I have a number of 2D (possibly intersecting) polygons which I rendered using OpenGL ES on the screen. All the polygons are completely contained within the screen. What is the most timely way to find the percentage area of the union of these polygons to the total screen area? Timeliness is required as I have a requirement for the coverage area to be immediately updated whenever a polygon is shifted.
Currently, I am representing each polygon as a 2D array of booleans. Using a point-in-polygon function (from a geometry package), I sample each point (x,y) on the screen to check if it belongs to the polygon, and set polygon[x][y] = true if so, false otherwise.
After doing that to all the polygons in the screen, I loop through all the screen pixels again, and check through each polygon array, counting that pixel as "covered" if any polygon has its polygon[x][y] value set to true.
This works, but the performance is not ideal as the number of polygons increases. Are there any better ways to do this, using open-source libraries if possible? I thought of:
(1) Unioning the polygons to get one or more non-overlapping polygons. Then compute the area of each polygon using the standard area-of-polygon formula. Then sum them up. Not sure how to get this to work?
(2) Using OpenGL somehow. Imagine that I am rendering all these polygons with a single color. Is it possible to count the number of pixels on the screen buffer with that certain color? This would really sound like a nice solution.
Any efficient means for doing this?
If you know background color and all polygons have other colors, you can read all pixels from framebuffer glReadPixels() and simply count all pixels that have color different than background.
If first condition is not met you may consider creating custom framebuffer and render all polygons with the same color (For example (0.0, 0.0, 0.0) for backgruond and (1.0, 0.0, 0.0) for polygons). Next, read resulting framebuffer and calculate mean of red color across the whole screen.
If you want to get non-overlapping polygons, you can run a line intersection algorithm. A simple variant is the Bentley–Ottmann algorithm, but even faster algorithms of O(n log n + k) (with n vertices and k crossings) are possible.
Given a line intersection, you can unify two polygons by constructing a vertex connecting both polygons on the intersection point. Then you follow the vertices of one of the polygons inside of the other polygon (you can determine the direction you have to go in using your point-in-polygon function), and remove all vertices and edges until you reach the outside of the polygon. There you repair the polygon by creating a new vertex on the second intersection of the two polygons.
Unless I'm mistaken, this can run in O(n log n + k * p) time where p is the maximum overlap of the polygons.
After unification of the polygons you can use an ordinary area function to calculate the exact area of the polygons.
I think that attempt to calculate area of polygons with number of pixels is too complicated and sometimes inaccurate. You can see something similar in stackoverflow answer about calculation the area covered by a polygon and if you construct regular polygons see area of a regular polygon ,
Related
I have two dataframes. One contains a column of Polygons, taken from an image of polygon shapes. Each polygon has a set of coordinates. This dataframe also has a "segment-id" column. I have another dataframe, containing a column of Points, also with coordinates. These Points represent pixels from the same image of Polygon shapes, and therefore have the same coordinate system. I want to give every Point the "segment-id" of the Polygon which contains it. Every Polygon contains at least one Point.
Currently, I achieve this by using a nested for loop:
for i, row in enumerate(point_df.itertuples(), 0):
point = pixel_df.at[i, 'geometry']
for j in range(len(polygon_df)):
polygon = polygon_df.iat[j, 0]
if polygon.contains(point):
pixel_df.at[i, 'segment_id'] = polygon_df.at[j, 'segment_id']
else:
pass
This is extremely slow. For 100 Points, it takes around 10 seconds. I need a faster way of doing this. I have tried using apply but it is still super slow.
Hope someone can help me out, thanks very much.
For fast "is point inside polygon":
Preparation: In the code that obtains the data describing the polygons; using all the vertices, find the minimum and maximum y-coord, and minimum and maximum x-coord; and store that with the polygon's data.
1) Using the point's coords and the polygon's "minimum and maximum x and y" (pre-determined during preparation); do a "bounding box" test. This is just a fast way to find out if the point is definitely not inside the polygon (so you can skip the more expensive steps most of the time).
2) Set a "yes/no" flag to "no"
3) For each edge in the polygon; determine if a horizontal line passing through the point would intersect with the edge, and if it does determine the x-coord of the intersection. If the x-coord of the intersection is less than the point's x-coord, toggle (with NOT) the "yes/no" flag. Ignore "horizontal line passes through a vertex" during this step.
4) For each vertex, compare its y-coord with the point's y-coord. If they're the same you need to look at both edges coming from that vertex to determine if the edge's vertices are in the same y direction. If the edge's vertices are in the same y direction (if the edges form a 'V' shape or upside-down 'V' shape) ignore the vertex. Otherwise (if the edges form a '<' or '>' shape), if the vertex's x-coord is less than the point's x-coord, toggle the "yes/no" flag.
After all this is done; that "yes/no" flag will tell you if the point was in the polygon.
I am currently creating a feature and patterning it across a flat plane to get the maximum number of features to fit on the plane. I do this frequently enough to warrant building some sort of marcro for this if possible. The issue that I run into is I still have to manually set the spacing between the parts. I want to be able to create a feature and have it determine "best" fit spacing given an area while avoiding overlaps. I have had very little luck finding any resources describing this. Any information or links to potentially helpful resources on this would be much appreciated!
Thank you.
Before, you start the linear pattern bit:
Select the face2 of that feature2, get the outer most loop2 of edges. You can test for that using loop2.IsOuter.
Now:
if the loop has one edge: that means it's a circle and the spacing must superior to the circle's radius
if the loop has more that one edge, that you need to calculate all the distances between the vertices and assume that the largest distance is the safest spacing.
NOTA: If one of the edges is a spline, then you need a different strategy:
You would need to convert the face into a sketch and finds the coordinates of that spline to calculate the highest distances.
Example: The distance between the edges is lower than the distance between summit of the splines. If the linear pattern has the a vertical direction, then spacing has to be superior to the distance between the summit.
When I say distance, I mean the distance projected on the linear pattern direction.
I need to draw an enclosing polygon of a group of rectangles that are placed next to each other.
Let's think of text fields that share at least one edge (or part of it) with at least one of the other rectangles.
I can get the rectangles points coordinates, and so I basically have any data I need about them.
Can you think of a simple algorithm / procedure to draw a polygon (connected straight paths) around these objects.
Here's a demonstration of different potential cases (A, B, C, etc...). In example A I also drew a blue polygon which is the path that I need to draw, outlining the group of rectangles.
I've read here about convex hull and stuff like that but really, this looks like a far simpler problem.
One (beginning of) solution I thought of was that the points I actually need to draw through are only ones that are not shared by any pair of rectangles, meaning points that are vertices of more than one rectangle are redundant. What I couldn't find out was the order by which I need to draw lines from one to the next.
I currently work on objective c, but any other language or algo would be appreciated, including pseudo.
Thanks!
IMHO it should be like this. Make a list of edged and see if some are overlaying: This should be simple if the rectangles are aligned with the x,y axis. You just find the edges that have the vertexes on the same x or y and the other coordinates need to be in between. After this the remaining edges should form the outline.
Another method to find common edges is to break all rectangles along each x and y axis where you have vertices. This should look as if you are growing all lines to infinity. After this all common edges will have common vertices and can be eliminated.
You have two rows, and three different y-values. Let's say y0 is the top of the thing, y2 is the bottom end, and y1 marks the middle between both rows.
Each row has a maximum and a minimum x-value, let's say the top-row goes from x0_min to x0_max, and the bottom row from x2_min to x2_max. Given those values you just draw around the thing:
(x0_min,y0)->
(x0_max,y0)->
(x0_max,y1)->
(x2_max,y1)->
(x2_max,y2)->
(x2_min,y2)->
(x2_min,y1)->
(x0_min,y1)->
(x0_min,y0)
I'm needing to implement a Minkowski sum function that can return the Minkowski sum of either 2 circles, 2 convex polygons or a circle and a convex polygon. I found this thread that explained how to do this for convex polygons, but I'm not sure how to do this for a circle and polygon. Also, how would I even represent the answer?! I'd like the algorithm to run in O(n) time but beggars can't be choosers.
Circle is trivial -- just add the center points, and add the radii. Circle + ConvexPoly is nearly as simple: move each segment perpendicularly outward by the circle radius, and connect adjacent segments with circular arcs centered at the original poly vertices. Translate the whole by the circle center point.
As for how you represent the answer: Well, it depends on what you want to do with it. You could convert it to a NURBS if you just want to draw it with a vector drawing library. You could approximate the circular arcs with polylines if you just want a polygonal approximation. Or you might store it as is -- "this polygon, expanded by such-and-such a radius". That would be the best choice for things like raycasting, for instance. Or as a compromise, you could connect adjacent segments linearly instead of with circular arcs, and store it as the union of the (new) convex polygon and a list of circles at the vertices.
Oh, about ConvexPoly + ConvexPoly. That's the trickiest one, but still straightforward. The basic idea is that you take the list of segment vectors for each polygon (starting from some particular extremal point, like the point on each poly with the lowest X coordinate), then merge the two lists together, keeping it sorted by angle. Sum the two points you started with, then apply each vector from the merged vector list to produce the other points.
I have a large set of overlapping circles each at a random location with a specific radius.
type Circle =
struct
val x: float
val y: float
val radius: float
end
Given a new point with type
type Point =
struct
val x: float
val y: float
end
I would like to know which circles in my set enclose the new point. A linear search is trivial. I'm looking for a structure that can hold the circles and return the enclosing circles with better than O(N) for the presented point.
Ideally the structure should be fast for insertion of new circles and removal of circles as well.
I would like to implement this in F# but ideas in any language are fine.
For your information I'm looking to implement
http://takisword.wordpress.com/2009/08/13/bowyerwatson-algorithm/
but it would be an O(N^2) if I use the naive approach of scanning all circles for every new point.
If we assume that circles are distributed over some rectangle with area 1 and the average area of a circle is a then a quadtree with m levels will leave you with an area with size 1/2^m. This leaves
O(Na/2^m)
as the expected number of circles left in the remaining area.
However, we have done O(log(m)) comparisons to get to this point. This leaves the total number of comparisons as
O(log(m)) + O(N/2^m)
The second term will be constant if log(m) is proportional to N.
This suggests that a quadtree can cut things down to O(log n)
Quadtree is a structure for efficient plane search. You can use it to hold subdivision of the plane.
For example you can create quad tree with such properties:
1. Every cell of quadtree contains indices of circles, overlapping it.
2. Every cell does contain not more than K circles (for example 10) // may be broken
3. Height of tree is bounded by M (usually O(log n))
You can construct quadtree, by iterating overlapped cells, and if number of circles inside cell exceedes K, then subdivide that cell into four (if not exceeding max height). Also something should be considered in case of cell inside circles, because its subdivision is pointless.
When finding circles you should localise quadtree, then iterate through overlapping circles and find, those which contains point.
In case of sparse circle distribution search will be very efficient.
I have a bachelor thesis, where I adapted quadtree, for closest segment location, with expected time O(log n), I think similar approach could be used here
Actually you search for triangles whose circumcircles include the new point p. Thus your Delaunay triangulation is already the data structure you need: First search for the triangle t which includes p (google for 'delaunay walk'). The circumcircle of t certainly includes p. Then start from t and grow the (connected) area of triangles whose circumcircles include p.
Implementing it in a fast an reliable way is a lot of work. Unless you want to create a new library you may want to use an existing one. My approach for C++ is Fade2D [1] but there are also many others, it depends on your specific needs.
[1] http://www.geom.at/fade2d/html/