I am currently working on a hole detection problem in 3D point cloud data. I am referring to this paper "Detecting Holes in Point Set Surfaces" by Gerhard H Bendels, Ruwen Schnabel and Reinhard Klein. One of the criterions mentioned is an Angle Criterion in which we need to determine angles between consecutive points in a KD Tree(Radially Nearest Neighbors to a given point).
See Image:
Angle between points
I am using Open3D to extract a KD Tree but I believe it is giving me an unsorted list of points rather than a list of consecutive points.
See Image:
List of Nearest neighbors
The point below '______' is the point of interest and rest are it's neighbors. Now my question is,
How do I know which point is next to which point?
And if that's not possible to know, How can I find the angles as shown in the first image.
I just need the angles to find the boundary probability for each point, so an answer would really help me progress.
Thanks
What I've Tried so far..
I have tried generating vectors out of all the points and calculated the angles using dot product. But that seems wrong because I believe I may be calculating dot products between first and third point.
I am dealing with a reverse-engineering problem regarding road geometry and estimation of design conditions.
Suppose you have a set of points obtained from the measurement of positions of a road. This road has straight sections as well as curve sections. Straight sections are, of course, represented by lines, and curves are represented by circles of unknown center and radius. There are, as well, transition sections, which may be clothoids / Euler spirals or any other usual track transition curve. A representation of the track may look like this:
We know in advance that the road / track was designed taking this transition + circle + transition principle into account for every curve, yet we only have the measurement points, and the goal is to find the parameters describing every curve on the track, this is, the transition parameters as well as the circle's center and radius.
I have written some code using a nonlinear optimization algorithm, where a user can select start and end points and fit a circle that to the arc section between them, as it shows in next figure:
However, I don't find a suitable way to take the transition into account. After giving it some thought I came to think that this s because, given a set of discrete points -with their measurement error- representing a full curve, it is not entirely clear where to consider it "begins" and where it "ends" and, moreover, it is less clear where to consider the transition, the proper circle and the exit transition "begin" and "end".
Is there any work on this subject which I may have missed? is there a proper way to fit the whole transition + curve + transition structure into the set of points?
As far as I know, there's no method to fit a sequence clothoid1-circle-clothoid2 into a given set of points.
Basic facts are that two points define a straight, and three points define a unique circle.
The clothoid is far more complex, because you need: The parameter A, the final radius Rf, an initial point px,py, the radius Ri at that point, and the tangent T (angle with X-axis) at that point.
These are 5 data you may use to find the solution.
Due to clothoid coords are calculated by expanded Fresnel integrals (see https://math.stackexchange.com/a/3359006/688039 a little explanation), and then apply a translation & rotation, there's no an easy way to fit this spiral into a set of given points.
When I've had to deal with your issue, what I've done is:
Calculate the radius for triplets of consecutive points: p1p2p3, p2p3p4, p3p4p5, etc
Observe the sequence of radius. Similar values mean a circle, increasing/decreasing values mean a clothoid; Big values would mean a straight.
For each basic element (line, circle) find the most probably characteristics (angles, vertices, radius) by hand or by some regression method. Many times the common sense is the best.
For a spiral you may start with aproximated values, taken from the adjacent elements. These values may very well be the initial angle and point, and the initial and final radius. Then you need to iterate, playing with Fresnel and 'space change' until you find a "good" parameter A. Then repeat with small differences in the other values, those you took from adjacents.
Make the changes you consider as good. For example, many values (A, radius) use to be integers, without decimals, just because it was easier for the designer to type.
If you can make a small applet to do these steps then it's enough. Using a typical roads software helps, but doesn't avoid you the iteration process.
If the points are dense compared to the effective radii of curvature, estimate the local curvature by least square fitting of a circle on a small number of points, taking into account that the curvature is most of the time zero.
You will obtain a plot with constant values and ramps that connect them. You can use an estimate of the slope at the inflection points to figure out the transition points.
I need to find points (from a rather small dataset) which are close enough to a polyline. All coordinates are WGS84.
I think of some r-tree thing to reduce the data to just a few candidates which then have to be checked in more detail.
While i managed to do this using "great circle" arithmetic, i am sure this is too pedantic for the following reasons:
The segmentation of those polylines is quite high. A single segment of a polyline can be considered to be no longer than 10 km.
The points in question are not more than a few hundred meters away from segments.
The area in question is Europe, so the algorithm does not need to be valid for extreme (near pole?) conditions. Again: points don't need to be checked agains the whole polyline (which could be hundrets of kilometers). Only the "nearby" segments need to be considured.
Do i need to transform the WGS84 coordinates to
some local cartesian reference system
to a mercator system
Or can i even just calculate with "angle differences"? I know that this is just a matter of accuracy: I can accept an error which is below ~50 meters.
I highly appreciate your suggestions!
On how to measure distance from point to polyline:
you have to measure distances from all your points to all segments of a polyline.
See Distance from a point to a polygon
You can do without converting coordinates to cartesian (especially if the area is rather small, you don't mind 50 meters error and you don't need exact distances, just relative) See https://en.wikipedia.org/wiki/Decimal_degrees.
I would like to find two vertices of two meshes (1 vertex per mesh) that define the closest distance between them. Or the two triangles would be fine I guess.
However I'm not sure how to search for this in CGAL's documentation, I'm sure that this is doable with some existing tool (probably based on a 3d distance field and/or AABBs). Could I please get a hint (keywords/link) on what to look for?
I've been pointed to the Optimal Distances CGAL package, but it's not exactly what I want, since it outputs the distance and the coordinates, so finding the vertex ID is an additional computational overhead.
I've already implemented a collision detection with CGAL to find the triangle-triangle intersection in a triangle-soup, using AABB-trees. I guess that I should be somehow close to this, although now a simple soup with all me object-triangles wouldn't do the job.
The solution found was this:
CGAL's Optimal Distances package can give an approximation of the closest distance between the convex hulls of two meshes, without explicitly computing the hulls. As a result one gets the shortest distance between these hulls, and the coordinates of the 2 points that lie on them and define this distance.
Then these coordinates can be used as a search-query in kd-trees that contains the original vertices of the meshes in order to find the closest vertices.
In case one mesh is non-convex, the hull that CGAL is using is very approximate, so convex decomposition might be necessary. In such a case one would have to check distances for each convex part and then take the shortest distance.
The above would result in something like this:
enter link description here
I am not good with math,and i just need someone to point me in the right direction.
Latitude Longitude
N 36° 13.488' W 095° 54.295'
N 36° 13.488' W 095° 53.805'
Assume that all three are located on a flat plane, at the same elevation.
Assume that the curvature of the earth is not a factor.
Assume that there are exactly 69.1691 miles per degree of latitude.
Assume that there are 55.9588 miles per degree of longitude (Tulsa area only)..
I am trying to figure out what the last points coordinate is.
Can anyone help. I just dont know where to begin
There are numerous ways to find the third point. The angles in an equilateral triangle are all 60 degrees. The third point lies on the bisector of the line connecting the two points you have. Expressing the points and lines using vectors, rather than coordinates, helps.
But the truth is, if you do not have enough mathematical knowledge for this kind of problem, you probably ought not to be tackling it as a programming problem. How can you know you've done it right?