I'm working on a game where I create a random map of provinces (a la Risk or Diplomacy). To create that map, I'm first generating a series of semi-random points, then figuring the Delaunay triangulations of those points.
With that done, I am now looking to create a Voronoi diagram of the points to serve as a starting point for the province borders. My data at this point (no pun intended) consists of the original series of points and a collection of the Delaunay triangles.
I've seen a number of ways to do this on the web, but most of them are tied up with how the Delaunay was derived. I'd love to find something that doesn't need to be integrated to the Delaunay, but can work based off the data alone. Failing that, I'm looking for something comprehensible to a relative geometry newbie, as opposed to optimal speed. Thanks!
The Voronoi diagram is just the dual graph of the Delaunay triangulation.
So, the edges of the Voronoi diagram are along the perpendicular bisectors of the edges of the Delaunay triangulation, so compute those lines.
Then, compute the vertices of the Voronoi diagram by finding the intersections of adjacent edges.
Finally, the edges are then the subsets of the lines you computed which lie between the corresponding vertices.
Note that the exact code depends on the internal representation you're using for the two diagrams.
If optimal speed is not a consideration, the following psuedo code will generate a Voronoi diagram the hard way:
for yloop = 0 to height-1
for xloop = 0 to width-1
// Generate maximal value
closest_distance = width * height
for point = 0 to number_of_points-1
// calls function to calc distance
point_distance = distance(point, xloop, yloop)
if point_distance < closest_distance
closest_point = point
end if
next
// place result in array of point types
points[xloop, yloop] = point
next
next
Assuming you have a 'point' class or structure, if you assign them random colours, then you'll see the familiar voronoi pattern when you display the output.
After trying to use this thread as a source for answers to my own similar question, I found that Fortune's algorithm — likely because it is the most popular & therefore most documented — was the easiest to understand.
The Wikipedia article on Fortune's algorithm keeps fresh links to source code in C, C#, and Javascript. All of them were top-notch and came with beautiful examples.
Each of your Delaunay triangles contains a single point of the Voronoi diagram.
You can compute this point by finding the intersection of the three perpendicular bisectors for each triangle.
Your Voronoi diagram will connect this set of points, each with it's nearest three neighbors. (each neighbor shares a side of the Delaunay triangle)
How do you plan on approaching the edge cases?
Related
I am dealing with a reverse-engineering problem regarding road geometry and estimation of design conditions.
Suppose you have a set of points obtained from the measurement of positions of a road. This road has straight sections as well as curve sections. Straight sections are, of course, represented by lines, and curves are represented by circles of unknown center and radius. There are, as well, transition sections, which may be clothoids / Euler spirals or any other usual track transition curve. A representation of the track may look like this:
We know in advance that the road / track was designed taking this transition + circle + transition principle into account for every curve, yet we only have the measurement points, and the goal is to find the parameters describing every curve on the track, this is, the transition parameters as well as the circle's center and radius.
I have written some code using a nonlinear optimization algorithm, where a user can select start and end points and fit a circle that to the arc section between them, as it shows in next figure:
However, I don't find a suitable way to take the transition into account. After giving it some thought I came to think that this s because, given a set of discrete points -with their measurement error- representing a full curve, it is not entirely clear where to consider it "begins" and where it "ends" and, moreover, it is less clear where to consider the transition, the proper circle and the exit transition "begin" and "end".
Is there any work on this subject which I may have missed? is there a proper way to fit the whole transition + curve + transition structure into the set of points?
As far as I know, there's no method to fit a sequence clothoid1-circle-clothoid2 into a given set of points.
Basic facts are that two points define a straight, and three points define a unique circle.
The clothoid is far more complex, because you need: The parameter A, the final radius Rf, an initial point px,py, the radius Ri at that point, and the tangent T (angle with X-axis) at that point.
These are 5 data you may use to find the solution.
Due to clothoid coords are calculated by expanded Fresnel integrals (see https://math.stackexchange.com/a/3359006/688039 a little explanation), and then apply a translation & rotation, there's no an easy way to fit this spiral into a set of given points.
When I've had to deal with your issue, what I've done is:
Calculate the radius for triplets of consecutive points: p1p2p3, p2p3p4, p3p4p5, etc
Observe the sequence of radius. Similar values mean a circle, increasing/decreasing values mean a clothoid; Big values would mean a straight.
For each basic element (line, circle) find the most probably characteristics (angles, vertices, radius) by hand or by some regression method. Many times the common sense is the best.
For a spiral you may start with aproximated values, taken from the adjacent elements. These values may very well be the initial angle and point, and the initial and final radius. Then you need to iterate, playing with Fresnel and 'space change' until you find a "good" parameter A. Then repeat with small differences in the other values, those you took from adjacents.
Make the changes you consider as good. For example, many values (A, radius) use to be integers, without decimals, just because it was easier for the designer to type.
If you can make a small applet to do these steps then it's enough. Using a typical roads software helps, but doesn't avoid you the iteration process.
If the points are dense compared to the effective radii of curvature, estimate the local curvature by least square fitting of a circle on a small number of points, taking into account that the curvature is most of the time zero.
You will obtain a plot with constant values and ramps that connect them. You can use an estimate of the slope at the inflection points to figure out the transition points.
I have a rectangle R of side lenghts lx (horizontal) and lz (vertical) and area of A = lx*lz. In this rectangle, there are N randomly distributed points. I would like to split A into N sub-areas, where each sub-area is determined based on circles of growing radius around each point of the point cloud. As soon as two growing circles intersect, the resulting radical line should demark the local border between the sub-areas of two points. The borders of rectangle R should additionally delimit the sub-areas.
The expected result is sketched in this Figure:
Expected sub-areas in an example with 6 points
The approximate procedure is artistically illustrated in this short Youtube movie:
https://www.youtube.com/watch?v=BXNvcQTNWXM&ab_channel=stopmotionkim
To find the sub-areas, I am looking for an approximate or exact algorithm that (first priority) is efficient and scales well with N (better than O[N^2], ideally O[N], although I doubt that this is possible), and (second priority), if approximate, is as accurate as possible.
Does anybody know how to do this or has a hint how one could start? Thanks a lot for your help!
What you are looking for is Clipped Voronoi Diagram. It can be calculated in O(nlogn). I suggest you to look at those papers/courses to have a better idea of the underlying theory and computational methods:
Efficient Computation of 3D Clipped Voronoi Diagram
Computational Geometry
For a Python implementation, see this post
I would like to find two vertices of two meshes (1 vertex per mesh) that define the closest distance between them. Or the two triangles would be fine I guess.
However I'm not sure how to search for this in CGAL's documentation, I'm sure that this is doable with some existing tool (probably based on a 3d distance field and/or AABBs). Could I please get a hint (keywords/link) on what to look for?
I've been pointed to the Optimal Distances CGAL package, but it's not exactly what I want, since it outputs the distance and the coordinates, so finding the vertex ID is an additional computational overhead.
I've already implemented a collision detection with CGAL to find the triangle-triangle intersection in a triangle-soup, using AABB-trees. I guess that I should be somehow close to this, although now a simple soup with all me object-triangles wouldn't do the job.
The solution found was this:
CGAL's Optimal Distances package can give an approximation of the closest distance between the convex hulls of two meshes, without explicitly computing the hulls. As a result one gets the shortest distance between these hulls, and the coordinates of the 2 points that lie on them and define this distance.
Then these coordinates can be used as a search-query in kd-trees that contains the original vertices of the meshes in order to find the closest vertices.
In case one mesh is non-convex, the hull that CGAL is using is very approximate, so convex decomposition might be necessary. In such a case one would have to check distances for each convex part and then take the shortest distance.
The above would result in something like this:
enter link description here
There are N points on a 2D grid (x,y). I need to find the shortest path, from point A to point B, but I can only travel from one point to another and I can't travel between two points if the distance between them is farther than a distance D. I thought it might be solved by using some kind of modified Dijkstra's algorithm, but I'm not sure how, because I've never implemented it before, just studied it on Wiki.
Well, Dijkstra finds shortest paths in graphs. So just consider the grid points to be nodes in a graph with edges between each node S and all other nodes T such that dist(S, T) <= D. You don't have to actually construct the graph because the edges are easily determined as needed by Dijkstra. Just check all nodes in a square around S with radius D. A S-T edge exists iff (Sx - Tx)^2 (Sy - Ty)^2 <= D^2.
Wiki explanation is sufficient for this.
Dijkstra's algorithm takes 3 inputs. The Graph, Starting node and Ending node.
To construct the graph just do this
For i 1..n in points
For j i+1..n in points
if(dist(points[i],points[j])<=D)
add j to childs of i
add i to childs of j
After constructing the graph, perform dijkstra.
The subtlety of a question like this lies in a critical definition - what is the measure of distance in your grid?
The are many different shortest path problems and solutions, and they are studied throughout mathematics. They are each characterised by the 'topology' of the area being searched. Consider a few distinct topologies with their own solutions:
A one sided piece of paper
Suppose your grid represents coordinates on a piece of paper - the shortest path is easy to find, as it is simply a straight line between those points.
The surface of the moon
If your grid represents locations on the moon in terms of latitude and longitude, the shortest path is an arc along the moon's surface - If you drove "in a straight line" between two points on the moon, you would be travelling in an arc, because of the moon's curvature.
Road Intersections
If you want to find the distance between two intersections in a grid of roads, where the traffic on each road has a different speed, and you can only travel along the roads, then you can find the shortest path using Dijkstra's algorithm.
One way road intersections
A slight variation of the above - we only need to consider roads in one direction. There might not be any paths in this case.
Summary
To give a good solution, we need to understand the topology of your grid. If the distance is pythagerous's theorem than that indicates euclidean geometry (like in the piece of paper example), so the solution is a straight line.
Is it possible you mean that you can travel between any two points if the are closer than D - like flying a plane between airports, for example?
EDIT: I didn't see your comment because you didn't use #. In your case your grid is like the airports a plane can fly between. The shortest path is found using Dijkstra's algorithm - the immediate neighbours of a point are all points closer than D. Find them, represent it all as a graph, and use Dijkstra's algorithm.
I would suggest using the formula to find the distance between 2 points i.e sqrt((x2-x1)^2+(y2-y1)^2). This distance is always the shortest between 2 points.
I'm trying to find/make an algorithm to compute the intersection (a new filled object) of two arbitrary filled 2D objects. The objects are defined using either lines or cubic beziers and may have holes or self-intersect. I'm aware of several existing algorithms doing the same with polygons, listed here. However, I'd like to support beziers without subdividing them into polygons, and the output should have roughly the same control points as the input in areas where there are no intersections.
This is for an interactive program to do some CSG but the clipping doesn't need to be real-time. I've searched for a while but haven't found good starting points.
I found the following publication to be the best of information regarding Bezier Clipping:
T. W. Sederberg, BYU, Computer Aided Geometric Design Course Notes
Chapter 7 that talks about Curve Intersection is available online. It outlines 4 different approaches to find intersections and describes Bezier Clipping in detail:
https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=1000&context=facpub
I know I'm at risk of being redundant, but I was investigating the same issue and found a solution that I'd read in academic papers but hadn't found a working solution for.
You can rewrite the bezier curves as a set of two bi-variate cubic equations like this:
∆x = ax₁*t₁^3 + bx₁*t₁^2 + cx₁*t₁ + dx₁ - ax₂*t₂^3 - bx₂*t₂^2 - cx₂*t₂ - dx₂
∆y = ay₁*t₁^3 + by₁*t₁^2 + cy₁*t₁ + dy₁ - ay₂*t₂^3 - by₂*t₂^2 - cy₂*t₂ - dy₂
Obviously, the curves intersect for values of (t₁,t₂) where ∆x = ∆y = 0. Unfortunately, it's complicated by the fact that it is difficult to find roots in two dimensions, and approximate approaches tend to (as another writer put it) blow up.
But if you're using integers or rational numbers for your control points, then you can use Groebner bases to rewrite your bi-variate order-3 polynomials into a (possibly-up-to-order-9-thus-your-nine-possible-intersections) monovariate polynomial. After that you just need to find your roots (for, say t₂) in one dimension, and plug your results back into one of your original equations to find the other dimension.
Burchburger has a layman-friendly introduction to Groebner Bases called "Gröbner Bases: A Short Introduction for Systems Theorists" that was very helpful for me. Google it. The other paper that was helpful was one called "Fast, precise flattening of cubic Bézier path and offset curves" by TF Hain, which has lots of utility equations for bezier curves, including how to find the polynomial coefficients for the x and y equations.
As for whether the Bezier clipping will help with this particular method, I doubt it, but bezier clipping is a method for narrowing down where intersections might be, not for finding a final (though possibly approximate) answer of where it is. A lot of time with this method will be spent in finding the mono-variate equation, and that task doesn't get any easier with clipping. Finding the roots is by comparison trivial.
However, one of the advantages of this method is that it doesn't depend on recursively subdividing the curve, and the problem becomes a simple one-dimensional root-finding problem, which is not easy, but well documented. The major disadvantage is that computing Groebner bases is costly and becomes very unwieldy if you're dealing with floating point polynomials or using higher order Bezier curves.
If you want some finished code in Haskell to find the intersections, let me know.
I wrote code to do this a long, long time ago. The project I was working on defined 2D objects using piecewise Bezier boundaries that were generated as PostScipt paths.
The approach I used was:
Let curves p, q, be defined by Bezier control points. Do they intersect?
Compute the bounding boxes of the control points. If they don't overlap, then the two curves don't intersect. Otherwise:
p.x(t), p.y(t), q.x(u), q.y(u) are cubic polynomials on 0 <= t,u <= 1.0.
The distance squared (p.x - q.x) ** 2 + (p.y - q.y) ** 2 is a polynomial on (t,u).
Use Newton-Raphson to try and solve that for zero. Discard any solutions outside 0 <= t,u <= 1.0
N-R may or may not converge. The curves might not intersect, or N-R can just blow up when the two curves are nearly parallel. So cut off N-R if it's not converging after after some arbitrary number of iterations. This can be a fairly small number.
If N-R doesn't converge on a solution, split one curve (say, p) into two curves pa, pb at t = 0.5. This is easy, it's just computing midpoints, as the linked article shows. Then recursively test (q, pa) and (q, pb) for intersections. (Note that in the next layer of recursion that q has become p, so that p and q are alternately split on each ply of the recursion.)
Most of the recursive calls return quickly because the bounding boxes are non-overlapping.
You will have to cut off the recursion at some arbitrary depth, to handle weird cases where the two curves are parallel and don't quite touch, but the distance between them is arbitrarily small -- perhaps only 1 ULP of difference.
When you do find an intersection, you're not done, because cubic curves can have multiple crossings. So you have to split each curve at the intersecting point, and recursively check for more interections between (pa, qa), (pa, qb), (pb, qa), (pb, qb).
There are a number of academic research papers on doing bezier clipping:
http://www.andrew.cmu.edu/user/sowen/abstracts/Se306.html
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.61.6669
http://www.dm.unibo.it/~casciola/html/research_rr.html
I recommend the interval methods because as you describe, you don't have to divide down to polygons, and you can get guaranteed results as well as define your own arbitrary precision for the resultset. For more information on interval rendering, you may also refer to http://www.sunfishstudio.com