How to draw mutually touching/kissing spheres in Autocad, Blender or PoV-Ray...? so that the incremental spheres are in closest proximity with the first sphere drawn. Just like magnetic spheres are clumped together with maximum density and having minimum volume.
It really depends on a lot of details, but if you want to do something like this, but in 3D
then you should search for the Apollonian Gasket.
Even if you have something else in mind, it has a plenty of variations you should look into.
The 2D algorithm is available here or here for AutoCAD. And from the same site in 3D, again for AutoCAD.
Related
I'm working on a project to detect the position and orientation of a paper plane.
To collect the data, I'm using an Intel Realsense D435, which gives me accurate, clean depth data to work with.
Now I arrived at the problem of detecting the 2D paper plane silhouette from the 3D point cloud data.
Here is an example of the data (I put the plane on a stick for testing, this will not be in the final implementation):
https://i.stack.imgur.com/EHaEr.gif
Basically, I have:
A 3D point cloud with points on the plane
A 2D shape of the plane
I would like to calculate what rotations/translations are needed to align the 2D shape to the 3D point cloud as accurate as possible.
I've searched online, but couldn't find a good way to do it. One way would be to use Iterative Closest Point (ICP) to first take a calibration pointcloud of the plane in a known orientation, and align it with the current orientation. But from what I've heard, ICP doesn't perform well if the pointclouds aren't kind of already closely aligned at the start.
Any help is appreciated! Coding language doesn't matter.
Does your 3d point cloud have outliers? How many in what way?
How did you use ICP exactly?
One way would be using ICP, with a hand-crafted initial guess using
pcl::transformPointCloud (*cloud_in, *cloud_icp, transformation_matrix);
(to mitigate the problem that ICP needs to be close to work.)
What you actually want is the plane-model that describes the position and orientation of your point-cloud right?
A good estimator of your underlying function can be found with: pcl::ransac
pcl::ransace model consensus
You can then get the computedModel coefficents.
Now finding the correct transformation is just: How to calculate transformation matrix from one plane to another?
I am developing some computer vision algorithms for vehicle applications.
I am in front of a problem and some help would be appreciated.
Let say we have a calibrated camera attached to a vehicle which captures a frame of the road forward the vehicle:
Initial frame
We apply a first filter to keep only the road markers and return a binary image:
Filtered image
Once the road lane are separated, we can approximate the lanes with linear expressions and detect the vanishing point:
Objective
But what I am looking for to recover is the equation of the normal n into the image without any prior knowledge of the rotation matrix and the translation vector. Nevertheless, I assume L1, L2 and L3 lie on the same plane.
In the 3D space the problem is quite simple. In the 2D image plane, since the camera projective transformation does not keep the angle properties more complex. I am not able to find a way to figure out the equation of the normal.
Do you have any idea about how I could compute the normal?
Thanks,
Pm
No can do, you need a minimum of two independent vanishing points (i.e. vanishing points representing the images of the points at infinity of two different pencils of parallel lines).
If you have them, the answer is trivial: express the image positions of said vanishing points in homogeneous coordinates. Then their cross product is equal (up to scale) to the normal vector of the 3D plane said pencils define, decomposed in camera coordinates.
Your information is insufficient as the others have stated. If your data is coming from a video a common way to get a road ground plane is to take two or more images, compute the associated homography then decompose the homography matrix into the surface normal and relative camera motion. You can do the decomposition with OpenCV's decomposeHomographyMatmethod. You can compute the homography by associating four or more point correspondences using OpenCV's findHomography method. If it is hard to determine these correspondences it is also possible to do it with a combination of point and line correspondences paper, however this is not implemented in OpenCV.
You do not have sufficient information in the example you provide.
If you are wondering "which way is up", one thing you might be able to do is to detect the line on the horizon. If K is the calibration matrix then KTl will give you the plane normal in 3D relative to your camera. (The general equation for backprojection of a line l in the image to a plane E through the center of projection is E=PTl with a 3x4 projection matrix P)
A better alternative might be to establish a homography to rectify the ground-plane. To do so, however, you need at least four non-collinear points with known coordinates - or four lines, no three of which may be parallel.
I need to generate a tetrahedral (volume) mesh of thin-walled object object. Think of objects like a bottle or a plastic bowl, etc, which are mostly hollow. The volumetric mesh is needed for an FEM simulation. A surface mesh of the outside surface of the object is available from measurement, using e.g. octomap or KinectFusion. Therefore the vertex spacing is relatively regular. The inner surface of the object can be calculated from the outside surface by moving all points inside, since the wall thickness is known.
So far, I have considered the following approaches:
Create a 3D Delaunay triangulation (which would destroy the existing surface meshes) and then remove all tetrahedra which are not between the two original surfaces. For this check, it might be required to create an implicit surface representation of the 2 surfaces.
Create a 3D Delaunay triangulation and remove tetrahedra which are "inside" (in the hollow space) or "outside" (of the outer surface) with Alphashapes.
Close the outside and inside meshes and load them into tetgen as the outside hull and as a hole respectively.
These approaches seem to be a bit inelegant to me, and they still have some pitfalls. I would probably need several libraries/tools for them. For 1 and 2, probably tetgen or another FEM meshing tool would still be required to create well-conditioned tetrahedra. Does anyone have a more straight-forward solution? I guess this should also be a common problem in 3D printing.
Concerning tools/libraries, I have looked into PCL, meshlab and tetgen so far. They all seem to do only part of the job. Ideally, I would like to use only open source libraries and avoid tools which require manual intervention.
One way is to:
create triangular mesh of surface points,
extrude (move) that surface to inner for a given thickness. That produces volume (triangular prism) mesh of a wall,
each prism can be split in three tetrahedrons.
The problem I see is aspect ratio.
A single layer of tetrahedra will not reproduce shell or bending behavior very well. A single element through the thickness will already require a large mesh. Putting more than one will likely break the bank in order to keep aspect ratios and angles acceptable.
I'd prefer brick or thick shell elements to tetrahedra in this case. I think the modeling will be easier and the behavior will be more faithful to the physics.
Currently I'm working on a little project just for a bit of fun. It is a C++, WinAPI application using OpenGL.
I hope it will turn into a RTS Game played on a hexagon grid and when I get the basic game engine done, I have plans to expand it further.
At the moment my application consists of a VBO that holds vertex and heightmap information. The heightmap is generated using a midpoint displacement algorithm (diamond-square).
In order to implement a hexagon grid I went with the idea explained here. It shifts down odd rows of a normal grid to allow relatively easy rendering of hexagons without too many further complications (I hope).
After a few days it is beginning to come together and I've added mouse picking, which is implemented by rendering each hex in the grid in a unique colour, and then sampling a given mouse position within this FBO to identify the ID of the selected cell (visible in the top right of the screenshot below).
In the next stage of my project I would like to look at generating more 'playable' terrains. To me this means that the shape of each hexagon should be more regular than those seen in the image above.
So finally coming to my point, is there:
A way of smoothing or adjusting the vertices in my current method
that would bring all point of a hexagon onto one plane (coplanar).
EDIT:
For anyone looking for information on how to make points coplanar here is a great explination.
A better approach to procedural terrain generation that would allow
for better control of this sort of thing.
A way to represent my vertex information in a different way that allows for this.
To be clear, I am not trying to achieve a flat hex grid with raised edges or platforms (as seen below).
)
I would like all the geometry to join and lead into the next bit.
I'm hope to achieve something similar to what I have now (relatively nice undulating hills & terrain) but with more controllable plateaus. This gives me the flexibility of cording off areas (unplayable tiles) later on, where I can add higher detail meshes if needed.
Any feedback is welcome, I'm using this as a learning exercise so please - all comments welcome!
It depends on what you actually want and what you mean by "more controlled".
Do you want to be able to say "there will be a mountain on coordinates [11, -127] with radius 20"? Complexity of this this depends on how far you want to go. If you want just mountains, then radial gradients are enough (just add the gradient values to the noise values). But if you want some more complex shapes, you are in for a treat.
I explore this idea to great depth in my project (please consider that the published version is just a prototype, which is currently undergoing major redesign, it is completely usable a map generator though).
Another way is to make the generation much more procedural - you just specify a sequence of mathematical functions, which you apply on the terrain. Even a simple value transformation can get you very far.
All of these methods should work just fine for hex grid. If artefacts occur because of the odd-row shift, then you could interpolate the odd rows instead (just calculate the height value for the vertex from the two vertices between which it is located with simple linear interpolation formula).
Consider a function, which maps the purple line into the blue curve - it emphasizes lower located heights as well as very high located heights, but makes the transition between them steeper (this example is just a cosine function, making the curve less smooth would make the transformation more prominent).
You could also only use bottom half of the curve, making peaks sharper and lower located areas flatter (thus more playable).
"sharpness" of the curve can be easily modulated with power (making the effect much more dramatic) or square root (decreasing the effect).
Implementation of this is actually extremely simple (especially if you use the cosine function) - just apply the function on each pixel in the map. If the function isn't so mathematically trivial, lookup tables work just fine (with cubic interpolation between the table values, linear interpolation creates artefacts).
Several more simple methods of "gamification" of random noise terrain can be found in this paper: "Realtime Synthesis of Eroded Fractal Terrain for Use in Computer Games".
Good luck with your project
I am writing a physics simulation using Ogre and MOC.
I have a sphere that I shoot from the camera's position and it travels in the direction the camera is facing by using the camera's forward vector.
I would like to know how I can detect the point of collision between my sphere and another mesh.
How would I be able to check for a collision point between the two meshes using MOC or OGRE?
Update: Should have mentioned this earlier. I am unable to use a 3rd party physics library as we I need to develop this myself (uni project).
The accepted solution here flat out doesn't work. It will only even sort of work if the mesh density is generally high enough that no two points on the mesh are farther apart than the diameter of your collision sphere. Imagine a tiny sphere launched at short range on a random vector at a huuuge cube mesh. The cube mesh only has 8 verts. What are the odds that the cube is actually going to hit one of those 8 verts?
This really needs to be done with per-polygon collision. You need to be able to check intersection of polygon and a sphere (and additionally a cylinder if you want to avoid tunneling like reinier mentioned). There are quite a few resources for this online and in book form, but http://www.realtimerendering.com/intersections.html might be a useful starting point.
The comments about optimization are good. Early out opportunities (perhaps a quick check against a bounding sphere or an axis aligned bounding volume for the mesh) are essential. Even once you've determined that you're inside a bounding volume, it would probably be a good idea to be able to weed out unlikely polygons (too far away, facing the wrong direction, etc.) from the list of potential candidates.
I think the best would be to use a specialized physics library.
That said. If I think about this problem, I would suspect that it's not that hard:
The sphere has a midpoint and a radius. For every point in the mesh do the following:
check if the point lies inside the sphere.
if it does check if it is closer to the center than the previously found point(if any)
if it does... store this point as the collision point
Of course, this routine will be fairly slow.
A few things to speed it up:
for a first trivial reject, first see if the bounding sphere of the mesh collides
don't calc the squareroots when checking distances... use the squared lengths instead.(much faster)
Instead of comparing every point of the mesh, use a dimensional space division algorithm (quadtree / BSP)for the mesh to quickly rule out groups of points
Ah... and this routine only works if the sphere doesn't travel too fast (relative to the mesh). If it would travel very fast, and you sample it X times per second, chances are the sphere would have flown right through the mesh without every colliding. To overcome this, you must use 'swept volumes' which basically makes your sphere into a tube. Making the math exponentially complicated.