Tiling the faces of a polygon/polytope

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In summary, tiling the faces of a polygon or polytope involves covering the surface of the shape with smaller shapes, without any gaps or overlaps. This process is important in fields such as mathematics, geometry, and architecture, as it allows for the study of symmetry, patterns, and spatial relationships. Tiling a polygon and tiling a polytope differ in that the former is done on a 2-dimensional shape, while the latter is done on a 3-dimensional shape. Not all polygons and polytopes can be tiled, as it depends on the size and shape of the tiles being used. Real-life applications of tiling include creating designs for various materials and in architectural designs, as well as in mathematics for studying symmetry and
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cr34m3
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(This topic is related to computational geometry - I hope this is the correct forum :) )

What I would like to achieve is the generation of a mesh of data points (of some resolution) on the faces of a polygon/polytope. I specifically need points ON the faces as I would like to do a transformation of the initial shape through some equation.

The problem is closer to linear programming than geometry, per say, as the initial shapes are defined by inequalities. I'll use a simple example of what I want to achieve:
For the inequalities;
x,y,z > 0
x,y,z < 1
x+y+z < 2
a unit-cube is generated from which the one corner is 'cut off'.
The vertices of this shape are;
(0 1 0), (0 0 0), (1 0 0), (0 0 1), (0 1 1), (1 0 1), (1 1 0)
and its clear that it has 7 faces (4 triangular- and 3 square-shaped). It is on these faces that I would like to generate extra data points (seeing that I only have the 7 vertices at the moment).

My example uses just 3 dimensions, but eventually I'll go up to even more, so my approach needs to be rather generic.

At the moment I am stumped, as all the routines/algorithms I've looked at are very data-driven and focus little on extra data generation. Any suggestions, ideas or pointers to other sources will be greatly appreciated.

Ideally I'd be looking for Python modules, but if writing a wrapper for some C/C++ code gets me what I want then so be it.

Thanks

André
 
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Hello André,

Thank you for your post and for bringing up this interesting topic. You are correct in thinking that this forum is a suitable place to discuss computational geometry.

From what I understand, you are trying to generate a mesh of data points on the faces of a polygon or polytope, where the initial shape is defined by inequalities. This is indeed a problem that falls under the realm of computational geometry.

One approach to solving this problem would be to use a technique called "sampling-based meshing." This involves randomly sampling points on the faces of the polygon/polytope and then using these points to generate a mesh. The resolution of the mesh can be controlled by adjusting the number of points sampled.

Another approach would be to use a technique called "boundary meshing," which involves creating a mesh that conforms to the boundaries of the polygon/polytope. This can be done by creating a mesh of triangles or quadrilaterals on each face of the polygon/polytope, and then connecting these meshes together along their shared edges.

There are several Python modules that can help with mesh generation, such as PyMesh and MeshPy. However, as you mentioned, it is also possible to use a wrapper for C/C++ code to achieve your desired result.

I hope this helps and provides some direction for your project. If you have any further questions or need more specific guidance, please feel free to ask. Best of luck with your research!
 

1. What is tiling the faces of a polygon/polytope?

Tiling the faces of a polygon or polytope is the process of covering the surface of the shape with smaller shapes, called tiles, in a way that leaves no gaps or overlaps. This can be done with regular or irregular tiles and can result in a variety of patterns and designs.

2. Why is tiling the faces of a polygon/polytope important?

Tiling the faces of a polygon or polytope is important in fields such as mathematics, geometry, and architecture. It allows for the study and understanding of symmetry, patterns, and spatial relationships. In architecture, tiling can be used for aesthetic purposes or to create efficient and practical designs.

3. What is the difference between tiling a polygon and tiling a polytope?

Tiling a polygon involves covering the surface of a 2-dimensional shape with smaller shapes, while tiling a polytope involves covering the surface of a 3-dimensional shape with smaller shapes. Polytopes can have a variety of faces, including polygons, and the process of tiling them can be more complex than tiling a simple polygon.

4. Can any polygon or polytope be tiled?

Not all polygons and polytopes can be tiled. For example, a regular pentagon cannot be tiled with only regular polygons, but it can be tiled with a combination of regular and irregular polygons. The ability to tile a shape also depends on the size and shape of the tiles being used.

5. What are some real-life applications of tiling the faces of a polygon/polytope?

Tiling is used in various real-life applications, such as creating designs for tiles, fabrics, and wallpapers. It is also used in architectural designs, such as in Islamic art and in the construction of domes. In mathematics, tiling is used to study symmetry and to solve problems related to packing and covering shapes.

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