I am attempting to use Three.js to morph one geometry into another. Here's what I've done so far (see http://stemkoski.github.io/Three.js/Morph-Geometries.html for a live example).
I am attempting to morph from a small polyhedron to a larger cube (both triangulated and centered at the origin). The animating is done via shaders. Each vertex on the smaller polyhedron has two associated attributes, its final position and its final UV coordinate. To calculate the final position of each vertex, I raycasted from the origin through each vertex of the smaller polyhedron and found the point of intersection with the larger cube. To calculate the final UV value, I used barycentric coordinates and the UV values at the vertices of the intersected face of the larger cube.
That led to a not awful but not great first attempt. Since (usually) none of the vertices of the larger cube were the final position of any of the vertices of the smaller polyhedron, big chunks of the surface of the cube were missing. So next I refined the smaller polyhedron by adding more vertices as follows: for each vertex of the larger cube, I raycasted toward the origin, and where each ray intersected a face of the smaller polyhedron, I removed that triangular face and added the point of intersection and three smaller faces to replace it. Now the morph is better (this is the live example linked to above), but the morph still does not fill out the entire volume of the cube.
My best guess is that in addition to projecting the vertices of the larger cube onto the smaller polyhedron, I also need to project the edges -- if A and B are vertices connected by an edge on the larger cube, then the projections of these vertices on the smaller polyhedron should also be connected by an edge. But then, of course it is possible that the projected edge will cross over multiple pre-existing triangles in the mesh of the smaller polyhedron, requiring multiple new vertices be added, retriangularization, etc. It seems that what I actually need is an algorithm to calculate a common refinement of two triangular meshes. Does anyone know of such an algorithm and/or examples (with code) of morphing (between two meshes with different triangularizations) as described above?
As it turns out, this is an intricate question. In the technical literature, the algorithm I am interested in is sometimes called the "map overlay algorithm"; the mesh I am constructing is sometimes called the "supermesh".
Some useful works I have been reading about this problem include:
Morphing of Meshes: The State of the Art and Concept.
PhD. Thesis by Jindrich Parus
http://herakles.zcu.cz/~skala/MSc/Diploma_Data/REP_2005_Parus_Jindrich.pdf
(chapter 4 especially helpful)
Computational Geometry: Algorithms and Applications (book)
Mark de Berg et al
(chapter 2 especially helpful)
Shape Transformation for Polyhedral Objects (article)
Computer Graphics, 26, 2, July 1992
by James R. Kent et al
http://www.cs.uoi.gr/~fudos/morphing/structural-morphing.pdf
I have started writing a series of demos to build up the machinery needed to implement the algorithms discussed in the literature referenced above to solve my original question. So far, these include:
Spherical projection of a mesh # http://stemkoski.github.io/Three.js/Sphere-Project.html
Topological data structure of a THREE.Geometry # http://stemkoski.github.io/Three.js/Topology-Data.html
There is still more work to be done; I will update this answer periodically as I make additional progress, and still hope that others have information to contribute!
Related
I'm doing 3D Delaunay, with the incremental method. I've tested it in 2D with an initial triangle for inserting the vertices and it works great, but if I use a triangle for 3D, some vertices do not fall into any circumscribed sphere therefore they don't get inserted.
I've tried with a tetrahedron but if the first node falls into the four of the faces, all vertices create new edges towards this new vertex, and deletes all of the initial triangles.
Whichever shape you take, you will always have to deal with side effects.
The best shape is no shape.
This is what we are doing in the CGAL library
http://www.cgal.org
Look at the manual, chapters "2D triangulations" and "3D triangulations".
See also or the journal paper https://hal.inria.fr/inria-00167199/
You can read my answer for this question (Bowyer-Watson algorithm: how to fill "holes" left by removing triangles with super triangle vertices). If the supertriangle is too small sometimes you end with circumcircle outside of the supertriangle. You can try a point-in-polygon test to avoid it.
I am looking for a method / algorithm that will allow me to merge several adjacent coplanar faces on a 3d mesh into a single face. I am hoping that this will optimize my mesh generation program, because right now it generates many 'little' triangles. When I look at the final 3d object on the screen, I can see that they all are oriented in the same direction and they could be replaced with one bigger triangle that encompasses the whole lot! I hope that is clear what I am trying to do. Thanks for your help.
I would suggest you project the faces in a single plane and than apply an algorithm for polygon uninon a plane. After that "unproject" and that's it. Always try to reduce dimensions when possible.
Your task is a special case of mesh simplification (or decimation), where the algorithm is only allowed to reduce some mesh elements without introducing any error in object's shape. And probably the most famous algorithm here is Surface Simplification Using Quadric Error Metrics.
It searches for edges in the mesh that can be contracted in a single vertex (which position is automatically selected for each edge) so that it minimizes quadratic error associated with that contraction (in your case the error is zero).
Let us consider a simple example of cube's face subdivided in 8 triangles:
Left: magenta edge is selected for contraction, remaining vertex will be located in the bottom point of the edge.
Center: after the first contraction, next magenta edge is selected, after contraction it will become a vertex in cube's corner.
Right: the final result of simplification (after contractions on other cube's faces as well), where no more coplanar triangles can be merged (at least in larger triangles).
The illustration above was prepared in MeshInspector application.
I am trying to make a quadrilateral mesh from a surface mesh (which is mostly triangular) generated by Mathematica. I am not looking for high quality mesher but a simple work around algorithm. I use GMSH for doing it externally. We can make use of Mathematic's CAD import capabilities to generate 3D geometries that are understood by the Mathematica kernel.
We can see the imported Geometry3D objects and the plots of number of sides in each polygons they consist of. It become visible that the polygons that form the mesh are not always triangles.
Name3D=RandomChoice[ExampleData["Geometry3D"][[All,2]],6];
AllPic=
Table[
Vertex=ExampleData[{"Geometry3D",Name3D[[i]]},"VertexData"];
Polygons=ExampleData[{"Geometry3D",Name3D[[i]]},"PolygonData"];
GraphicsGrid[
{{ListPlot[#,Frame-> True,PlotLabel->Name3D[[i]] ]&#(Length[#]&/#Polygons),
Graphics3D[GraphicsComplex[Vertex,Polygon[Polygons]],Boxed-> False]}}
,ImageSize-> 300,Spacings-> {0,0}],
{i,1,Length#Name3D}];
GraphicsGrid[Partition[AllPic,2],Spacings-> {0,0}]
Now what I am looking for is an algorithm to form a quadrilateral mesh from that polygon information available to MMA. Any easy solution is very much welcome. By easy solution I mean which is not going to work in a very general setting (where mesh constitutes of polygons with sides more than 5 or 6) and which might be quite inefficient compared to commercial software. But one can see that there are not many quadrilateral surface mesh generator available other than few expensive commercial one.
BR
this will produce quads regardless of the input topology:
insert one vertex in the center of each face
insert one vertex at the midpoint of each edge
insert edges connecting each face's center vertex with it's edges' midpoint vertices
Starting with a 3D mesh, how would you give a rounded appearance to the edges and corners between the polygons of that mesh?
Without wishing to discourage other approaches, here's how I'm currently approaching the problem:
Given the mesh for a regular polyhedron, I can give the mesh's edges a rounded appearance by scaling each polygon along its plane and connecting the edges using cylinder segments such that each cylinder is tangent to each polygon where it meets that polygon.
Here's an example involving a cube:
Here's the cube after scaling its polygons:
Here's the cube after connecting the polygons' edges using cylinders:
What I'm having trouble with is figuring out how to deal with the corners between polygons, especially in cases where more than three edges meet at each corner. I'd also like an algorithm that works for all closed polyhedra instead of just those that are regular.
I post this as an answer because I can't put images into comments.
Sattle point
Here's an image of two brothers camping:
They placed their simple tents right beside each other in the middle of a steep walley (that's one bad place for tents, but thats not the point), so one end of each tent points upwards. At the point where the four squares meet you have a sattle point. The two edges on top of each tent can be rounded normally as well as the two downward edges. But at the sattle point you have different curvature in both directions and therefore its not possible to use a sphere. This rules out Svante's solution.
Selfintersection
The following image shows some 3D polygons if viewed from the side. Its some sharp thing with a hole drilled into it from the other side. The left image shows it before, the right after rounding.
.
The mass thats get removed from the sharp edge containts the end of the drill hole.
There is someething else to see here. The drill holes sides might be very large polygons (lets say it's not a hole but a slit). Still you only get small radii at the top. you can't just scale your polygons, you have to take into account the neighboring polygon.
Convexity
You say you're only removing mass, this is only true if your geometry is convex. Look at the image you posted. But now assume that the viewer is inside the volume. The radii turn away from you and therefore add mass.
NURBS
I'm not a nurbs specialist my self. But the constraints would look something like this:
The corners of the nurbs patch must be at the same position as the corners of the scaled-down polygons. The normal vectors of the nurb surface at the corners must be equal to the normal of the polygon. This should be sufficent to gurarantee that the nurb edge will be a straight line following the polygon edge. The normals also ensure that no visible edges will result at the border between polygon and nurbs patch.
I'd just do the math myself. nurbs are just polygons. You'll have some unknown coefficients and your constraints. This gives you a system of equations (often linear) that you can solve.
Is there any upper bound on the number of faces, that meet at that corner?
You might you might employ concepts from CAGD, especially Non-Uniform Rational B-Splines (NURBS) might be of interest for you.
Your current approach - glueing some fixed geometrical primitives might be too inflexible to solve the problem. NURBS require some mathematical work to get used to, but might be more suitable for your needs.
Extrapolating your cylinder-edge approach, the corners should be spheres, resp. sphere segments, that have the same radius as the cylinders meeting there and the centre at the intersection of the cylinders' axes.
Here we have a single C++ header for generating triangulated rounded 3D boxes. The code is in C++ but also easy to transplant to other coding languages. Also it's easy to be modified for other primitives like quads.
https://github.com/nepluno/RoundCornerBox
As #Raymond suggests, I also think that the nepluno repo provides a very good implementation to solve this issue; efficient and simple.
To complete his answer, I just wrote a solution to this issue in JS, based on the BabylonJS 3D engine. This solution can be found here, and can be quite easily replaced by another 3D engine:
https://playground.babylonjs.com/#AY7B23
I've got a bunch of overlapping triangles from a 3D model projected into a 2D plane. I need to merge each island of touching triangles into a closed, non-convex polygon.
The resultant polygons shouldn't have any holes in them (since the source data doesn't).
Many of the source triangles share (floating point identical) edges with other triangles in the source data.
What's the easiest way to do this? Performance isn't particularly important, since this will be done at design time.
Try gpc, or the General Polygon Clipper Library.
Imagine the projection onto a plane as a "view" of the model (i.e. the direction of projection is the line of sight, and the projection is what you see). In that case, the borders of the polygons you want to compute correspond to the silhouette of the model.
The silhouette, in turn, is a set of edges in the model. For each edge in the silhouette, the adjacent faces will have normals that either point away from the plane or toward the plane. You can check this be taking the dot product of the face normal with the plane normal -- look for edges whose adjacent face normals have dot products of opposite signs with the projection direction.
Once you have found all the silhouette edges you can join them together into the boundaries of the desired polygons.
Generally, you can find more about silhouette detection and extraction by googling terms like mesh silouette finding detection. Maybe a good place to start is here.
I've also found this[1] approach, which I will be trying next.
[1] 2d outline algorithm for projected 3D mesh