I am trying to read Fortune's description of his additively weighted Voronoi algorithm but I am experiencing some difficulty in understanding it.
I imagine the algorithm as being like that for regular voronoi diagrams with beach line parabolas of different sizes as to take into account the point weights but if that's the case I am not sure how the edges of the resulting weighted voronoi diagram aren't straight lines.
Can someone explain the additively weighted version of Fortune's algorithm?
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Given a polyhedron defined by a matrix of 3-Dimensional vertices and its faces(delaunay triangles), I want to be able to create a smooth 3-D object.
Is there any software that has built a built in function that would allow me to do this?
If not, I have found a paper that seems to describe what I want, but I am unable to fully understand the math. http://graphics.berkeley.edu/papers/Turk-MIS-2002-10/Turk-MIS-2002-10.pdf.
Here is an examples of what I am looking for.
Rabbit
One solution for "smoothing" geometry, if we state the problem a bit more formally, is to perform mean curvature flow on your mesh. Here are some search terms - "curve-shortening flow", "mean curvature flow", "willmore flow", "conformal curvature flow" ...
Image source: Keenan Crane. Context and permission
"Smoothness of a surface or curve is very hard to define. (For an empirical test on what people perceive as smooth see http://www.levien.com/phd/thesis.pdf#page=23).
If you only care about perceived smoothness, for example, smoother appearance while rendering in high resolution etc., an easier approach would be Catmull-Clark subdivision scheme.
The geometric intuition is quite simple. In the case of a 2D curve, in every instance, every point on a curve moves according to some function of the curvature at that point. If we let the curve or surface move like this for some time, it will start smoothing out areas with high curvature more and more, eventually becoming a circle (or a sphere in 3d) and then collapse to a point. So for smoothing usually we have to preserve areas or volumes.
One way to define it is in terms of some energy, and our goal is to minimise this energy on the mesh. For example willmore flow minimises the willmore energy. Sometimes this process is called fairing.
I am not aware of a prepackaged library or tool, that's freely available and open source for curvature flow.
Algorithms
2D only
K.Mikula, D.Sevcovic, "Tangentially stabilized Lagrangian algorithm for elastic curve evolution driven by intrinsic Laplacian of curvature",
pdf
2D and 3D
https://www.youtube.com/watch?v=Jhqlmcms04M.
Keenan Crane's page has more information on this and more examples too.
http://www.cs.cmu.edu/~kmcrane/Projects/ConformalWillmoreFlow/
2D and 3D (level set method)
https://math.berkeley.edu/~sethian/2006/level_set.html
I have been studying DDA and Bresenham algorithms for line drawing and am curious about one thing.In both the algorithms,we consider a pixel grid to be of unit size and perform further steps.My question is if I change my grid size to say 0.5*0.5 instead of 1*1 grid,will there be any changes reflected in the way both algorithms work.If yes,can somebody enlighten me as to what those changes will be in each algorithm respectively.Very curious to know.Thanks for your answers in advance.Anybody who can help,please do so,since it urgent.I have an exam and I want this concept clarified.Please.Thanks:)
The curve is in fact the trajectory of a bus, the curve is represented by many (up to a few thousand) discrete points on the curve (the points were recorded by a GPS device installed on the bus).
Input a point P, I need to find the closest point on the curve to the point P. The point P is usually no more than 30m away from the trajectory of the bus. Note, the closest point isn't necessary a point recorded by the GPS device, it could be a point somewhere between two recorded points.
First I need an algorithm to recover the trajectory from those recorded points. It would be great if the interpolated curve could show sharp turns made by the bus. Which curve is best for such task ? Is Bezier curve good enough ? And finally I have to calculate the closest point on the curve, of course the algorithm completely depends on the kind of curve chosen.
I'm doing some research, and don't have much knowledge in curve interpolation, so any suggestions are welcome.
For computing the trajectory from recorded points, I recommended using the centripetal or chord-length Catmull-Rom splines. See link for more details. Catmll-Rom splines are in fact special cubic Hermite curves, which can be easily converted into cubic Bezier curves. Please note that the result from Catmull-Rom spline is a G1 curve only in general. If you want the trajectory to be with higher continuity (such as C2), you can go with natural cubic splines or general B-spline interpolation. Whatever approach you take, it is advised to keep the spline's degree no higher than 5. Degree 3 is a popular choice.
Once you have the mathematical representation for the trajectory, you can compute the minimum distance between a given point P and the trajectory. In general, the squared distance between point P and a curve C(t) is represented as D(t) = |P-C(t)|^2. The minimum of D will happen at where its first derivative is zero, which means we have to find the root for the following equation:
dD/dt = 2*(P-C(t)).C'(t) =0
When C(t) is of degree 3, dD/dt will be of degree 5. This is the reason why it is recommended to use a low degree curve earlier.
There are many literatures or online materials talking about how to find the root of a polynomial (of any degree) efficiently and robustly. Here is another SO post that might be useful.
Can someone help me to understand the purpose of delaunay triangulation? what is it use for?
Also I got to know that delaunay triangulation is the dual graph of voronoi diagram but not really fully understand it. Thanks so much
The Delaunay triangulation is a "good" triangulation in that it finds triangles that are not too skewed. It is unique and guarantees that the circumcircle of any triangle is empty. It is relatively fast to build, requiring no more than O(N Log(N)) operations.
It is used where triangulations are used, for instance for interpolation on irregular 2D point sets.
It is said to be the dual of the Voronoi diagram because vertexes (sites) are in one-to-one correspondence with the faces (regions) and vice versa. And when two Voronoi regions share an edge, the corresponding sites are linked by a triangle side.
Currently I am looking for an efficient algorithm to compute the intersection of two triangle meshes. I have searched over the internet, but haven't found valuable materials. The book Real-Time Collision Detection is a helpful book but is too complex for my task. I also found the post:Triangle to triangle collision detection in 3D. However I hope to find a detailed description about the algorithm.
Regards
Jogging
Well it depends on meshes size, testing each triangle in each mesh against the other is only valid in small meshes since it has n^2 complexity.
To work around that most algorithms use
Spatial portioning
first to subdivide the space into smaller ones and then tackles each one separately.
For spatial portioning most algorithms use
OcTrees
or BSPTrees however if you don't need to complicate things you can just subdivide the space into n boxes then check triangle triangle intersection in each box