I have a function whose range is a non-convex, simply connected region in R3. When I sample the function I know if the resulting point is on the surface of the region or not. I'd like to triangulate those samples subject to the surface constrains, i.e., the resulting tetrahedra should not
"hide" surface points. The hull would not be convex, of course.
I searched around for a library. So far I found Triangle, but it only works in R2. I also found TetGen, which works in R3, but it requires to provide
the surface triangulation (which I don't have). Also, as far as I can see, these C/C++ libraries do not have Python bindings.
Any suggestions? Thanks!
You may take a look at CGAL it has python-bindings.
One side note: If you need the surface triangulation (which seems to be a 2D problem), you may take each face, project it to 2D, triangulate it, and back to your 3D face.
EDIT due to comment: CGAL does "only" 3D triangulation. 3D constrained triangulation requires Steiner points. Since not every input can be triangulated in 3D (Schönhardt polyhedron as the classical counter-example).
Maybe you can take a look at MeshPy it looks like it has what you are looking for: "MeshPy provides Python interfaces to three well-regarded mesh generators, Triangle by J. Shewchuk, TetGen by Hang Si, and gmsh by Christophe Geuzaine and Jean-Francois Remacle."
Related
I have vector graphics. (In my first case, it's the epigraph of a function whose formula is given. So it is a shape whose outline is given by a parametric curve.)
I want to rasterize this image with anti-aliasing. So I want raster graphics, i.e. a numpy array. I want to obtain this array in a low-level way, avoiding libraries that are meant for object-oriented interactive GUI visualizations with plot axes, etc.. I just want an array. The only problem with doing something like Y,X=np.ogrid(...) and then picture = Y>f(X) is that that's not anti-aliased. (Note that blurring that binary picture is worse than a good dedicated anti-aliasing algorithm.) How to rasterize with anti-aliasing in Python without any overkill GUI-centered libraries?
If the curve is given by an implicit equation F(x,y)=0, evaluate the value of the function at the four corners of every pixel. If the signs are the same, the pixel is wholly outside or inside. If the signs vary, the area inside the polygon formed by the corners and the points along the edges where the function vanishes (find these by a mere linear interpolation) tells you the mixture of background and foreground colors (alpha blending coefficient).
Tracing the polygon isn't that difficult: traverse the four edges of the square and keep the positive vertices and zero points in the order you meet them. You will get from a triangle to an hexagon. The area is obtained by the shoelace formula.
The case of a parametric function is a little harder. You need to find the intersections of the curve with the grid lines, and perform the area estimation in all cells that are traversed. For this, draw the curve as a polyline (this is called flattening), and slice the polyline with horizontals, then verticals.
Manim might be able to rasterize epigraphs and parametric curves well and fast. Its community edition seems actively maintained.
Edits/comments with details are welcome.
I currently have 2 paths generated by a motion planning algorithm in Python. I interpolated into two 3D Polylines with equidistant points. Next I need to find whether these two paths of different lengths intersect (OR even ideally the point(s) at which they get near a certain distance of each other, if any). This is to eventually simulate a collision avoidance algorithm.
After much searching I have found multiple libraries and algorithms that can return points/line intersections of two Polylines, however they are all in 2D. I tried using the Bentley-Ottmann algorithm/python library, the Shapely library, the Geometry3D library, and other 2D solutions found on SO, but none have worked in a 3D environment. Next I tried using a 2D geometry library to find the intersections in the x-y, x-z, and y-z planes individually as shown in the graphs below showing a simple case, but I'm not sure where to go from there or whether that's even the right approach. The Polylines would usually contain 300+ points.
Any help pointing me towards a 3D algorithm or simple adjustments to make to the 2D libraries work in 3D would be really appreciated!
3D graph view, x-y view, z-y view.
I'm just a newbie in Blender.
Going to create an jigsaw puzzled sphere model, like wikipedia one, or these plastic 3D puzzles you have probably saw.
For now, i have created Python script which creates arbitrary plain 2D puzzles with Bezier curves, which later can be easy be converted to a mesh
But how to wrap it around a sphere ?
PS. Just had an idea -
to unwrap cube on the puzzle plane, copy edges as negative as shown below
(there no copy of edges on the picture yet).
Then with affine transformations, transform each 2D cube face to respective 3D place, and then apply Object->Transform->To Sphere modifier.
What do you think ? Is there a better way to create puzzled sphere ?
Thanks for your attention !
EDIT: You know, there is a dodecahedron, which can be also assembled from pentagon faces
Transformed code to blender add-on
https://bitbucket.org/ios29A/blender_puzzle_generator, maybe it will help someone
Actually 30Kb of Python code, and cube faces are transformed by hands
Just finished it with affine transformations of cube faces.
See a heart from 7x7x7 cube, so here is plain->cube->sphere->lattice transform
I think this method makes it possible to create any 3D shape from squares, even not sphere ones.
Was going to print it in plastic before metal.
Here it is 3х3х3 and was made right before 14' February
It's much simpler than you may imagine.
Download the addon from the given link above (https://bitbucket.org/ios29A/blender_puzzle_generator).
Install it (refer to the documentation).
In Blender, create a cuboid puzzle (add-curve-puzzle. Choose cuboid in the shape option).
Convert the curve into a mesh (object-convert to-mesh...)
Select the cuboid and enter edit mode (tab)
Select all (A)
Mesh-transform-to sphere.
Move your mouse.
I have this object/point cloud,rendered with pyopengl and pygame.
My object is a numpy array of the co-ordinates of the point. I wish to generate a 3d triangular mesh of this object, also it would be nice if you could decrease the number of triangles.
I have tried scipy.spatial.Delaunay and it doesnt generate triangles for 3d objects.
Dual Contouring would probably work well here, it's an algorithm that takes voxelized data and turns it into a mesh. I don't understand it trivially enough to outline it here, but basically you'd take your array of points and place them into a 3D grid array where if that grid cell contains a point it's set to equal 1 (full), and if it doesn't it is set to 0 (empty), you would then run the DC algorithm on this grid and it would output a mesh. The nice thing about this algorithm is it supports internal cavities and concave shapes.
Here's some links I found that may help you if you decide to use DC:
Basic Dual Contouring Theory
http://ngildea.blogspot.com/2014/11/implementing-dual-contouring.html
This is the github repo to the source I used when I implemented this algorithm in Unity3D:
https://github.com/nickgildea/DualContouringSample
As far as I am aware there is no inbuilt polygon functionality for Python. I want to create a 3D map and figured that polygons would be the best way to go about it.
Not wanting to reinvent the wheel I did some googling and found that there's a lot of Python stuff out there, but I couldn't find what I wanted. Thus before I reinvent the wheel (or invent it a whole), does anybody know of a Polygon system for Python?
Note that it does need to be 3D (I found quite a few 2D ones). Note also that I am not interested in the displaying of them but in storing them and the datastructure within Python.
Thanks
One of the most complete geography/mapping systems available for Python that I know about is GeoDjango. This works on top of the Django, an MVC framework. With it comes a large collection of polygon, line and distance calculation tools that can even take into account the curvature of the earth's surface if need be.
With that said, the quickest way I can think of to produce a 3D map is using a height map. Create a two dimensional list of tuples containing (x, y, z) coordinates. Each tuple represents an evenly spaced point on a grid, mapped out by the dimensions of the array. This creates a simple plane along the X and Z axes; the ground plane. The polygons that make up the plane are quads, a polygon with four sides.
Next, to produce the three dimensional height, simply give each point a Y value. This will create peaks and valleys in your ground plane.
How you render this will be up to you, and converting your grid of points into a polygon format that something like OpenGL can understand may take some work, but have a look at Visual Python, its the simplest 3D library I've seen for Python.
I think you mean Polyhedron, not Polygon .. and you might wanna look at vpython
CGAL is a C++ geometry library which, amongst other things, models Polyhedra (3D flat-surfaced shapes)
It has Python bindings available. The documentation for the Polygon3 class is here:
http://cgal-python.gforge.inria.fr/Manual/CGAL.Polyhedron.html#Polyhedron_3