How to decompose the polygon with self intersections to the set of simple polygons?
The input polygon P = {p1, ... pn} is given by the set of n vertices with the CCW orientation. I would like to perform a decposition to a set of m polygons P1, ..., Pm.
A simple walking along the segments from the intersection to the next one does not bring any effect; there are 2 segments with the same start point represented by the intersection point.
Probably, some lexicographical sort of edges may help...
Calculate all intersections, make new nodes and divide edges at intersections, for every node create list of adjacent edges.
Start from some point. Walk using the most CCW edge from current vertex (relative to the last edge). Add traversed edges to polygon and remove them (or mark). When you return to the same vertex, close polygon.
Repeat from the first vertex still having edges.
Related
Let's say we have a class/struct Point
class Point
{
int x;
int y;
}
and a class Polygon that contains list of Points
class Polygon
{
List<Point> points;
Path(List<Point> points)
{
//some implementation
}
}
I am interested in finding the Minimal bounding rectangle points of the Polygon(https://en.wikipedia.org/wiki/Minimum_bounding_rectangle). The found minimal bounding rectangle sides might not be parallel to both axis , so I am trying to find an algorithm written in Java,C#,C++ .Can anyone propose or link such a solution, thanks!
It is possible to construct minimal bounding box (both min-area and min-perimeter) using Rotating Calipers approach.
You can find description at this wayback archive page
The minimum area rectangle enclosing a convex polygon P has a side
collinear with an edge of the polygon.
The above result dramatically restricts the range of possible
rectangles. Not only is it not necessary to check all directions, but
only as many as the polygon's number of edges.
Illustrating the above result: the four lines of support (red), with
one coinciding with an edge, determine the enclosing rectangle (blue).
A simple algorithm would be to compute for each edge the corresponding
rectangle collinear with it. But the construction of this rectangle
would imply computing the polygon's extreme points for each edge, a
process that takes O(n) time (for n edges). The entire algorithm would
then have quadratic time complexity.
A much more efficient algorithm can be found. Instead of recomputing
the extreme points, it is possible to update them in constant time,
using the Rotating Calipers. Indeed, consider a convex polygon, with a
two pair of lines of support through all four usual extreme points in
the x and y directions. The four lines already determine an enclosing
rectangle for the polygon. But unless the polygon has a horizontal or
vertical edge, this rectangle is not a candidate for the minimum area.
However, the lines can be rotated until the above condition is
satisfied. This procedure is at the heart of the following algorithm.
The input is assumed to be a convex polygon with n vertices given in
clockwise order.
Compute all four extreme points for the polygon, and call them xminP,
xmaxP, yminP ymaxP.
Construct four lines of support for P through all four points. These
determine two sets of "calipers".
If one (or more) lines coincide with an edge, then compute the area of
the rectangle determined by the four lines, and keep as minimum.
Otherwise, consider the current minimum area to be infinite.
Rotate the lines clockwise until one of them coincides with an edge of
its polygon.
Compute the area of the new rectangle, and compare it to the current
minimum area. Update the minimum if necessary, keeping track of the
rectangle determining the minimum.
Repeat steps 4 and 5, until the lines have been rotated an angle
greater than 90 degrees. Output the minimum area enclosing rectangle.
Because two pairs of "calipers" determine an enclosing rectangle, this
algorithm considers all possible rectangles that could have the
minimum area. Furthermore, aside from the initialization, there are
only as many steps in the algorithm's main loop as there are vertices.
Thus the algorithm has linear time complexity.
You can do this as follows:
Find the convex hull of the polygon points. A popular method is the Graham scan.
For each edge of the convex hull, find the minimum bounding rectangle that must have one side overlapping with that edge, so that the convex hull's edge is a subsegment of that rectangle's side.
Find the two perpendicular sides of the rectangle by walking along the convex hull vertices, and projecting the vertex to the first side, retaining the two that have all other projections between them. The final opposite side can be found in a similar manner. The vertices that lie on the rectangle, should be the starting point for when this step is executed again for the next edge on the convex hull (step 2).
Calculate the size of these rectangles, and retain the minimal one.
The complexity of this algorithm is determined by the first step, which is O(nlogn). The second step is O(n), as you have a selection of one edge and 3 vertices that rotates around the convex hull, visiting each edge once, and each vertex 3 times.
I am working with several convex polygons that overlap each other and I need to combine them back together to form one single polygon that may be convex or concave.
The problem is always as follows:
1) The polygons that I need to merge together are always convex.
2) The vertices of each polygon are defined in clockwise order.
3) The polygons are never in any specific order.
4) The final polygon can only be simple convex or concave polygon, i.e. no self-intersection, no duplicate vertices or holes in the shape.
Here is an example of the kind of polygons that I am working with.
![overlapping convex polygons]"image removed")
My current approach is to start from the first polygon and vertex by vertex I loop through all vertices of all of the polygons to find overlap. If there is no overlap, I store the vertex for the final outline and continue.
Upon finding overlapping vertices, I determine which polygon to continue to by measuring the angles of the possible paths and by choosing the one that leads towards the outside of the shape.
This method works until I encounter polygons that do not have vertices overlapping each other, but instead one polygon's vertex is overlapping another polygon's side, as is the case with the rectangle in the image.
I am currently planning on solving these situations by running line intersect checks for all shapes that I have not yet processed, but I am convinced that this cannot be the easiest or the best method in terms of performance.
Does someone know how I should approach this problem in a more efficient manner and/or universal manner?
I solved this issue and I'm posting the answer here in case someone else runs into this issue as well.
My first step was to implement a pre-processing loop based on trincot's suggestions.
I calculated the minimum and maximum x and y bounds for each individual shape.
I used these values to determine all overlapping shapes and I stored a simple array for each shape that I could later use to only look at shapes that can overlap each other.
Then, for the actual loop that determines the outline of the final polygon:
I start from the first shape and simply compare its vertices to those of the nearby shapes. If there is at least one vertex that isn't shared by another vertex, it must be on the outer edge and the loop starts from there. If there are only overlapping vertices, then I add the first shape to a table for all checked shapes and repeat this process with another shape until I find a vertex that is on the outer edge.
Once the starting vertex is found, the main loop will check the vertices of the starting shape one by one and measure how far from the given vertex is from every nearby shapes' edges. If the distance is zero, then the vertex either overlaps with another shape's vertex or the vertex lies on the side of another shape.
Upon finding the aforementioned type of vertex, I add the previous shape's number to the table of checked shapes so that it isn't checked again. Then, I check if there are other shapes that share this particular vertex. If there are, then I determine the outermost shape and continue from there, starting back from step 2.
Once all shapes have been checked, I check that all non-overlapping vertices from the starting shape were indeed added to the outline. If they weren't, I add them at the end.
There may be computationally faster methods, but I found this one to be simple to write, that it meets all of my requirements and it is fast enough for my needs.
Given a vertex, you could speed up the search of an "overlapping" vertex or edge as follows:
Finding vertices
Assuming that the coordinates are exact, in the sense that if two vertices overlap, they have exactly the same x and y coordinates, without any "error" of imprecision, then it would be good to first create a hash by x-coordinate, and then for each x-entry you would have a hash by y-coordinate. The value of that inner hash would be a list of polygons that have that vertex.
That structure can be built in O(n) time, and will allow you to find a matching vertex in constant time.
Only if that gives no match, you would go to the next algorithm:
Finding edges
In a pre-processing step (only once), create a segment tree for these polygons where a "segment" corresponds to a min/max x-coordinate range for a particular polygon.
Given a vertex, use the segment tree to find the polygons that are in the right x-coordinate range, i.e. where the x-coordinate of the vertex is within the min/max range of x-coordinates of the polygon.
Iterate those polygons, and eliminate those that do not have an y-coordinate range that has the y-coordinate of the vertex.
If no polygons remain, the vertex does not participate in any edge of another polygon.
You cannot get more than one polygon here, since that would mean another polygon shares the vertex, which is a case already covered by the hash-based algorithm.
If you get just one polygon, then continue your search by going through the edges of that polygon to find a match -- which is what you already planned on doing (line intersect check), but now you would only need to do it for one polygon.
You could speed that line intersect check up a little bit by first filtering the edges to those that have the right x-range. For convex polygons you would end up with at most two edges. At most one of those two will have the right y-range. If you get such an edge, check whether the vertex is really on that edge.
Suppose you have a set of triangles, like the one shown in the image below, where black is a triangle edge, red is a triangle point, green is the polygon that needs to be found, and blue is the polygon's points.
The polygon described may or may not be concave. The triangles in it will always be adjacent (share all three points with the other triangles in the set).
What sort of algorithms exist to generate the polygon that such a set of triangles describes? The polygon should be in the form of a list of points in clockwise or counter-clockwise order.
How About Following A simple GrahamScan Algo... That's should Do the Trick.
Assume you have N distinct points Pi and M triangles. We define each triangle by 3 indices i, j and k to the points. Each triangle will have 3 edges defined as E(i,j), E(j,k) and E(i,k). The way to find the "polygon" is as follows:
1) Loop thru all triangles. For each triangle, add the 3 edges into a set. Edges with two identical point indices should be considered as the same edge. Namely, E(i,j) = E(j,i). Once encounter such case, remove both E(i,j) and E(j,i) from the set as these are the interior edges.
2) The remaining edges in the set should be the edges forming the polygon.
3) Sort the edges in the set by the point indices as follows:
(3a) Pick any edge from the set, say E(i,j).
(3b) Add indices i and j into a std::vector, then remove E(i,j) from the set.
(3c) Find the edge from the set that shares the last point index in the std::vector (which is j now). Denote this edge as E(j,k). Add index k into the std::vector and remove E(j,k) from the set.
(3d) Repeat step (3c) until the set contains no edges. The point indices in the std::vector will be the points order for the polygon.
If you only have M triangles and 3*M (x, y) values for the triangle vertices, then you will need to do some pre-processing to remove identical points and re-define the triangles in terms of the point indices as stated above.
Say I have a polygon. It can be a convex one or not, it doesn't matter, but it doesn't have holes. It also has "inner" vertices and edges, meaning that it is partitioned.
Is there any kind of popular/known algorithm or standard procedures for when I want to check if a point is inside that kind of polygon?
I'm asking because Winding Number and Ray Casting aren't accurate in this case
Thanks in advance
You need to clarify what you mean by 'inner vertices and edges'. Let's take a very general case and hope that you find relevance.
The ray casting (point in polygon) algorithm shoots off a ray counting the intersections with the sides of the POLYGON (Odd intersections = inside, Even = outside).
Hence it accurately gives the correct result regardless of whether you start from inside the disjoint trapezoidal hole or the triangular hole (inner edges?) or even if a part of the polygon is completely seperated and/or self intersecting.
However, in what order do you feed the vertices of the polygon such that all the points are evaluated correctly?
Though this is code specific, if you're using an implementation that is counting every intersection with the sides of the polygon then this approach will work -
- Break the master polygon into polygonal components. eg - trapezoidal hole is a polygonal component.
- Start with (0,0) vertex (doesn't matter whether (0,0) actually lies wrt your polygon) followed by the first component' vertices, repeating its first vertex after the last vertex.
- Include another (0,0) vertex.
- Include the next component , repeating its first vertex after the last vertex.
- Repeat the above two steps for each component.
- End with a final (0,0) vertex.
2 component eg- Let the vertices of the two components be (1x,1y), (2x,2y), (3x,3y) and (Ax,Ay), (Bx,By), (Cx,Cy). Where (Ax,Ay), (Bx,By), (Cx,Cy) could be anything from a disjoint triangular hole, intersecting triangle or separated triangle.
Hence , the vertices of a singular continous polygon which is mathematically equivalent to the 2 components is -
(0,0),(1x,1y),(2x,2y),(3x,3y),(1x,1y),(0,0),(Ax,Ay),(Bx,By),(Cx,Cy),(Ax,Ay),(0,0)
To understand how it works, try drawing this mathematically equivalent polygon on a scratch pad.-
1. Mark all the vertices but don't join them yet.
2. Mark the repeated vertices separately also. Do this by marking them close to the original points, but not on them. (at a distance e, where e->0 (tends to/approaches) ) (to help visualize)
3. Now join all the vertices in the right order (as in the example above)
You will notice that this forms a continuous polygon and only becomes disjoint at the e=0 limit.
You can now send this mathematically equivalent polygon to your ray casting function (and maybe even winding number function?) without any issues.
I have two polygons and I want to get the minimal distance and the points between this distance is measured. Of course such a point could very well lie on the edge between two nodes.
Here is an example:
I am looking for an algorithm that gives me the green distance and the two points.
If the polygons do not intersect, you could do this:
If you have polygon A and polygon B and A[i] and B[j] are the vertexes of A and B respectively. Then you could compute the closest distance from A[i] to each segment of B (you could use something like this, but take into account that you'd be working with segments, so you have to work with the starting and ending point of the segment).
Then you have to do the same but from all B[j] to all segments of A.
And finally take the smallest one.
Just remember my previous comment: consider the starting and ending point of the segment while computing the shortest distance to the line this segment is in, because the intersection point could be out of the segment. Look here to check this last thing. If the point is out keep the closest edge of the segment)
Regards