I have a 2D layer/section containing the 2D polygons colored in shades of green.
I have another layer containing the 2D polygons colored in shades of blue.
I intend to figure out how different are the two layers. Maybe extracting the layers differential. Can anybody provide an algorithm, library or a hint? I took a look at CGAL, but not quite sure which package to try out.
Unclear the kind of comparison you want to do.
Assuming you want the geometric intersection of the triangulations, an easy solution is to intersect all polygons pairwise. (If the number of triangles is large, use some acceleration technique to avoid all comparisons.)
If you need the triangulated intersection, you can use a clipping algorithm (Sutherland–Hodgman) and triangulate the resulting polygons, which can range in shape from triangles to hexagons. Fortunately, they are convex.
Related
I got an outline (list of points) for a plane I want to generate. The plane is quite big and I need evenly distributed vertices inside the outline. Each vertex has a color value from red to green to visualize some data in the plane. I need to visualize the data as precise as possible in real time.
My idea was to simply create a grid and adjust all the vertices outside of the outline. This turned out to be quite complex.
This is a quick example what I want to achieve.
Is there any algorithm that solves this problem?
Is there another way to generate a mesh from an outline with evenly distributed vertices?
It sounds like you want to do something like this:
1) First generate a triangulate your polygon to create a mesh. There are plenty of options: https://en.wikipedia.org/wiki/Polygon_triangulation
2) Then while any of the edges in the mesh are too long (meaning that the points at either end might be too far apart), add the midpoint of the longest edge to the mesh, dividing the adjacent triangles into 2.
The results is a mesh with every point within a limited distance of other points in every direction. The resulting mesh will not necessarily be optimal, in that it may have more points than are strictly required, but it will probably satisfy your needs.
If you need to reduce the number of points and thin triangles, you can apply Delaunay Triangulation flipping around each candidate edge first: https://en.wikipedia.org/wiki/Delaunay_triangulation#Visual_Delaunay_definition:_Flipping
Although not totally clear from the question, the marching cubes algorithm, adapted to two dimensions, comes to mind. A detailed descriptione of the two-dimensional version can be found here.
Delaunay meshing can create evenly distributed vertices inside a shape. The image below shows a combined grid- and Delaunay-mesh. You may have a look here.
I need an algorithm for splitting a (convex) polygon (let's call it base polygon). The polygon should be splitted into several smaller polygons by another polygon's edges (let's call this one the splitting polygon).
I know there exist algorithms for clipping a polygon (f.e. the Sutherland-Hodgman algorithm), but these algorithms discard the vertices that are lying outside of the splitting polygon instead of creating new polygons with them. I don't want to clip a polygon, i want to split it into several small parts.
I know the answer seems quite obvious because I would just have to extend the existing algorithms.
The problem is that I can't figure out a nice and performant way of doing this.
Are there existing algorithms that describe how to best split a polygon in a performant way?
There has to be a simple solution for this problem that I can't figure out at the moment.
You can see this problem as that of finding the intersection of the subject polygon and the complement of the window polygon. So you can use the standard Surtherland-Hodgman algorithm for that purpose, taking two precautions:
swap the roles of the subject and the window (actually your window is not convex as you consider its complement, only the subject is convex),
embed the window polygon in a large bounding box that covers both polygons, and consider the box as a polygon with a hole.
Example: the subject polygon is the rectangle and the window is the pentagon. Form the green polygon (larger rectangle with a hole) and clip it inside the rectangular (convex) window. The result of the clipping is in blue.
With some extra care, it should be possible to do without the large bounding box.
I want to do the following: I have some faces in the 3D space as polygons. I have a projection direction and a projection plane. I have a convex clipping polygon in the projection plane. I wnat to get a polygon representing the shaddow of all the faces clipped on the plane.
What I do till now: I calculate the projections of the faces as polygons in the projection plane.
I could use the Sutherland–Hodgman algorithm to clip all the singe projected polygons to clip to the desired area.
Now my question: How can I combine the projected (maybe clipped) polygons together? Do I have to use algorithms like Margalit/Knott?
The algorithm should be quite efficient because it has to run quite often. So what algorithm do you suppose?
Is it maybe possible to modify the algorithm of Sutherland–Hodgman to solve the merging problem?
I'm currently implementing this algorithm (union of n concave polygons) using Bentley–Ottmann to find all edge intersections and meanwhile keeping track of the polygon nesting level on both sides of edge segments (how many overlapping polygons each side of the line is touching). Edges that have a nesting level of 0 on one side are output to the result polygon. It's fairly tricky to get done right. An existing solution with a different algorithm design can be found at:
http://sourceforge.net/projects/polyclipping/
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.
I've got a bunch of overlapping triangles from a 3D model projected into a 2D plane. I need to merge each island of touching triangles into a closed, non-convex polygon.
The resultant polygons shouldn't have any holes in them (since the source data doesn't).
Many of the source triangles share (floating point identical) edges with other triangles in the source data.
What's the easiest way to do this? Performance isn't particularly important, since this will be done at design time.
Try gpc, or the General Polygon Clipper Library.
Imagine the projection onto a plane as a "view" of the model (i.e. the direction of projection is the line of sight, and the projection is what you see). In that case, the borders of the polygons you want to compute correspond to the silhouette of the model.
The silhouette, in turn, is a set of edges in the model. For each edge in the silhouette, the adjacent faces will have normals that either point away from the plane or toward the plane. You can check this be taking the dot product of the face normal with the plane normal -- look for edges whose adjacent face normals have dot products of opposite signs with the projection direction.
Once you have found all the silhouette edges you can join them together into the boundaries of the desired polygons.
Generally, you can find more about silhouette detection and extraction by googling terms like mesh silouette finding detection. Maybe a good place to start is here.
I've also found this[1] approach, which I will be trying next.
[1] 2d outline algorithm for projected 3D mesh