Is there any simple (or not) algorithm that is capable of creating polygons from closed path?
Assume w have following path:
0,0; 2,0, 2,1; 1,1;
1,2; 2,2; 2,3; 0,3;
I need to be able to create polygon indexes for OpenGL vertex buffer. Language I'm using is C#.
Someone suggested me Convex Hull, but it's not the thing I'm looking for, cause I have a shape already. I know that this could be a trivial issue, but seriously, I can't find any description or something to point mi in the right direction.
Edit:
Answer 1 suggest to select a point and connect it to other not connected points, this works fine for presented in answer shape but it will not work for shape i posted, shape above looks like this:
Converting to triangles is either easy or hard, depending on what your requirements are and how well you want to do it.
If your polygon is convex, the easiest way is to use GL_TRIANGLES with the indicies
0, 1, 2, 0, 2, 3, 0, 3, 4, ...
It will look like this:
The situation for concave is more work. An algorithm that also works for concave polygons (not ones with holes!) is the Ear-clipping method, described on wikipedia (there is more on that page.)
Things get really interesting when you want a "good" triangulation: avoid skinny or small triangles, reduce number of triangles etc., and then you can trade off quality for speed. I won't go into any advanced algorithms here; searching for polygon triangulation on Google will get you lots of info.
As for normals, if your polygon is planar (it most likely "should" be) then take two non co-incident edges and cross product them (there are two normals and you probably only want one: you need to choose the order in which you cross product (clockwise or anticlockwise) based on the right hand rule.
Related
I am getting coordinates - lat/lon and I want to check if these coordinates are in the continental United States or not. Is there a easy way to do it in C#? I can convert the coordinates into MGRS or UTM.
Thanks!
Oh wow, they have it just for you:
http://econym.org.uk/gmap/states.xml
All the coords of the US states!
Build up a polygon and apply any polygon-contains-point algorithm.
The classic algorithm is ray-casting, and its even pretty simple. Let me know if you have any trouble with it.
Now, you have two options:
Use the data to build a polygon for each state, and check a point with each one of them. If none match, it is not in the US.
However, there is a problem with that approach - I don't know how the data was gathered, but its possible that there are very little gaps between states, or even slight overlaps. So if you only care if its generally in the US or not, I suggest the second approach:
Use the data to build a polygon for each state, and an algorithm to combine those polygons to one (like union). Save that polygon and use that as with the 1st approach.
I wonder if there's any described algorithm that can convert isochrones into approximate area to show a range of some feature (in my problem this feature is a road network).
Example. I have something like on the image beneath:
It's a simple network (where I can arrive from the start point in X minutes or going Y kilometers). I have information of all the nodes and links. Now I need to create an isochrone map that show an approximate range where I can arrive.
Problems:
Convex hull - sucks because of too general approximation,
I can create buffors on roads - so I will get some polygon that shows range, but I will also have the holes by roads that connect into circles.
What I need to obtain is something like this:
I've found some potentially useful information HERE, but there are only some ideas how it could be done. If anyone has any concept, please, help me to solve my problem.
Interesting problem, to get better answers you might want to define exactly what will this area that shows the range (isochrone map) be used for? For example is it illustrative? If you define what kind of approximation you want it could help you solve the problem.
Now here are some ideas.
1) Find all the cycles in the graph (see link), then eliminate edges that are shared between two cycles. Finally take the convex hull of the remaining cycles, this together with all the roads, so that the outliers that do not form cycles are included, will give a good approximation for an isochrome map.
2) A simpler solution is to define a thickness around each point of every road, this thickness should be inversely proportional to how long it takes to arrive at that point from the starting point. I.e. the longer it takes to arrive at the point the less thick. You can then scale the thickness of all points until all wholes are filled, and then you will have an approximate isochrome map. One possible way of implementing this is to run an algorithm that takes all possible routes simultaneously from the starting point, branching off at every new intersection, while tracking how long it took to arrive at each point. During its execution, at every instant of time all previously discovered route should be thickened. At the end you can scale this thickness so as to fill all wholes.
Hopefully this will be of some help. Good luck.
I have solved the problem (it's not so fast and robust, but has to be enough for now).
I generated my possible routes using A* (A-Star) algorithm.
I used #Artur Gower's idea from point one to eliminate cycles and simplify my geometry.
Later I decided to generate 2 types of gemetries (1st - like on the image, 2nd - simple buffers):
1st one:
3. Then I have removed the rest of unnecessary points using Douglas-Peucker algorithm (very fast!).
4. In the end I used Concave Hull algorithm (aka Alpha-Shapes or Non-Convex Hull).
2nd one:
3. Apply a buffer to the existing geometry and take the exterior ring (JTS library made that really easier:)).
I have 2D world maps that are basically Mercator-Like projections, (If you walk west long enough you end up east of where you started)
Question I have: Can you use A* for computing paths on these types of maps as well?
I can't think of any reason why you couldn't (I'm thinking that you would simply represent the edge map nodes such that the North, South, East, Wed, "border" nodes simply connected to the opposite side).
Thanks in advance, if anyone has seen something like this before or can give me a few hints I would appreciate it.
Pathfinding algorithms don't really care about global topology of the map. The only tricky part is to get a good estimator for A* but using the 3D distance should be ok if your map is indeed a surface in a 3d space and step cost is step length.
Your map can have all sort of strange "connections" (including for example knotted bridges) and this will not be a problem if you implement A* correctly.
I can't imagine why a Mercator-Like projections would cause a problem for A*, as long as your heuristic function approximates distances correctly. I think something along the below function should work fine
float heuristic(point from, point to, size mapsize) {
float x = from.x - to.x;
if (abs(x) > mapsize.x/2)
x = mapsize.x - x;
float y = from.y - to.y;
if (abs(y) > mapsize.y/2)
y = mapsize.y - y;
return sqrt(x*x+y*y);
}
Edited: I realize know I was misled by the non-graph theoretical) use of the word edge (where the question title strongly suggested a graph algorithm question :))
Why do you suppose there are no edges? There are many logical discrete locations that you could model, and limited connections between (i.e. not where a wall is :)). There you go: you have your edges.
What you probably mean is that you don't want to represent your edges in data (which you don't have to, but still there are the logical edges that connect locations/points.)
That said:
you ask whether someone has seen things like this before. I vaguely recall seeing something relevant to this in Knuths Dancing Links article (DLX) which is an implementation technique for A* algorithms.
http://en.wikipedia.org/wiki/Dancing_Links
original publication [PDF]
The article specifically treats states as 'cells' (in a grid) with east/west/north/south links. It's been a long time so I don't quite recall how you would map (no pun intended) your problem on that algorithm.
The dance steps. One good way to implement algorithm X is to represent each 1 in the
matrix A as a data object x with five fields L[x]; R[x]; U [x]; D[x]; C[x]. Rows of the matrix
are doubly linked as circular lists via the L and R fields ("left" and "right"); columns are
doubly linked as circular lists via the U and D fields ("up" and "down"). Each column
list also includes a special data object called its list header.
I'm thinking of creating a program to let me play or solve slitherlink puzzles, like on krazydad.com. It consists of tiles of 4, 5, 6, 7 and 8 sides. All but the seven sided tiles seem to have sides with the same length, with the sides between two seven sided tiles (and therefore connecting five-sided tiles to 4 sided tiles) having sides of approximately 70% of the normal length. As you can see in the picture below, octagons are surrounded by alternating pentagons and hexagons. These are attached to others a by the far sides of the hexagons. Attached to the tips of the pentagons are smaller lines connecting to squares connecting to other groups. Around the squares are then formed seven-sided figures with two short sides. I think the outer edge is defined by just omitting tiles that are too far away from the center.
For a data structure I think I need a graph connecting all nodes. I can let the user click to place a solid line on the closest link, and I can check for loops or too many lines entering a node fairly easily. I'll also need to create tiles and associate lines to them, with inner lines being assigned to both tiles, but treated as one line.
As for setting it up, I am thinking of manually figuring out the points and defining the minimal set of repeated tiles (1 8, 4 5, 4 6, 4 7 and 1 4), then placing them next to each other. When placing, I would check for existing close points to each one I'm placing and combine them if found. Then I would need to check for duplicate lines and merge them as well.
Is there an easier or cleaner way to A) generate the tiling or B) merge the nodes and links when doing my tiling?
some observations that might help:
if you join the centres of neighbouring polygons you have a delaunay triangulation (1).
the dual (2) of the delaunay triangulation is the graph above (with slightly different edge lengths, but you can adjust that if necessary)
there's a discussion here (3) of how to generate graphs from delaunay triangulations
so, putting that together, you could:
generate the centres of the tiles (see below)
construct the delaunay triangulation from the tile centres (by joining neigbours).
find the dual to get the graph you want (the process of finding the dual should be supported by a good graph library)
to generate the pattern of tile centres, take the minimal set and start with the centre 8. for each 90 degree rotation about there, add the (rotated) minimal set (plus an 8 in addition to the one you're rotating around), removing duplicates. then do the same on the 8s that you have added (either recurse or use a stack).
once you have the centres, i'm not sure what the best way to connect neighbours would be - you want some efficient way of generating a set of candidates. but it's not a hard problem, just fiddly (a "fancy" solution would be quadtree or spatial hashes, but just a crude binning would probably be enough).
This seems non-trivial (it gets asked quite a lot on various forums), but I absolutely need this as a building block for a more complex algorithm.
Input: 2 polygons (A and B) in 2D, given as a list of edges [(x0, y0, x1, y2), ...] each. The points are represented by pairs of doubles. I do not know if they are given clockwise, counter-clockwise or in any direction at all. I do know that they are not necessarily convex.
Output: 3 polygons representing A, B and the intersecting polygon AB. Either of which may be an empty (?) polygon, e.g. null.
Hint for optimization: These polygons represent room and floor boundaries. So the room boundary will normally fully intersect with the floor boundary, unless it belongs to another floor on the same plane (argh!).
I'm kind of hoping someone has already done this in c# and will let me use their strategy/code, as what I have found so far on this problem is rather daunting.
EDIT: So it seems I'm not entirely chicken for feiling faint at the prospect of doing this. I would like to restate the desired output here, as this is a special case and might make computation simpler:
Output: First polygon minus all the intersecting bits, intersection polygons (plural is ok). I'm not really interested in the second polygon, just its intersection with the first.
EDIT2: I am currently using the GPC (General Polygon Clipper) library that makes this really easy!
Arash Partow's FastGEO library contains implementations of many interesting algorithms in computational geometry. Polygon intersection is one of them. It's written in Pascal, but it's only implementing math so it's pretty readable. Note that you will certainly need to preprocess your edges a little, to get them into clockwise or counterclockwise order.
ETA: But really, the best way to do this is to not do this. Find another way to approach your problem that doesn't involve arbitrary polygon intersections.
If you are programming in .NET Framework, you may want to take a look at SqlGeometry class available in .NET assemblies shipped as Microsoft SQL Server System CLR Types
The SqlGeometry class provides STIntersection method
SqlGeometry g1 = SqlGeometry.Parse("POLYGON ((...))");
SqlGeometry g2 = SqlGeometry.Parse("POLYGON ((...))");
SqlGeometry intersection = g1.STIntersection(g2);
What I think you should do
Do not attempt to do this yourself if you can possibly help it. Instead, use one of the many available polygon intersection algorithms that already exist.
I was strongly considering the following codebase on the strength of their demonstration code and the fact that they mentioned their handling of most/all of the weird cases. You would need to donate an amount (of you/your company's choice) if you use it commercially, but it's worth it to get a robust version of this kind of code.
http://www.cs.man.ac.uk/~toby/gpc/
What I actually did was to use a polygon-intersection algorithm that is part of the Java2D libraries. You can possibly find something similar in MS's own C# libraries to use.
There are other options out there as well; look for "polygon clipper" or "polygon clipping", since the same basic algorithms that handle polygon intersection also tend to be usable for the general clipping cases.
Once you actually have a polygon clipping library, you just need to subtract polygon B from polygon A to get your first piece of output, and intersect polygons A and B to get your second piece of output.
How to roll your own, for the hopelessly masochistic
When I was considering rolling my own, I found the Weiler-Atherton algorithm to have the most potential for general polygon-cutting. I used the following as a reference:
http://cs1.bradley.edu/public/jcm/weileratherton.html
http://en.wikipedia.org/wiki/Weiler-Atherton
The details, as they say, are too dense to include here, but I have no doubt that you'll be able to find references on Weiler-Atherton for years to come. Essentially, you split all the points into those that are entering the final polygon or exiting the final polygon, then you form a graph out of all the points, and then walk the graph in the appropriate directions in order to extract all the polygon pieces you want. By changing the way you define and treat the "entering" and "exiting" polygons, you can achieve several possible polygon intersections (AND, OR, XOR, etc.).
It's actually fairly implementable, but like with any computational geometry code, the devil is in the degeneracies.
You may also want to have a look at the NetTopologySuite or even try importing it into Sql server 2008 & it's spatial tools.
Clipper is an open source freeware polygon clipping library (written in Delphi and C++) that does exactly what you're asking - http://sourceforge.net/projects/polyclipping/
In my testing, Clipper is both significantly faster and far less prone to error than GPC (see more detailed comparisons here - http://www.angusj.com/delphi/clipper.php#features). Also, while there's source code for both Delphi and C++, the Clipper library also includes a compiled DLL to make it very easy to use the clipping functions in other (Windows) languages too.
A polygon is fully described by an ordered list of points (P1, P2, ..., Pn). The edges are (P1 - P2), (P2 - P3), ..., (Pn - P1). If you have two polygons A and B which overlaps, there will be a point An (from the list on points describing polygon A) which lies within the area surrounded by polygon B or vice versa (a point of B lies in A). If no such point is found, then the polygons does not overlap. If such a point is found (i.e. Ai) check the adjacent points of the polygon A(i-1) and A(i+1). Repeat until you find a point outside the area or all points are checked (then the first polygon lies completly within the second polygon). If you found a point outside then you can calculate the crossing point. Find the corresponding edge of polygon B and follow it with resersed roles = now check if the points of polygon B lie within polygon A. This way you can build a list of points which describe the overlapping polygon. If needed you should check if the polygons are identical, (P1, P2, P3) === (P2, P3, P1).
This is just an idea and there maybe better ways. If you find a working and tested solution I would recommend that you use it...
narozed
Try to use GIS tools for that, such as ArcObjects, TopologySuite, GEOS, OGR, etc. I'm not sure if all of these distributions are availuable to .net, but they all do the same.
This academic paper explains how to do this.
If you dare to take a look to other people's GPL C++ code, you can see how do they do it in Inkscape:
http://bazaar.launchpad.net/~inkscape.dev/inkscape/trunk/view/head:/src/2geom/shape.cpp (line 126)