I have several 2 dimensional circles that I want to draw a border around. I've done this using a convex hull before, but my goal is to make the border almost like a surrounding "blob". I attached a picture to show what I mean.
Essentially, I want the border to outline the circles, and be pulled slightly into the middle of the area if no circles are present. The center shape shows my current train of thought -- create normal lines for each circle, and somehow merge them into a complete shape.
Summed up, I have 2 questions:
1. Are there any existing algorithms to do this?
2. If not, are there any algorithms that would help me merge the circle outlines into a single larger path?
Thank you!
"Everything You Always Wanted to Know About Alpha Shapes But Were Afraid to Ask" is for you http://cgm.cs.mcgill.ca/~godfried/teaching/projects97/belair/alpha.html
One way to get this border could be to simply compute the distance to the centers of circles: for a given point this distance is the minimum of the distances from this point to all the centers of the given circles. Then sample this distance function over a regular grid. And finally extract the f-level of this function as a collection of polylines with an isocurve extraction algorithm (like Marching Squares). f should be the radius of the circles augmented with the desired margin.
Related
I have a situation where I have a set of pixels that make up the border of a quadrilateral (very close to square). I'm trying to determine the location of the corners as best as possible and have been struggling for a while now. My first thought was to determine the straight lines of the border and then calculate the corner points, but I don't have access to OpenCV or other image processing libraries, unfortunately.
Below are three cases where the black outline is the image boundary and the red outline is the quadrilateral boundary. I have a list of all of the pixels that make up the red boundary and the red boundary thickness may vary.
My initial thought was that I could just find the pixel that is closest to each of the four image boundaries, however this won't quite work for the first case where the inner quadrilateral isn't tilted.
Any thoughts on how to tackle this problem would be great. I'm coding in dart, but am looking for a psuedocode answer that I can implement myself.
(I have seen this post, which is similar to my problem, but I think there should be a simpler solution for my problem since I have access to all of the boundary points of the quadrilateral)
Having a list of all rectangle boundary pixels, you can use simple methods like this:
Calculate gravity center of rectangle (just sum X- and Y- coordinates of pixels and divide by their number) - it is diagonal intersection.
Find the farthest pixels - they are corners.
In case of bad quality of data set (empty places, excessive pixels) center calculation might be inexact. So you can apply Hough transform to extract sides (as lines) and calculate their intersections.
I have a shape (in black below) and a point inside the shape (red below). What's the algorithm to find the closest distance between my red point and the border of the shape (which is the green point on the graph) ?
The shape border is not a series of lines but a randomly drawn shape.
Thanks.
So your shape is defined as bitmap and you can access the pixels.
You could scan ever growing squares around your point for border pixels. First, check the pixel itself. Then check a square of width 2 that covers the point's eight adjacent pixels. Next, width 4 for the next 16 pixels and so on. When you find a border pixel, record its distance and check against the minimum distance found. You can stop searching when half the width of the square is greater than the current minimum distance.
An alternative is to draw Bresenham circles of growing radius around the point. The method is similar to the square method, but you can stop immediately when you have a hit, because all points are supposed to have the same distance to your point. The drawback is that this method is somewhat inaccurate, because the circle is only an approximation. You will also miss some pixels along the disgonals, because Bresenham circles have artefacts.
(Both methods are still quite brute-force and in the worst case of a fully black bitmap will visit every node.)
You need a criterion for a pixel on the border. Your shape is antialiassed, so that pixels on the border are smoothed by making them a shade of grey. If your criterion is a pixel that isn't black, you will chose a point a bit inside the shape. If you cose pure white, you'll land a bit outside. Perhaps it's best to chose a pixel with a grey value greater than 0.5 as border.
If you have to find the closest border point to many points for the same shape, you can preprocess the data and use other methods of [nearest-neighbour serach].
As always, it depends on the data, in this case, what your shapes are like and any useful information about your starting point (will it often be close to a border, will it often be near the center of mass, etc).
If they are similar to what you show, I'd probably test the border points individually against the start. Now the problem is how you find the border without having to edge detect the entire shape.
The problem is it appears you can have sharply concave borders (think of a circle with a tiny spike-like sliver jutting into it). In this case you just need to edge detect the shape and test every point.
I think these will work, but don't hold me to it. Computational geometry seems to be very well understood, so you can probably find a pro at this somewhere:
Method One
If the shape is well behaved or you don't mind being wrong try this:
1- Draw 4 lines (diving the shape into four quandrants). And check the distance to each border. What i mean by draw is keep going north until you hit a white pixel, then go south, west, and east.
2- Take the two lines you have drawn so far that have the closest intersection points, bisect the angle they create and add the new line to your set.
3- keep repeating step two until are you to a tolerance you can be happy with.
Actually you can stop before this and on a small enough interval just trace the border between two close points checking each point between them to refine the final answer.
Method Two (this wil work with the poorly behaved shapes and plays well with anti-aliasing):
1- draw a line in any direction until he hit the border (black to white). This will be your starting distance.
2- draw a circle at this distance noting everytime you go from black to white or white to black. These are your intersection points.
As long as you have more than two points, divide the radius in half and try again.
If you have no points increase your radius by 50% and try again (basically binary search until you get to two points - if you get one, you got lucky and found your answer).
3- your closet point lies in the region between your two points. Run along the border checking each one.
If you want to, to reduce the cost of step 3 you can keep doing step 2 until you get a small enough range to brute force in step 3.
Also to prevent a very unlucky start, draw four initial lines (also east, south, and west) and start with the smallest distance. Those are easy to draw and greatly reduce your chance of picking the exact longest distance and accidentally thinking that single pixel is the answer.
Edit: one last optimization: because of the symmetry, you only need to calculate the circle points (those points that make up the border of the circle) for the first quadrant, then mirror them. Should greatly cut down on computation time.
If you define the distance in terms of 'the minimum number of steps that need to be taken to reach from the start pixel to any pixel on the margin', then this problem can be solved using any shortest path search algorithm like bread first search or even better if you use A* search algorithm.
I'm trying to workout how to efficiently calculate the layout of a dynamic/random view.
The view will contain a number of circles. Each circle has a predetermined size. These sizes are all between a given maximum and minimum size.
I'm trying to find an efficient way of laying out the circles within a rect with a couple of conditions.
The circles mustn't overlap with the edge of the rect and the circles must have a minimum "spacing" between them.
The first method I came up with is to randomly generate coordinate pairs and place the biggest circle. Then randomly generate more coordinate pairs until a suitable one is generated for the next circle. And the next, and the next, and so on until all are drawn.
The problems with this are that it could potentially take a long time to complete. Each subsequent circle will take longer to place as there are fewer places that it can go.
Another problem is that it could be impossible to layout the view.
I'm sure there must be more efficient ways of doing this but I'm not sure where to begin.
The Formula must deal between the smallest possible square they need or from a other point of view, with an arrangement with the smallest possible density between the edgepoints. But your problem could be solved by sorting the circles by size and then start with the largest and arrange them step by step to the smallest because the larger are more bulky and the smaller fit easier in small corners by there nature.
Build triangles of circles, where 3 circles have a discribing space they use together. This triangle has 3 corners right? :) so messure/calc the angle of that corners and the corner with the nearest 90degree angle should be placed in a square corner, better to say the three circles should be places mirrored so that the circle with the fittest 90degree corner behind is the one who goes in the corner. If a 4th circle fits into the rest of this triangle, wonderful, if not you place exact this circle in it which is taken minimum outerspace of that triangle. because its not said that the next smaller circle is the one who fit's perfect, which also means you have a stack of not placed circles until one is found who fits better. after you find one you go upwards your circle-stack again and try to fit the next of it in one of the corners or you try to build the next triangle. and here you see how complex this is, damn! http://en.wikipedia.org/wiki/Malfatti_circles
But anyway, the more one of this triangles corners is 90° the less space it would take with this 3 circles in it.
An other concept could be to think about larger circles like space who leftover a triangle in one of its corners in relation to the rectangle. This triangle has a maximum space availible for a smaller circle. If there is no perfectly fitting circle your taken square-space grows up. So as far as i think about to handle this problem with imagined triangles to compare with fitting circles in it or taking triangle ranges from the square is the solutions base.
I have a kind of cutting problem. There is an irregular polygon that doesn't have any holes and a list of standard sized of rectangular tiles and their values.
I want an efficient algorithm to find the single best valued tile that fit in this polygon; or an algorithm that just says if a single tile can fit inside the polygon. And it should run in deterministic time for irregular polygons with less than 100 vertices.
Please consider that you can rotate the polygon and tiles.
Answers/hints for both convex and non-convex polygons are appreciated.
Disclaimer: I've never read any literature on this, so there might be a better way of doing this. This solution is just what I've thought about after having read your question.
A rectangle has two important measurements - it's height and it's width
now if we start with a polygon and a rectangle:
1: go around the perimeter of the polygon and take note of all the places the height of the rectangle will fit in the polygon (you can store this as a polygon*):
2: go around the perimeter of the new polygon you just made and take note of all the places the width of the rectangle will fit in the polygon (again, you can store this as a polygon):
3: the rectangle should fit within this new polygon (just be careful that you position the rectangle inside the polygon correctly, as this is a polygon - not a rectangle. If you align the top left node of the rectangle with the top left node of this new polygon, you should be ok)
4: if no area can be found that the rectangle will fit in, rotate the polygon by a couple of degrees, and try again.
*Note: in some polygons, you will get more than one place a rectangle can be fitted:
After many hopeless searches, I think there isn't any specific algorithm for this problem. Until, I found this old paper about polygon containment problem.That mentioned article, present a really good algorithm to consider if a polygon with n points can fit a polygon with m points or not. The algorithm is of O(n^3 m^3(n+m)log(n+m)) in general for two transportable and rotatable 2D polygon.
I hope it can help you, if you are searching for such an irregular algorithm in computational geometry.
This might help. It comes with the source code written Java
http://cgm.cs.mcgill.ca/~athens/cs507/Projects/2003/DanielSud/
I am interested in using shapes like these:
Usually a tangram is made of 7 shapes(5 triangles, 1 square and 1 parallelogram).
What I want to do is fill a shape only with tangram shapes, so at this point,
the size and repetition of shapes shouldn't matter.
Here's something I manually tried:
I am a bit lost on how to approach this.
Assuming I have a path (an ordered list/array of points of the outline),
I imagine I should try to do some sort of triangulation.
Is there such a thing as Deulanay triangulation with triangles constrained to 45 degrees
right angled triangles ?
A more 'brute' approach would be to add a bunch of triangles(45 degrees) and use SAT
for collision detection to 'fix' overlaps, and hopefully gaps will be avoided.
Since the square and parallelogram can be made of triangles(45 degrees) too, I imagine there
would be a nice clean geometric solution, right ?
How do I pack triangles(45 degrees) inside an arbitrary shape ?
Any ideas are welcome.
A few random thoughts (maybe they help you find a better solution) if you're using only the original sizes of the shapes:
as you point out, all shapes in the tangram can be made composed of e.g. the yellow or pink triangle (d-g-c), so try also thinking of a bottom-up approach such as first trying to place as many yellow triangles into your shape and then combine them into larger shapes if possible. In the worst case, you'll end up with a set of these smallest triangles.
any kind triangulation of non-polygons (such as the half-moon in your example) probably does not work very well...
It looks like you require that the shapes can only have a few discrete orientations. To find the best fit of these triangles into the given shape, I'd propose the following approximate solution: draw a grid of triangles (i.e. a square grid with diagonal lines) across the shape and take those triangles which are fully contained. This most likely will not give you the optimal coverage but then you could repeatedly shift the grid by a tenth of the grid size in horizontal and vertical direction and see whether you'll find something which covers a larger fraction of the original shape (or you could go in steps of 1/2 then 1/4 etc. of the original grid size in the spirit of a binary search).
If you allow any arbitrary scaling of the shapes you could approximate any (reasonably smooth ?) shape to arbitrary precision by adding smaller and smaller shapes. E.g. if you have a raster image, you can e.g. choose the size of the yellow triangle such that two of them make a pixel on the image and then you can represent any such raster image.