Discretization
Meshes.discretize
— Functiondiscretize(geometry, method)
Discretize geometry
with discretization method
.
Meshes.discretizewithin
— Functiondiscretizewithin(boundary, method)
Discretize geometry within boundary
with boundary discretization method
.
Meshes.triangulate
— Functiontriangulate(object)
Triangulate object
of parametric dimension 2 into triangles using an appropriate triangulation method.
Meshes.DiscretizationMethod
— TypeDiscretizationMethod
A method for discretizing geometries into meshes.
Meshes.BoundaryDiscretizationMethod
— TypeBoundaryDiscretizationMethod
A method for discretizing geometries based on their boundary.
FanTriangulation
Meshes.FanTriangulation
— TypeFanTriangulation()
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(resolution = (800, 400))
viz(fig[1,1], hexagon)
viz(fig[1,2], mesh, showfacets = true)
fig
RegularDiscretization
Meshes.RegularDiscretization
— TypeRegularDiscretization(n1, n2, ..., np)
A method to discretize primitive geometries using regular samples along each parametric dimension. The number of samples n1
, n2
, ..., np
is passed to RegularSampling
.
sphere = Sphere((0.,0.,0.), 1.)
mesh = discretize(sphere, RegularDiscretization(10,10))
fig = Mke.Figure(resolution = (400, 400))
viz(fig[1,1], mesh, showfacets = true)
fig
Dehn1899
Meshes.Dehn1899
— TypeDehn1899()
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/Twoearstheorem.
Because the algorithm relies on recursion, it is mostly appropriate for polygons with small number of vertices.
References
- 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),
(0.22926679, 0.47329807)])
mesh = discretize(polyarea, Dehn1899())
fig = Mke.Figure(resolution = (800, 400))
viz(fig[1,1], polyarea)
viz(fig[1,2], mesh, showfacets = true)
fig
FIST
Meshes.FIST
— TypeFIST([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, FIST())
fig = Mke.Figure(resolution = (800, 400))
viz(fig[1,1], polyarea)
viz(fig[1,2], mesh, showfacets = 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),
(0.18142937, 0.54681134)]
inners = [[(0.87789994, 0.32551613), (0.5614043, 0.540334), (0.9494598, 0.39622766), (0.87789994, 0.32551613)],
[(0.2799388, 0.52516246), (0.38555774, 0.32233855), (0.36943135, 0.30108362), (0.2799388, 0.52516246)]]
polyarea = PolyArea(outer, inners)
mesh = discretize(polyarea, FIST())
fig = Mke.Figure(resolution = (800, 400))
viz(fig[1,1], polyarea)
viz(fig[1,2], mesh, showfacets = true)
fig