I've just finished reading a book named "Computational Geometry Algorithms and Applications". The algorithm introduced in this book is very helpfull for my future work.
But algorithm in this book only concerned about straight line segments. what i want to known is the same algorithm that can deal with both straight lines and conic arcs.
Such as find intersections of mixed line segments and conic arcs; offset polygon with conic arcs; find convex hull of concave polygon with conic arc edge...
3rd party libs, like CGAL can deal with problems like this, but i want to known the details of the algorithm. what's book or materials should i refer to?
In general, computational geometry with curved arcs is more complicated and less explored. But not unexplored, and often similar techniques suffice. One place to look is CGAL, as you know; and LEDA, especially here:
(Added): In response to the request for literature references, you could start with the paper below, and search backward in time via its references, and forward in time via Google Scholar (which reports it is cited by 79 papers):
Eric Berberich, Arno Eigenwillig, Michael Hemmer, Susan Hert, Kurt Mehlhorn, Elmar Schömer
"A Computational Basis for Conic Arcs and Boolean Operations on Conic Polygons."
Lecture Notes in Computer Science Volume 2461, 2002, pp 174-186.
(Springer link)
Related
Disclaimer: I'm not 100% on whether this is a well-formed question, so please feel free to comment and suggest improvements. I'll be actively looking out for ways to improve this question.
I have a triangle mesh, let's say the Stanford Bunny. Now, I want to raycast a ray from a source point in 3D along a 3D direction vector, and identify just the first intersection of that ray with the triangle mesh.
I already have a naive implementation cooked up. However, I'm looking for a more advanced implementation. In particular, I'll be casting many millions of rays in many directions, so I'm looking for a multi-threaded or GPU-accelerated implementation.
I have to believe that there must be some pretty complete projects online, as raycasting triangle meshes is a fundamental part of 3D computer graphics. However, I can't find anything beyond personal projects, which leads me to believe that I am using the wrong search terms, or something pretty simple along those lines.
I am looking for suggestions on existing tools that can raytrace polygonal meshes.
If all you need to do is find the distance to the mesh for millions of rays. Then it might be a good idea to look up CUDA raytracing tutorial online. This will show you how to cast many millions of rays. In most tutorials, raytracing is used to render to the screen with the camera matrix. However, this is not necessary. Simply adjust the rays starting parameters to what you need them to be such as 3D vector and position. Then output the data back to the CPU. Be weary of the bandwidth between the GPU and CPU sending millions of intersection points between the CPU and GPU can make the program run exceptionally slow.
I am currently trying to construct the area covered by a device over an operating period.
The first step in this process appears to be constructing a polygon of the covered area.
Since the pattern is not a standard shape, convex hulls overstate the covered area by jumping to the largest coverage area possible.
I have found a paper that appears to cover the concept of non-convex hull generation, but no discussions on how to implement this within a high level language.
http://www.geosensor.net/papers/duckham08.PR.pdf
Has anyone seen a straight forward algorithm for constructing a non-convex hull or concave hull or perhaps any python code to achieve the same result?
I have tried convex hulls mainly qhull, with a limited edge size with limited success.
Also I have noticed some licensed libraries that will not be able to be distributed, so unfortunately thats off the table.
Any better ideas or cookbooks?
You might try looking into Alpha Shapes. The CGAL library can compute them.
Edit: I see that the paper you linked references alpha shapes, and also has an algorithm listing. Is that not high level enough for you? Since you listed python as a tag, I'm sure there are Delaunay triangulation libraries in Python, which I think is the hardest part of implementing the algorithm; you just need to make sure you can modify the resulting triangulation output. The boundary query functions can probably be implemented with associative arrays.
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 am studying the rasterization algorithm and try to make a list of papers which were seminal in this area. For example "A Parallel Algorithm for Polygon Rasterization" would be one.
The one or group of papers I am looking for at the moment, are the papers that introduced the concept of interpolating vertex attributes (RGB, n, st, etc.) across the surface of a triangle using the inverse projection method.
Basically, my goal is to get back to the source of the technique.
Any other fundamental/seminal paper you could actually recommend in this area would be helpful as well. Thanks
To answer the question in part, the Wikipedia article on Gouraud shading mentions Gouraud's PhD thesis and apparently a follow-up paper as sources.
Gouraud, Henri (1971). Computer Display of Curved Surfaces, Doctoral
Thesis. University of Utah. Gouraud, Henri (1971).
"Continuous shading
of curved surfaces". IEEE Transactions on Computers C–20 (6): 623–629.
doi:10.1109/T-C.1971.223313.
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