. They run from 0 to 1 along two adjecent edges of the patch. The control points are k_ij.
The B functions are weight functions for each control point;
Suppose you want to approximate a Bézier surface by a grid of 10x10 points. To do that you have to evaluate the function p for u and v running from 0 to 1 in 10 steps (generating the steps is easily done with numpy.linspace).
For each (u,v) pair, p returns a 3D point.
If you want to visualise these points, you could use mplot3d from matplotlib.
By "compact manifold" do you mean a lower dimensional function like a trajectory or a surface that is embedded in 3d? You have several alternatives for the surface-problem in R depending on how "parametric" or "non-parametric" you want to be. Regression splines of various sorts could be applied within the framework of estimating mean f(x,y) and if these values were "tightly" spaced you may get a relatively accurate and simple summary estimate. There are several non-parametric methods such as found in packages 'locfit', 'akima' and 'mgcv'. (I'm not really sure how I would go about statistically estimating a 1-d manifold in 3-space.)
Edit: But if I did want to see a 3D distribution and get an idea of whether is was a parametric curve or trajectory, I would reach for package:rgl and just plot it in a rotatable 3D frame.
If you are instead trying to form the convex hull (for which the word interpolate is probably the wrong choice), then I know there are 2-d solutions and suspect that searching would find 3-d solutions as well. Constructing the right search strategy will depend on specifics whose absence the 2 comments so far reflects. I'm speculating that attempting to model lower and higher order statistics like the 1st and 99th percentile as a function of (x,y) could be attempted if you wanted to use a regression effort to create boundaries. There is a quantile regression package, 'rq' by Roger Koenker that is well supported.
I'm performing a simulation of a simple queue using SimPy. One of the questions about the system is what is the distribution of the waiting times by a visitor. What I do is draw a normalized histogram of the sample I get during the simulation process.
This distribution is not purely continuous, we have a non-zero probability of the waiting time being exactly zero, hence the peak near the left end. I want it to be somehow obvious from the picture, what is the actual probability of hitting 0 exactly. Right now the height of the peak does not visualize that properly, the height is even higher than one (the reason is that many points are hitting a small segment near zero).
So the question is the general visualization technique of such distributions that are mixtures of a continuous and a discrete one.
(based on the discussion in the comments to OP).
For a distribution of some variable, call it t, being a mixture of a discrete and and continuous components, I'd write the pdf a sum of a set of delta-peaks and a continuous part,
p(t) = \sum_{a} p_a \delta(t-t_a) + f(t)
where a enumerates the discrete values t_a and p_a are probabilities of t_a, and f(t) is the pdf for the continuous part of the distribution, so that f(t)dt is the probability for t to belong to [t,t+dt).
Notice that the whole thing is normalized, \int p(t) =1 where the integral is over the approprite range of t.
Now, for visualizing this, I'd separate the discrete components, and plot them as discrete values (either as narrow bins or as points with droplines etc). Then for the rest, I'd use the histogram where you know the correct normalization from the equation above: the area under the histogram should sum up to 1-\sum_a p_a.
I'm not claiming this being the way, it's just what I'd do.
I am trying to make a python script that will output a force based on a measured angle. The inputs are time, the curve and the angle, but I am having trouble using interpolation to fit the force to the curve. I looked at scipy.interpolate, but I'm not sure it will help me because the points aren't evenly spaced.
numpy.interp does not require your points to be evenly distributed. I'm not certain if you mean by "The inputs are time, the curve and the angle" that you have three independent variables, if so you will have to adapt it quite a bit... But for one-variable problems, interp is the way to go.