# Generating a nice mesh on a sphere S^2

**URL:** <https://discourse.julialang.org/t/generating-a-nice-mesh-on-a-sphere-s-2/73075>\
**Category:** Visualization\
**Tags:** question\
**Created:** [December 14, 2021, 2:18pm UTC](https://discourse.julialang.org/t/generating-a-nice-mesh-on-a-sphere-s-2/73075 "2021-12-14T14:18:45Z")\
**Posts on this page:** 9\
**Page:** 1

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**Author:** ![BVPs](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bvps/32/2190_2.png) [@BVPs](https://discourse.julialang.org/u/BVPs)\
**Post date:** [December 14, 2021, 2:18pm UTC](https://discourse.julialang.org/t/generating-a-nice-mesh-on-a-sphere-s-2/73075/1 "2021-12-14T14:18:45Z")

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I would like to generate a nice mesh/grid (i.e., as homogeneous/isotropic as possible) on the unit sphere S^2 in R^3. In particular, I would like to generate two possible grid systems: 1) based on triangles; or 2) based on the Goldberg polyhedra (i.e., the so-called Geodesic Grid). I am wondering whether there are Julia packages to generate those meshes/grids.  
Thanks a lot in advance!

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**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [December 14, 2021, 2:56pm UTC](https://discourse.julialang.org/t/generating-a-nice-mesh-on-a-sphere-s-2/73075/2 "2021-12-14T14:56:36Z")

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[Gmsh.jl](https://github.com/JuliaFEM/Gmsh.jl) can generate triangular meshes for spheres (via the `addSphere(x,y,z,radius)` function).

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**Author:** ![BVPs](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bvps/32/2190_2.png) [@BVPs](https://discourse.julialang.org/u/BVPs)\
**Post date:** [December 14, 2021, 5:02pm UTC](https://discourse.julialang.org/t/generating-a-nice-mesh-on-a-sphere-s-2/73075/3 "2021-12-14T17:02:22Z")

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Thanks a lot, @stevengj ! I’ll try that package.

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**Author:** ![PetrKryslUCSD](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/petrkryslucsd/32/215825_2.png) [@PetrKryslUCSD](https://discourse.julialang.org/u/PetrKryslUCSD)\
**Post date:** [December 14, 2021, 6:20pm UTC](https://discourse.julialang.org/t/generating-a-nice-mesh-on-a-sphere-s-2/73075/4 "2021-12-14T18:20:36Z")

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Do you want quadrilaterals or triangles? Either can be generated with FinEtools. For example [https://github.com/PetrKryslUCSD/FinEtoolsMeshing.jl/blob/8f2460a191e46fa3779fd2ac6a6662654aae9a8e/test/test\_meshing.jl#L2963](https://github.com/PetrKryslUCSD/FinEtoolsMeshing.jl/blob/8f2460a191e46fa3779fd2ac6a6662654aae9a8e/test/test_meshing.jl#L2963)

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**Author:** ![Perrin\_Meyer](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/perrin_meyer/32/15686_2.png) [@Perrin\_Meyer](https://discourse.julialang.org/u/Perrin_Meyer)\
**Post date:** [December 14, 2021, 7:32pm UTC](https://discourse.julialang.org/t/generating-a-nice-mesh-on-a-sphere-s-2/73075/5 "2021-12-14T19:32:22Z")

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Here is the GMSH geo file I use to generate spherical (and ellipsoidal meshes) (borrowed from [http://bempp.com/](http://bempp.com/))

> // from bempp shapes.py  
> //  
> // Return an ellipsoid grid.  
> //  
> // Parameters  
> // ----------  
> // r1 : float  
> // Radius of first major axis  
> // r2 : float  
> // Radius of second major axis  
> // r3 : float  
> // Radius of third major axis  
> // origin : tuple  
> // Tuple specifying the origin of the ellipsoid  
> // h : float  
> // Element size.  
> // “”"  
> // stub = “”"
> 
> max\_freq = 2000;  
> C = 343.2;
> 
> wav = C / max\_freq;
> 
> lc = wav / 6; // Characteristic Length – i.e. basic mesh size parameter
> 
> r1 = 0.25;  
> r2 = 0.25;  
> r3 = 0.25;
> 
> orig0 = 0.0;  
> orig1 = 0.0;  
> orig2 = 0.0;
> 
> cl = lc;  
> cl2 = lc;
> 
> Point(1) = {orig0,orig1,orig2,cl};  
> Point(2) = {orig0+r1,orig1,orig2,cl};  
> Point(3) = {orig0,orig1+r2,orig2,cl2};  
> Point(4) = {orig0-r1,orig1,orig2,cl};  
> Point(5) = {orig0,orig1-r2,orig2,cl2};  
> Point(6) = {orig0,orig1,orig2-r3,cl2};  
> Point(7) = {orig0,orig1,orig2+r3,cl2};
> 
> Ellipse(1) = {2,1,2,3};  
> Ellipse(2) = {3,1,4,4};  
> Ellipse(3) = {4,1,4,5};  
> Ellipse(4) = {5,1,2,2};  
> Ellipse(5) = {3,1,3,6};  
> Ellipse(6) = {6,1,5,5};  
> Ellipse(7) = {5,1,5,7};  
> Ellipse(8) = {7,1,3,3};  
> Ellipse(9) = {2,1,2,7};  
> Ellipse(10) = {7,1,4,4};  
> Ellipse(11) = {4,1,4,6};  
> Ellipse(12) = {6,1,2,2};
> 
> Line Loop(13) = {2,8,-10};  
> Ruled Surface(14) = {13};
> 
> Line Loop(15) = {10,3,7};  
> Ruled Surface(16) = {15};
> 
> Line Loop(17) = {-8,-9,1};  
> Ruled Surface(18) = {17};
> 
> Line Loop(19) = {-11,-2,5};  
> Ruled Surface(20) = {19};
> 
> Line Loop(21) = {-5,-12,-1};  
> Ruled Surface(22) = {21};
> 
> Line Loop(23) = {-3,11,6};  
> Ruled Surface(24) = {23};
> 
> Line Loop(25) = {-7,4,9};  
> Ruled Surface(26) = {25};
> 
> Line Loop(27) = {-4,12,-6};  
> Ruled Surface(28) = {27};
> 
> Surface Loop(29) = {28,26,16,14,20,24,22,18};
> 
> // GMSH did NOT like this compound constraint…  
> // Compound Surface{28,26,16,14,20,24,22,18};
> 
> Volume(30) = {29};  
> Physical Surface(1) = {14};  
> Physical Surface(2) = {16};  
> Physical Surface(3) = {18};  
> Physical Surface(4) = {20};  
> Physical Surface(5) = {22};  
> Physical Surface(6) = {24};  
> Physical Surface(7) = {26};  
> Physical Surface(8) = {28};
> 
> // To generate a curvilinear mesh and optimize it to produce provably valid  
> // curved elements (see A. Johnen, J.-F. Remacle and C. Geuzaine. Geometric  
> // validity of curvilinear finite elements. Journal of Computational Physics  
> // 233, pp. 359-372, 2013; and T. Toulorge, C. Geuzaine, J.-F. Remacle,  
> // J. Lambrechts. Robust untangling of curvilinear meshes. Journal of  
> // Computational Physics 254, pp. 8-26, 2013), you can uncomment the following  
> // lines:  
> //  
> // Mesh.ElementOrder = 2;  
> // Mesh.HighOrderOptimize = 2;
> 
> Mesh.Algorithm = 2;  
> Mesh.ElementOrder = 2;  
> Mesh.HighOrderOptimize = 2;

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**Author:** ![Perrin\_Meyer](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/perrin_meyer/32/15686_2.png) [@Perrin\_Meyer](https://discourse.julialang.org/u/Perrin_Meyer)\
**Post date:** [December 14, 2021, 7:36pm UTC](https://discourse.julialang.org/t/generating-a-nice-mesh-on-a-sphere-s-2/73075/6 "2021-12-14T19:36:16Z")

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![gmshsphere](https://global.discourse-cdn.com/julialang/original/3X/c/7/c7a01e3e8bcdf16c88bc48cd62af3fc266c738ff.png)

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**Author:** ![PetrKryslUCSD](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/petrkryslucsd/32/215825_2.png) [@PetrKryslUCSD](https://discourse.julialang.org/u/PetrKryslUCSD)\
**Post date:** [December 14, 2021, 8:18pm UTC](https://discourse.julialang.org/t/generating-a-nice-mesh-on-a-sphere-s-2/73075/7 "2021-12-14T20:18:48Z")

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![image](https://global.discourse-cdn.com/julialang/original/3X/4/0/40fc49051090781b69aea1899d93a4dd128f8e95.jpeg)

> <https://github.com/PetrKryslUCSD/FinEtoolsMeshing.jl/blob/14e8fdde23503050852442defb9150039c8e2a3c/test/test_meshing.jl#L3412>

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**Author:** ![juliohm](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/juliohm/32/215266_2.png) [@juliohm](https://discourse.julialang.org/u/juliohm)\
**Post date:** [December 14, 2021, 10:45pm UTC](https://discourse.julialang.org/t/generating-a-nice-mesh-on-a-sphere-s-2/73075/8 "2021-12-14T22:45:16Z")

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Here is a solution in pure Julia:

```julia
using Meshes

# sphere geometry
sphere = Sphere((0.,0.,0.), 1.)

# polygonal mesh with quadrangles + triangles
mesh = discretize(sphere, RegularDiscretization(30, 30))

# triangulate the mesh so that it contains only triangles
tmesh = triangulate(mesh)

# you can refine a mesh further with triangles
ttmesh = refine(tmesh, TriRefinement())

# or refine it further with quadrangles
qmesh = refine(tmesh, QuadRefinement())

```

Here is a visualization of all these meshes:

```julia
using MeshViz

# choose a Makie backend
import GLMakie as Mke

fig = Mke.Figure()
viz(fig[1,1], mesh, showfacets = true)
viz(fig[1,2], tmesh, showfacets = true)
viz(fig[1,3], ttmesh, showfacets = true)
viz(fig[1,4], qmesh, showfacets = true)
fig

```

 ![image](https://global.discourse-cdn.com/julialang/original/3X/1/0/10a9d08bd3d6797ead2c74430150d1a3c6e31ebf.jpeg)

More discretization methods are available, check the docs: [Discretization · Meshes.jl](https://juliageometry.github.io/Meshes.jl/stable/algorithms/discretization.html)

We are always happy to review PRs with new methods.

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**Author:** ![BVPs](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bvps/32/2190_2.png) [@BVPs](https://discourse.julialang.org/u/BVPs)\
**Post date:** [December 14, 2021, 10:52pm UTC](https://discourse.julialang.org/t/generating-a-nice-mesh-on-a-sphere-s-2/73075/9 "2021-12-14T22:52:42Z")

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Thank you everyone for suggesting all these packages! I’ll check them all, and decide which one I would like to use.