Given a set of lines I want to filter lines based on their distances to different points. Since there can be potentially many lines, I don't want to calculate the distance between a given point and each line. I'd prefer to somehow filter out those lines that are far away anyway. I thought about using a quadtree which is good to prefilter nearby points to a given point. However it seems inconvinient to build a quadtree containing lines since every leaf in the tree has to contain all lines that cut the leaf's corresponding quad in the plane. Have some of you had a similar problem? Do you know common solutions for similar problems?
Thanks in advance!
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I am not sure if this algorithm exists, much appreciated if someone can provide me the just the Algorithm's name, then I can Google it up.
Basically let's say I have N Points within a polygon (both convex and concave), and I would like to have a way/algorithm to split this polygon into N polygons, that each of these N polygon contains 1 point only.
Thanks.
I'm reluctant to post this as an answer, but it won't fit in the comments.
In the GIS world, this is sometimes referred to as voronoi algorithm. Most GIS tools, like ESRI ArcMap can generate veronoi polgons from a set of points. For your use case I think you can create a veronoi polygon set from your points using the package in the link below (it it's compatible), then take that output, and do some fancy spatial joining to replace your polygon with multiple polygons.
Here is a link to the wikipedia page describing the concept
http://en.wikipedia.org/wiki/Voronoi_diagram
also delaunoy triangulation is another approach you might want to look at
http://www.spatialdbadvisor.com/oracle_spatial_tips_tricks/283/application-of-delaunay-triangulation-and-inverse-distance-weighting-idw-in-oracle
here's another link that has the st_veronoi function mentioned with a link to the above.
http://www.spatialdbadvisor.com/source_code/223/geoprocessing-package-documentation
the basis of this package appears to be java JTS, which is apparently being compiled within java stored procs in oracle. JTS is the "standard" for geometry operations in Java. I think I'm going to give it a try myself.
Bear in mind, I have only done this using a tool like ArcGIS, not using anything i mentioned above.... so HTH and I'm not leading you down a rat hole.
I can't give you a name but can describe three different algorithms
I'm going to call the set of points you are given "targets" to simplify my solution beacuse I want to call arbitrary locations on the plain "points":
You're going to be doing quite a lot of arithmetic on 2-vectors
my algorithm to partition the polygon is simple: find the nearest target.
the set of points nearest to any target will have straight edges. the vertices will be equidistant to three (or more) of the targets (or be where the edge intersects the boundary polygon),
your algorithm might go like this:
cross the original set of targets with itself twice to produce a set of triples rejecting those that don't copntain three distinct targets.
for each set of three find the point equidistant from all three targets if that point is closer to any other target reject it.
eventually you'll have (at most) n-2 vertices, then you just need to work out how the edges join up. which you can do this by looking at which targets spawned each vertex.
now you need to add the edges which end at infinity take a cross of targets and itself
and find the halfway points between each pair of targets, any points that don't have eactly two nearest targets can be rejected, each of these ponts represents a line (perpendicular bisector) and it will end at one a vertex or at infinity
finally trim the map using the boundary polygon, you may want to drop one of the edges from any fragment that does not contain a target
another way
on the other hand you could use a fractal partitioning scheme to divide the polygon into chunks dividing each chunk smaller until it contains a single polygon, the results will be less aesthetically pleasing but looks weren't a design requirement AFAICT.
eg the fractal mapping used for IP addresses.
then having converted coordinates into numbers into divide this into chunks at convenient points, (IE by dropping unneeded trailing 1's)
another way
measure the extent of set of targets if it is wider than it is high draw a line vertically dividing it in half else draw horizontally.
if the lit hots one of the targets adjust it so that it misses.
repeat for each half until the extet is zero (which means a single point)
You didn't mention any restriction on the shapes of the containing polygons, so I'll assume you don't care. I'll also assume we're in two dimensions, though this can easily be extended. The idea is simple: create new polygons by slicing up your initial polygon with vertical strips halfway between points adjacent on the x-axis. In cases where points share an x-coordinate, split the strip containing them with vertical slices between the points on the y-axis.
Use markg's suggestions if long, thin slices don't work for you.
I´ve got the following theoretical problem:
I have an amount n of cuboids in 3-dimensional space.
They are aligned to the coordinate-system, so that one cuboid can be described via a point (x,y,z) and dimensions (dimX,dimY,dimZ).
I want to organize these cuboids in a way that I´m able to check if a newly inserted cuboid intersecs with one of the existing (collision detection).
To do this I decided to use hierarchical bounding-boxes.
So in sum I have a binary-tree-structure of bounding volumes.
Insertion is then done by determining recursively the distance to both children (=the distance between the two centers of two cuboids) and inserting in the path with the smallest distance.
Collision detection works similar, but we take all bounding volumes in a sub-path which are intersecting a given cuboid.
The tricky part is, how to balance this tree to get better performance if some cuboids are very close to each other and others are far away.
So far, I´ve found no way to use e.g. an AVL-tree because then I´d have to be able to compare two cuboids in some way that does not break the conditions on which collision detection depends.
P.S.: I know there are libraries to do this, but I want to understand the principles of collision detection e.g. in games in detail and therefore want to implement this by myself.
I´ve now tried it with space-partitioning instead of object-partitioning. That´s not exactly what I wanted to do but I found much more helpful information about it, e.g.: https://en.wikipedia.org/wiki/Kd-tree
With this information it should be possible to implement it.
I am searching for an algorithm which calculates, from a list of lines, the area a given point (P) is in. The lines are stored with x and y coordinates, the resulting area schould be stored as a polygon which may has holes in it.
Here two images to illustrate what I mean:
Many CAD applications use such algorithms to let the user hatch areas. I don't know how to call such a method so I don't really know what to search for.
Edit 1: To clearify what I'm looking for:
I don't need an algorithm to calculate the area of a polygon, I need an algorithm which returns the polygon which is formed by lines around P. If P is outside any possible polygon, the algorithm fails.
(I also edited the images)
Edit 2: I read the accepted answer of this possible duplicate and it seams I'm looking for a point location algorithm.
I also stumpled across this site which does exactly explains what I'm looking for and more important it led me to the open-source CGAL library which provides the functionality to do such things. Point 4.6 of the CGAL manual gives an example how to use this library to form a region from a bunch of line segments.
One way to do it is to imagine a line from P to infinity. For each polygon test each edge to see if it crosses the line. Count up how many lines cross. If its even then the point is outside the polygon, if its odd then the point is inside the polygon.
This is a fairly standard mathematical result. If you want to get a little deeper into the subject have a look at https://en.wikipedia.org/wiki/Winding_number.
I did a fairly similar things earlier this week Detecting Regions Described by Lines on HTML5 Canvas
this calculates a the polygon around a point given a set of lines. You can see a working example at the jsfiddle mentioned in the answer
the difference is that it uses infinite mathematical lines rather than line segments, but it could easily be modified for this example.
An outline algorithm
First construct two data-structures one of line-segments one of intersection-points of line-segments. In each record which two lines give each intersection, so you can flip from
one to the other. You can get this by pair-wise intersection of line-segments.
It might be easiest to cut the segments at the intersection points so each segment only have two solutions on it, one at each end. We assume we have done this.
Prune the data-structures, remove all line-segments with no intersections and only one segment - these cannot contribute.
Find possible starting lines. You could calculate the distance from each line to the point and take the smallest. You could check for intersections with a line from the point to infinity.
Walk around the polygon anti-clockwise. With you starting line find the most anti-clockwise end. At that point find the most anti-clockwise segment. Follow that and repeat. It may happen that closed loop is formed, in which case discard all the segments in the loop. Continue until you get back to the starting point.
Check if the polygon actually encloses the point. If so we are done. If not discard segments in the polygon and start again.
I am trying to fit more than one line to a list of points in 2D. My points are quite low in number (16 or 32).
These points are coming from a simulated environment of a robot with laser range finders attached to its side. If the points lie on a line it means that they detected a wall, if not, it means they detected an obstacle. I am trying to detect the walls and calculate their intersection, and for this I thought the best idea is to fit lines on the dataset.
Fitting one line to a set of points is not a problem, if we know all those points line on or around a line.
My problem is that I don't know how can I detect which sets of points should be classified for fitting on the same line and which should not be, for each line. Also, I don't even now the number of points on a line, while naturally it would be the best to detect the longest possible line segment.
How would you solve this problem? If I look at all the possibilities for example for groups of 5 points for all the 32 points then it gives 32 choose 5 = 201376 possibilities. I think it takes way too much time to try all the possibilities and try to fit a line to all 5-tuples.
So what would be a better algorithm what would run much faster? I could connect points within limit and create polylines. But even connecting the points is a hard task, as the edge distances change even within a single line.
Do you think it is possible to do some kind of Hough transform on a discrete dataset with such a low number of entries?
Note: if this problem is too hard to solve, I was thinking about using the order of the sensors and use it for filtering. This way the algorithm could be easier but if there is a small obstacle in front of a wall, it would distract the continuity of the line and thus break the wall into two halves.
A good way to find lines in noisy point data like this is to use RANSAC. Standard RANSAC usage is to pick the best hypothesis (=line in this case) but you can just as easy pick the best 2 or 4 lines given your data. Have a look at the example here:
http://www.janeriksolem.net/2009/06/ransac-using-python.html
Python code is available here
http://www.scipy.org/Cookbook/RANSAC
The first thing I would point out is that you seem to be ignoring a vital aspect of the data, which is you know which sensors (or readings) are adjacent to each other. If you have N laser sensors you know where they are bolted to the robot and if you are rotating a sensor you know the order (and position) in which the measurements are taken. So, connecting points together to form a piecewise linear fit (polylines) should be trivial. Once you had these correspondances you could take each set of four points and determine if they can be modeled effectively by only 2 lines, or something.
Secondly, it's well known that finding a globally optimal fit for even two lines to an arbitrary set of points is NP-Hard as it can be reduced to k-means clustering, so I would not expect to find an efficient algorithm for this. When I used the Hough transform it was for finding corners based on pixel intensities, in theory it is probably applicable to this sort of problem but it is just an approximation and it is probably going to take a fair bit of work to figure out and implement.
I hate to suggest it but it seems like you might benefit by looking at the problem in a slightly different way. When I was working in autonomous navigation with a laser range finder we solved this problem by discretizing the space into an occupancy grid, which is the default approach. Of course, this just assumes the walls are just another object, which is not a particularly outrageous idea IMO. Beyond that, can you assume you have a map of the walls and you are just trying to find the obstacles and localize (find the pose of) the robot? If so, there are a large number of papers and tech reports on this problem.
A part of the solution could be (it's where I would start) to investigate the robot. Questions such as:
How fast is the robot turning/moving?
At what interval is the robot shooting the laser?
Where was the robot and what was its orientation when a point was found?
The answers to these questions might help you better than trying to find some algorithm that relies on the points only. Especially when there are so very few.
The answers to these questions give you more information/points. E.g., if you know the robot was at a specific position when it detected a point, there are no points in between the position of the robot and the detected point. that is very valuable.
For all the triads, fit a line through them and compute how much the line is deviating or not from the points.
Then use only the good (little-deviating) triads and merge them if they have two points at common, and the grow the sets by appending all triads that have (at least) two points in the set.
As the last step you can ditch the triads that haven't been merged with any other but have some common element with result set of at least 4 elements (to get away with crossing lines) but keep those triads that did not merge with anyone but does not have any element in common with sets (this would yield three points on the right side of your example as one of the lines).
I presume this would find the left line and bottom line in the second step, and the right line the third.
Just wondering if there was a nice (already implemented/documented) algorithm to do the following
boo! http://img697.imageshack.us/img697/7444/sdfhbsf.jpg
Given any shape (without crossing edges) and two points inside that shape, compute all the paths between the two points such that all reflections are perfect reflections. The path lengths should be limited to a certain length otherwise there are infinite solutions. I'm not interested in just shooting out rays to try to guess how close I can get, I'm interested in algorithms that can do it perfectly. Search based, not guess/improvement based.
I think you can do better than computing fans. Call your points A and B. You want to find paths of reflections from A to B.
Start off by reflecting A in an edge, and call the reflection A1. Can you draw a line from A1 to B that only hits that edge? If yes, that means you have a path from A to B that reflects on the edge. Do this for all the edges and you'll get all the single reflection paths that exist. It should be easy to construct these paths using the properties of reflections. Along the way, you need to check that the paths are legal, i.e. they do not cross other edges.
You can continue to find paths consisting of two reflections by reflecting all the first generation reflections of A in all the edges, and checking to see whether a line can be drawn from those points through the reflecting edge to B. Keep on doing this search until the distance of the reflected points from B exceeds a threshold.
I hope this makes sense. It should be easier than chasing fans and dealing with their breakups, even though you're still going to have to do some work.
By the way, this is a corner of a well studied field of billiards on tables of various geometries. Of course, a billiard ball bounces off the side of a table the same way light bounces off a mirror, so this is just another way of thinking of reflections. You can delve into this with search terms like polygonal billiards unfolding illumination, although the mathematicians tend to dwell on finding cases where there are no pool shots between two points on a polygonal table, as opposed to directly solving the problem you've posed.
Think not in terms of rays but fans. A fan would be all the rays emanating from one point and hitting a wall. You can then check if the fan contains the second point and if it does you can determine which ray hits it. Once a fan hits a wall, you can compute the reflected fan by transposing it's origin onto the outside of the wall - by doing this all fans are basically triangle shaped. There are some complications when a fan partially hits a wall and has to be broken into pieces to continue. Anyway, this tree of reflected fans can be traversed breadth first or depth first since you're limiting the total distance.
You may also want to look into radiosity methods, which is probably similar to what I've just described, but is usually done in 3d.
I do not know of any existing solutions for such a problem. Good luck to you if you find one, but incase you don't the first step to a complete but exponential (with regard to the line count) would be to break it into two parts:
Given an ordered subset of walls A,B,C and points P1, P2, calculate if a route is possible (either no solutions or a single unique solution).
Then generate permutations of your walls until you exceed whatever limit you had in mind.
The first part can be solved by a simple set of equations to find the necessary angles for each ray bounce. Then checking each line against existing lines for collisions would tell you if the path is possible.
The parameters to the system of equations would be
angle_1 = normal of line A with P1
angle_2 = normal of line B with intersection of line A
angle_3 = normal of line C with ...
angle_n = normal of line N-1 with P2
Each parameter is bounded by the constraints to hit the next line, which may not be linear (I have not checked). If they are not then you would probably have to pick suitable numerical non-linear solvers.
In response to brainjam
You still need wedges....
alt text http://img72.imageshack.us/img72/6959/ssdgk.jpg
In this situation, how would you know not to do the second reflection? How do you know what walls make sense to reflect over?