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Dual Marching Tetrahedra - Diamond Lattic Crystal (DMT-DLC)

A physical model - somewhat hard because the slopes are really all 45 degrees... Yellow lines show mating faces - there isn't really a gap between blue and red cells. This is 11 simple cells connnected together. This arrangment is fairly stable, and can continue alternating red-blue forever, in every direction.

Multitexture exmaple

A single cell - skew alignment...

Multitexture exmaple

A single cell - recti-linear aligned... this is the top of a 'cube' cell. THe corners that are elongated should be considered compressed into the points of the cube...

Multitexture exmaple

This is a Amalgamated Cell... this is actually proper alignment of points. (see below). This is a point-on view. All points look the same.

Multitexture exmaple

This is the rectangular view of the cell. This is a square-on view of a complex cell. This is an octet of cells... the center points (and outside corners) are missing.... This shows the alternating arrangement of cells in the lattice. The firat image above, the 4x4 grouping of simple cells, includes all the points and segments, but has been displaced.

Multitexture exmaple

This is the other edge-on view...

Multitexture exmaple

See also

(this Demo) Testing framework: https://d3x0r.github.io/IsoSurface-MultiTexture

Original Testing framework : http://mikolalysenko.github.io/Isosurface/ https://github.com/mikolalysenko/isosurface.git

Prior Updated Testing framework: https://d3x0r.github.io/MarchingTetrahedra/

see also https://en.wikipedia.org/wiki/Marching_tetrahedra

This adds texturing to the flat isosurface mesh calculation. It demonstrates a method of 3d volume repeating textures. (Texture generator is included in src/). Adds several options for scaling/stylizing data.

Marching Tetrahedra 3 (DLC-Volume) mode slices a cloud one cloud at a time; removes 3 calucations per spot; increasing size of working data; improves potential merged points in maximized/minimized conditions.

Marching Tetrahedra 3 (DLC-Plane) mode slices a cloud a plane at a time, generating a single calculation for all shared points. Duduces the point count.

There is a version 'marchintTetrahedra2.js' which is the simplest, naive implementation that works on a single group of 8 points computing the inner mesh; although longer, this appears simpler than the plane or volume meshing versions, although does significantly more work.

This yields really good results for opening spaces, does not suffer from any discontinuities.

This meshing implementation assumes that 0 and below is 'outside' and anything > 0 has density.

in the case of distance to surface as point cloud data, the density is -distance; but then the option to use zero as inside surface must also be selected.

All triangles are trivially aligned with their proper normals. The accompanying source code is complete.

The tetrahedral computation (tetCompute) in this code assumes the following.

The triangle mesher, given any input point, the point is either inside, or outside.

  • (If 0 is 'inside') If it is inside, the face generation is inverted, and all values inverted. In the case of 0 the state does not change from 'outside' to 'inside' so the origin point, which is ALWAYS outside, and is the source of computing the delta from 0 to points 1,2, or 3, then those points could also be 0, resulting in a slope that has no delta; this is forced to 0 as a result, so a 0 'inside' point is as close to itself as it can be.
  • (If 0 is 'outside') then, since the first point must be outside, if it is 0, then no inversion takes place. If it isn't 0, then the delta computation for the slope will never be 0.

The conditions and plane generation can also all be reversed, such that 0 is 'inside' is proper, and the mesher expects the first inpoint point to be inside... which reverse all of the subtractions and >= to < comparisons.

Inspired from cells of diamond crystal lattice...

![Diamond cubic lattice] https://en.wikipedia.org/wiki/Cubic_crystal_system#/media/File:Visualisation_diamond_cubic.svg

Instead of blank cells, I have alternating same-cells (the inside tetrahdron is rotated 90 degrees each time, so the edge deltas are computed along the same diagonal in mating cells.

My original idea was actually more of a marching - cuboctohedron https://en.wikipedia.org/wiki/Cuboctahedron which is formed from trying to densly stack equilateral tetrahedrons... and then scalling a large stack down, in the middle you end up with 90 degree cuts...

The cell is actually simpler than that; being a marching-cube, but with the middle plane of it skewed 0.5 to the right and 'forward'/'up'? where the values of the skewed plane are an average of the 8 points around it. This ended up being 14 tetrahedrons, with 4 pyramidal penta-hedrons sliced into 2 tetrahedron each... But slicing the pyramids caused a lot of confusion in mapping inverted/notinverted to the tetrahedra face emission.

the final result was only 5 tetrahedra, and no fabricated points. (the above also required a 3x3x2 matrix of values, so the 0.5 skew to the right 1 cell could be calculated) This only requires the 5 points of the cell much like yours.

I was iterating through the tet vertices to find one that was 'outside' and then just always build from an outside point; but depending on ordering of the vertices that can be tricky; I ordered mine so every other point is inverted, and a mod4 can be applied to wrap the point indexing, reducing it to just 1 if case , and its sub-cases...

Inflation

inflation

References

This paper inadvertantly describes this method, without directly defining the basis as tetrahedra, only tetrahedral regions are considered; (maybe a solution to the 5 point quads flipping) https://www.cse.wustl.edu/~taoju/research/interfree_paper_final.pdf

Found but not really used; this describes the edge-tree associated with the octree. An edge-tree is used in MT to determine the faces within each tetrahedron. http://www.cs.jhu.edu/~misha/MyPapers/SGP07a.pdf

This has a 'responsibility assignment' which delegates relavence between near points (?). https://www.academia.edu/38558591/Volume_octree_with_an_implicitly_defined_dual_grid

Dual Quaterions... (Dual Numbers) Mostly because it was new; was pursuring repsenting the normals as quaternions... https://cs.gmu.edu/~jmlien/teaching/cs451/uploads/Main/dual-quaternion.pdf

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