Setup
Function will need to provide the distance from a point to the closest edge of a polygon
Point is known to be inside of the polygon
Polygon can be convex or concave
Many points (millions) will need to be tested
Many separate polygons (dozens) will need to be ran through the function per point
Precalculated and persistently stored data structures are an option.
The final search function will be in C++
For the function implementation, I know a simple method would be to test the distance to all segments of the polygon using standard distance to line segment formulas. This option would be fairly slow at scale and I am confident there should be a better option.
My gut instinct is that there should be some very fast known algorithms for this type of function that would have been implemented in a game engine, but I'm not sure where to look.
I've found a reference for storing line segments in a quadtree, which would provide for very rapid searching and I think it could be used for my purpose to quickly narrow down which segment to look at as the closest segment and then would only need to calculate the distance to one line segment.
https://people.cs.vt.edu/~shaffer/Papers/SametCVPR85.pdf
I've not been able to locate any code examples for how this would work. I don't mind implementing algorithms from scratch, but don't see the point in doing so if a working, tested code base exists.
I've been looking at a couple quadtree implementations and I think the way it would work is to create a quadtree per polygon and insert each polygon's line segments with a bounding box into the quadtree for that polygon.
The "query" portion of the function I would be making would then consist of creating a point as a very small bounding box, which would then be used to search against the quadtree structure, which would then find only the very closest portions of the polygon.
http://www.codeproject.com/Articles/30535/A-Simple-QuadTree-Implementation-in-C
and
https://github.com/Esri/geometry-api-java/blob/master/src/main/java/com/esri/core/geometry/QuadTree.java
My real question would be, does this seem like a sound approach for a fast search time function?
Is there something that would work faster?
EDIT:
I've been looking around and found some issues with using a quadtree. The way quadtrees work is good for collision detection, but isn't setup to allow for efficient nearest neighbor searching.
https://gamedev.stackexchange.com/questions/14373/in-2d-how-do-i-efficiently-find-the-nearest-object-to-a-point
R-Trees look to be a better option.
https://en.wikipedia.org/wiki/R-tree
and
efficient way to handle 2d line segments
Based on those posts, R-trees look like the winner. Also handy to see that C++ Boost already has them implemented. This looks close enough to what I was planning on doing that I'll go ahead and implement it and verify the results.
EDIT:
Since i have implemented an PMR quadtree, I see now, that the nearest neighbour search is a bit more complex than I described.
If the quad search result for the search point would be empty then it gets more complex.
I remeber a description somewhere in Hannan Sammets:Multidimensional search structure.
Giving the answer below I had in mind searching for all objects withing a specified distance. This is easy for the PMR quadtree, but just finding the closest is more complex.
Edit End
I would not use a R-Tree.
The weak point (and the strong point!) on R-trees is the separation of the space into rectangles.
There are three algorithms known to do that separation but none is well suited for all situations.
R-trees are really complex to implement.
Why then do it? Just because R-Trees can be twice fast than a quad tree when perfectly implemented. The speed difference between a quadtree and a R-Tree is not relevant. The monetary difference is. (If you have working code for both I would use the PMR quadtree, if you have only code for the R-Tree then use that, If you have none use the PMR Quadtree)
Quad trees (PMR) always work, and they are simple to implement.
Using the PMR quad tree, you just find all segments related to the search point. The result will be a few segments, then you just check them and ready you are.
People that tell quad trees are not suited or neighbour search, do not know that there are hundreds of different quad trees. The non suitability is just true for a point quad tree, not for the PMR one, which stores bounding boxes.
I once remebered the compelx description of finding the neighbour points in a POINT-Quadtree. For the PMR-quadtree I had nothing to do (for a search within a specified rectangular interval), no code change, Just iterate the result and find the closest.
I think that there are even better solutions than Quad tree or R-Tree for your spefic questions, but the point is that the PMR always work. Just implement it one time and use if for all spatial searches.
Since there are so many more points to test than polygons, you could consider doing some fairly extensive pre-processing of the polygons in order to accelerate the average number of tests to find the nearest line segment per point.
Consider an approach like this (assumes polygons have no holes):
Walk the edges of the polygon and define line segments along each equidistant line
Test which side of the line segment a point is to restrict the potential set of closest line segments
Build an arithmetic coding tree with each test weighted by the amount of space that is culled by the half-space of the line segment. this should give good average performance in determining the closest segment for a point and open up the possibility of parallel testing over multiple points at once.
This diagram should illustrate the concept. The blue lines define the polygon and the red lines are the equidistant lines.
Notice that needing to support concave polygons greatly increase the complexity, as illustrated by the 6-7-8 region. Concave regions mean that the line segments that extend to infinity may be defined by vertices that are arbitrarily far apart.
You could decompose this problem by fitting a convex hull to the polygon and then doing a fast, convex test for most points and only doing additional work on points that are within the "region of influence" of the concave region, but I am not sure if there is a fast way to calculate that test.
I am not sure how great the quadtree algorithm you posed would be, so I will let someone else comment on that, but I had a thought on something that might be fast and robust.
My thought is you could represent a polygon by a KD-Tree (assuming the vertices are static in time) and then find the nearest two vertices, doing a nearest 2 neighbor search, to whatever the point is that lies in this polygon. These two vertices should be the ones that create the nearest line segment, regardless of convexity, if my thinking is correct.
Related
Given a huge collection of points (float64) in 2d space...
Is there a way to determine the nearest neighbour using a feature of OpenGL or DirectX?
I've implemented a kd-tree, which is still not fast enough.
A kd-tree should work just fine. But here's some hints.
I implemented a kd-tree once for a million point data set once. Here's what I learned out of it:
Did you try profiling your code? You might find that there are easy optimizations to make such as common helper functions needing to be forced inline.
Did you actually test your code to validate that it was culling out tree branches for partitions that are easily identified as "too far away". If you aren't careful, you can easily have a bug that does needless distance computations on points too far away.
Easiest thing: Where comparing linear distance between points, you don't need to take the SQRT of (x2-x1)*(y2-y1).
Most of the time spent in my code was just building the tree from the original data set, including multiple full sorts on each iteration deciding which axis was the best to partition on. An easier algorithm would be to just alternate between partitioning on the x and y axis for each tree branch and to cache the sorting order for each axis. It may not build the most optimal search tree, but the overall savings can be enormous.
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A simple algorithm for polygon intersection
I'm looking for an outline on how to quickly calculate the intersection of two arbitrarily oriented quadrilaterals (no preset corner angle or side length constraints). I am not looking to simply check whether they intersect, but wish to get the points making up the resulting intersecting region. I know that in general polygon intersection isn't a trivial problem and there are libraries available that do a good job.
But since in this special case where I'm only concerned with four sided shapes, I was wondering if there was a quick method I could use without including an entire additional library in my application.
So far all I've thought of is:
Run 'point in polygon' on both shapes with respect to each other
Intersect each edge of each polygon with each other
Do the above two steps definitively get me all the points that make up the resulting intersection region? Is there a better method to use?
Also it would be nice if I could get the correct ordering of the points that make up the resulting region. It's not mandatory -- if you are aware of any clever/quick ways of doing this bit (convex hull?) I'd appreciate any suggestions.
You didn't state wether the 2 quadriliterals are convex or not; if they are, you could use a regular convex polygon intersection algorithm such as http://www.iro.umontreal.ca/~plante/compGeom/algorithm.html
From what I can gather, it doesn't require any exotic datastructures or operations, so it shouldn't be difficult to implement.
Intersection of convex polygons is relatively easy. Google it, there's a lot of resources both on SO and elsewhere.
Not all quadrilaterals are convex though. Intersection of two non-convex quadrilaterals can result in several disconnected polygons, having just their points will give you very little, but if that's what you need go ahead and intersect each pair of edges. It will be much easier and faster than any general method
Even for convex shapes, the dumb brute-force method may be faster. You have to do some testing to find out what works best for you.
I'm terrible with math, but I have a situation where I need to find all points in a 3D space that are arbitrarily close to a vector being projected through that same space. The points can be stored in any fashion the algorithm calls for, not that I can think of any particularly beneficial ordering for them.
Are there any existing C++ algorithms for this feat? And if so (or not), what kind of mathematical concept does or would it entail, since I'd love to attempt to understand it and tie my brain into a pretzel.
( This algorithm would be operating on a space with perhaps 100,000 points in it, it would need to test around 1,000,000 vectors, and need to complete those vectors within 1/30th of a second. I of course doubt if any algorithm can perform this feat at all, but it'll be fun to see if that's true or not. )
You would probably want to store your points in some spatial data structure. The ones that come to mind are:
oct-trees
BSP trees
kd-trees
They have slightly different properties. An oct-tree divides the entire world up into 8 equally sized cubes, organized to themselves form a larger cube. Each of these cubes are then in turn split into 8, evenly sized, cubes. You keep splitting the cubes until you have less than some number of points in a cube. With this tree structure, you can quite easily traverse the tree, extracting all points that may intersect a given cube. Once you have that list of points, you can test them one at a time. Since your test geometry is a sphere (distance from a point) you would circumscribe a cube around the sphere and get the points that may intersect it. As an optimization, you may also inscribe a cube in your circle, and anything that for sure intersects that, you can simply include in your hit-set right away.
The BSP tree is a Binary space partitioning tree. It's a tree of planes in 3-space, forming a binary tree. The main problem of using this for your problem is that you might have to do a lot of square roots while traversing it, to find the distance to the planes. The principle is the same though, once you have fewer than some number of points you form a leaf with those points in it. All leaves in a BSP tree are convex polygons (except for the leaves that are along the perimeter, which will be infinitely large polygons). When building the BSP, you want to split the points in half for each step, to truly get O(log n) searches.
The kd-tree is a special case of BSP, where all planes are axis aligned. This typically speeds up tests against them quite significantly, but doesn't allow you to optimize the planes based on your set of points quite as well.
I don't know of any c++ libraries that implement these, but I'm sure there are plenty of them. These are fairly common techniques used in video games, so you might want to look at game engines.
It might help your understanding of octrees when you can think of it as a curve that fills the space traversing every coordinate only once and w/o crossing itself. The curve maps the 3d complexity to a 1d complexity. There are some of this monster curve, like the z curve, the hilbert curve, and the moore curve. The latter is a copy of 4 hilbert curves and has very good space fills quality. But isn't a search for the closest points not solved with dijkstra algorithm?
I've got a point in 2d image for example the red Dot in the given picture and a set of n points blue dot (x1,y1)...(xn,yn) and I want to find nearest point to (x0,y0) in a way better than trying all points. Like to have best possible solution. Would appreciate if you share any similar class if you have.
There are many approaches to this, the most common probably being using some form of space partitioning to speed up the search so that it is not O(n). For details, see Nearest neighbor search on Wikipedia.
Most solutions that we could suggest would depend on a little bit more knowledge, I am going to go out on a limb and say that unless you already know that you are short on time. I.e. there are tens of thousands of blue dots or you have to do thousands of these calculations in a short time. "Linear Search" will serve you well enough.
Don't bother calculating the actual distance, save yourself calculating the square root and use this as the "distance".
Most other methods use more complex data structures to sort the points in respect to their geometric arrangement. But are a lot harder to implement.
I know that I can use a KD-Tree to store points and iterate quickly over a fraction of them that are close to another given point. I'm wondering whether there is something similar for lines.
Given a set of lines L in 3D (to be stored in that data structure) and another "query line" q, I'd like to be able to quickly iterate through all lines in L that "are close enough" to q. The distance I'm planning to use is the minimal Euclidean distance between two points u and v where u is some point on the first line and v is some point on the second line. Computing that distance is not a problem (there's a nice trick involving the cross product).
Maybe you guys have a good idea or know where to look for papers, descriptions, etc...
TIA,
s.
Another option - and the most commonly used one for spatial indexing in disk-based database systems - is the R-Tree. It's a bit more complicated to implement than a KD-Tree, but it's generally considered to be faster, and has no problem indexing lines and polygons.
You can use KD-Trees for this as well.
It's possible to build a KD-Tree that works on primitives, not points. Many ray tracers do this to make triangle hit testing much faster. The best description I've seen is in this ray tracing tutorial.
A potentially faster, though not 100% accurate, solution, is to just keep a list of points per line segment, and insert these into a standard point-based KD-Tree. Find the nearest points, then have them tagged with the line segment, and use that to get the nearest lines. It's crude, but often very fast compared to other options. The "trick" is to find the right balance of keeping large spaces between points along the segment (faster) vs. breaking the segment into more points (slower, but more accurate).