Discretization
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Meshes.discretize — Function
discretize(geometry, [method])Discretize geometry with discretization method.
If the method is ommitted, a default algorithm is used with a specific number of elements.
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Meshes.discretizewithin — Function
discretizewithin(boundary, method)Discretize geometry within boundary with boundary discretization method.
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Meshes.simplexify — Function
simplexify(object)Discretize object into simplices using an appropriate discretization method.
Notes
This function is sometimes called "triangulate" when the object has parametric dimension 2.
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Meshes.DiscretizationMethod — Type
DiscretizationMethodA method for discretizing geometries into meshes.
FanTriangulation
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Meshes.FanTriangulation — Type
FanTriangulation()The fan triangulation algorithm for convex polygons. See https://en.wikipedia.org/wiki/Fan_triangulation.
hexagon = Hexagon((0.,0.), (1.,0.), (1.,1.),
(0.75,1.5), (0.25,1.5), (0.,1.))
mesh = discretize(hexagon, FanTriangulation())
fig = Mke.Figure(size = (800, 400))
viz(fig[1,1], hexagon)
viz(fig[1,2], mesh, showsegments = true)
fig
DehnTriangulation
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Meshes.DehnTriangulation — Type
DehnTriangulation()Max Dehns' triangulation proved in 1899.
The algorithm is described in the first chapter of Devadoss & Rourke 2011, and is based on a theorem derived in 1899 by the German mathematician Max Dehn. See https://en.wikipedia.org/wiki/Two_ears_theorem.
Because the algorithm relies on recursion, it is mostly appropriate for polygons with small number of vertices.
References
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Devadoss, S & Rourke, J. 2011. Discrete and computational geometry
# polygonal area
polyarea = PolyArea([(0.22926679, 0.47329807), (0.23094065, 0.44913536), (0.2569517, 0.38217533),
(0.3072999, 0.272418), (0.34814754, 0.18421611), (0.37949452, 0.11756973),
(0.4013409, 0.07247882), (0.41368666, 0.048943404), (0.42597583, 0.031655528),
(0.4382084, 0.0206152), (0.45038435, 0.015822414), (0.4625037, 0.017277176),
(0.47175184, 0.02439156), (0.47812873, 0.03716557), (0.4816344, 0.055599205),
(0.48226887, 0.07969247), (0.48172843, 0.10446181), (0.4800131, 0.12990724),
(0.47712287, 0.15602873), (0.47305775, 0.18282633), (0.47093934, 0.20558843),
(0.47076762, 0.22431506), (0.47254258, 0.23900622), (0.47626427, 0.24966191),
(0.47768936, 0.25845313), (0.47681788, 0.26537988), (0.4736498, 0.27044216),
(0.46818516, 0.27363995), (0.4613889, 0.27232954), (0.45326096, 0.2665109),
(0.44380143, 0.256184), (0.43301025, 0.24134888), (0.4246466, 0.22978415),
(0.41871038, 0.22148979), (0.4152017, 0.21646582), (0.4141205, 0.21471222),
(0.41227448, 0.21589448), (0.40966362, 0.22001258), (0.40628797, 0.22706655),
(0.40214747, 0.23705636), (0.40200475, 0.24653101), (0.40585983, 0.25549048),
(0.41371268, 0.2639348), (0.4255633, 0.2718639), (0.4378565, 0.28495985),
(0.4505922, 0.30322257), (0.46377045, 0.32665208), (0.47739124, 0.35524836),
(0.5046394, 0.36442512), (0.5455148, 0.35418236), (0.60001767, 0.32452005),
(0.66814786, 0.27543822), (0.7186763, 0.24664374), (0.75160307, 0.23813659),
(0.76692814, 0.2499168), (0.7646515, 0.28198436), (0.7769703, 0.29925033),
(0.8038847, 0.3017147), (0.84539455, 0.28937748), (0.9015, 0.26223865),
(0.94408435, 0.24899776), (0.9731477, 0.24965483), (0.98869, 0.26420987),
(0.9907113, 0.29266283), (0.9849871, 0.31338844), (0.97151726, 0.32638666),
(0.950302, 0.3316575), (0.9213412, 0.32920095), (0.8798396, 0.34078467),
(0.8257972, 0.36640862), (0.7592141, 0.40607283), (0.6800901, 0.4597773),
(0.6450007, 0.49104902), (0.6539457, 0.49988794), (0.7069251, 0.48629412),
(0.803939, 0.45026752), (0.877913, 0.4226481), (0.9288472, 0.40343583),
(0.9567415, 0.39263073), (0.961596, 0.39023277), (0.9419039, 0.40523484),
(0.89766514, 0.43763688), (0.8288798, 0.48743892), (0.7355478, 0.55464095),
(0.6655121, 0.60063523), (0.6187727, 0.6254217), (0.5953296, 0.62900037),
(0.5951828, 0.6113712), (0.57516366, 0.60261106), (0.53527224, 0.6027198),
(0.4755085, 0.6116975), (0.3958725, 0.6295441), (0.33913234, 0.6398651),
(0.30528808, 0.6426605), (0.2943397, 0.6379303), (0.30628717, 0.6256744),
(0.32149008, 0.6093727), (0.33994842, 0.5890249), (0.36166218, 0.5646312),
(0.38663134, 0.5361916), (0.3919681, 0.520893), (0.3776725, 0.5187355),
(0.34374446, 0.52971905), (0.29018405, 0.5538437), (0.25439468, 0.5678829),
(0.2363764, 0.5718367), (0.23612918, 0.56570506), (0.25365302, 0.549488),
(0.2733971, 0.5246488), (0.29536137, 0.49118724), (0.3195459, 0.4491035),
(0.34595063, 0.39839754), (0.3647463, 0.3590396), (0.37593287, 0.33102974),
(0.37951034, 0.31436795), (0.37547874, 0.30905423), (0.36070493, 0.3204269),
(0.33518887, 0.348486), (0.29893062, 0.3932315), (0.25193012, 0.45466346)])
mesh = discretize(polyarea, DehnTriangulation())
fig = Mke.Figure(size = (800, 400))
viz(fig[1,1], polyarea)
viz(fig[1,2], mesh, showsegments = true)
fig
HeldTriangulation
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Meshes.HeldTriangulation — Type
HeldTriangulation([rng]; shuffle=true)Fast Industrial-Strength Triangulation (FIST) of polygons.
This triangulation method is the method behind the famous Mapbox’s Earcut library. It is based on a ear clipping algorithm adapted for complex n-gons with holes. It has O(n²) time complexity where n is the number of vertices. In practice it is very efficient due to heuristics implemented in the algorithm.
The option shuffle is used to shuffle the order in which ears are clipped. It improves the quality of the triangles, which can be very sliver otherwise. Optionally, specify the random number generator rng.
References
mesh = discretize(polyarea, HeldTriangulation())
fig = Mke.Figure(size = (800, 400))
viz(fig[1,1], polyarea)
viz(fig[1,2], mesh, showsegments = true)
fig
DelaunayTriangulation
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Meshes.DelaunayTriangulation — Type
DelaunayTriangulation()Constrained Delaunay triangulation of polygons. Optionally, specify the random number generator rng.
References
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Cheng et al. 2012. Delaunay Mesh Generation
Notes
Wraps DelaunayTriangulation.jl. For any internal errors, file an issue at DelaunayTriangulation.jl
mesh = discretize(polyarea, DelaunayTriangulation())
fig = Mke.Figure(size = (800, 400))
viz(fig[1,1], polyarea)
viz(fig[1,2], mesh, showsegments = true)
fig
As can be seen in the following example, all discretization methods for Polygon automatically work in the presence of holes:
outer = [(0.18142937, 0.54681134), (0.38282228, 0.107781954), (0.43220532, 0.013640274),
(0.48068276, 0.019459315), (0.48322055, 0.11583236), (0.46696007, 0.2230227),
(0.48184678, 0.2656454), (0.45998818, 0.2784367), (0.4168235, 0.2190962),
(0.4124987, 0.21208182), (0.39593673, 0.2520411), (0.44333926, 0.28375763),
(0.4978224, 0.3981428), (0.7703431, 0.20181546), (0.7612364, 0.33008572),
(0.9856581, 0.2215304), (0.99374324, 0.3353423), (0.9688778, 0.38663587),
(0.59554976, 0.655444), (0.59496254, 0.58492756), (0.27641845, 0.656314),
(0.3242084, 0.6072907), (0.42408508, 0.49353212), (0.20984341, 0.59003067)]
inners = [[(0.87789994, 0.32551613), (0.5614043, 0.540334), (0.9494598, 0.39622766)],
[(0.2799388, 0.52516246), (0.38555774, 0.32233855), (0.36943135, 0.30108362)]]
polyarea = PolyArea([outer, inners...])
mesh = discretize(polyarea, DelaunayTriangulation())
fig = Mke.Figure(size = (800, 400))
viz(fig[1,1], polyarea)
viz(fig[1,2], mesh, showsegments = true)
fig
RegularDiscretization
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Meshes.RegularDiscretization — Type
RegularDiscretization(n1, n2, ..., np)A method to discretize primitive geometries with n1×n2×...×np elements sampled regularly along each parametric dimensions. The adequate number of points is calculated for each type of geometry and passed to RegularSampling.
sphere = Sphere((0.,0.,0.), 1.)
mesh = discretize(sphere, RegularDiscretization(10,10))
viz(mesh, showsegments = true)
ManualDiscretization
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Meshes.ManualDiscretization — Type
ManualDiscretization()Discretize geometries manually using indices of vertices.
box = Box((0., 0., 0.), (1., 1., 1.))
mesh = discretize(box, ManualDiscretization())
viz(mesh, colors = 1:nelements(mesh))