Ok, so we looked at some research done about tree generation.
Now one or two articles about some 3d techniques, which might at first glance be seen as irelevant to trees,
but you will see how it is relevant when I will discus blobmesh (metaballs, implicit surfaces) in tree generation.
So as we all know implicit surfaces solutions result ugly mesh topology, and here is a nice idea how to make topology “right”.

*(if my short overview of this paper is unclear, please have a look at original paper, sins i am not a mathematician, some stuff is hard to understand for me too.)

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One problem in 3-dimensional modeling for animation and film special defects is mesh topology. In order to correctly animate surfaces which undergo morphing transformations, the topology of the mesh, in other words the structure of polygons and the way they are interconnected, must be correct. This means it must follow certain rules or requirements. One of the requirements is to have a mesh which consist of quads polygons, rather then triangular polygons. This brings up another important term in 3-dimensional animation and modeling which is edge loops. This term is not scientific and used by modelers.

The edge loop is not exactly a technical term, but rather jargon used by artists. It is hard to determine what is an edge loop and what is not. The edge loop is a certain way polygons are arranged together. The idea is that the main visual or underlying structures of an object which is being modeled would be represented in a polygon arrangement. The polygons no longer define only the surface approximation of an object, but also reflect the biological or mechanical underlying structures of the represented object. When an object is animated it deforms, and in order for the object to deform realistically its structures (polygonal in this case) must represent the structures and movement which exist in real object. For example, a correct mesh topology of a human body should contain polygons which represent an approximation of the human body’s volume correctly, and the way the polygons are interconnected should represent the main muscle structures. This method provides more realistic human body animation, due to the more convincing muscle movements.

When 3-dimensional objects are scanned from real life objects and usually outputted from a 3d-scanner, the mesh is irregular and does not form or represent any certain structure. The vertexes in the mesh are distributed more or less equally along the complete surface. In film or animation, such objects are usually modeled manually (Boudon 2006). The scan is used only as a reference, and the modeler has to create an object from scratch, while creating a correct mesh topology. In other words, the scanner scans only the surface without any real ability to interpret inner structures of that object. Figure 25 illustrates this.

The same problem exists with implicit surfaces, since implicit surface polygonisation topology of a mesh is quite messy and it is very hard to animate such surfaces, unless they are polygonised for each and every single frame separately (see section 2.2.4, for a short description of implicit surfaces).

Some very interesting research has been done concerning Anisotropic Polygonal remeshing, or in other words retopologisation of polygonal meshes by Alliez, Cohen-Steiner, Devillers, L’evy and Desburn. In their paper “Anisotropic Polygonal remeshing”, the authors propose a method to correct bad or irregular mesh topology (Alliez et al. 2003).

This method employs natural anisotropy meaning dependent on the direction of a given surface and tries to mimic the way an artist would create a 3 dimensional object by using the minimum number of surface elements to create a detailed surface (Alliez et al. 2003). They developed an algorithm which inputs the existing mesh, and estimates the directional fields of a given surface at each vertex. Then using the calculation of the directional fields, the algorithm estimates the minimum amount of curves necessary. Meanwhile, the algorithm creates curves which are always parallel in order to create an effective mesh topology. In the intersection of the curves, the algorithm then generates vertices. After that algorithm generates a mesh using the created vertices and direction curves as guidelines (Alliez et al. 2003).

Anisotropic Polygonal RemeshingFig 25. Picture taken from: Pierre Alliez, David Cohen-Steiner, Olivier Devillers, Bruno Lévy, and Mathieu Desbrun 2003. Anisotropic Polygonal Remeshing. ACM SIGGRAPH 2003 Papers SIGGRAPH ’03, Volume 22 Issue 3

As shown in figure 25, the overall surface shape of the model remains the same, but the number of elements dramatically decreases. Also the topology of the mesh is much better and resembles one which one would expect was created manually. This method is very valuable in 3-dimensional modeling and could potentially serve for re-meshing virtual models generated by 3-dimensional scanners or irregular mesh topologies created by implicit surface polygonisation.

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