Meshes

Meshes can be constructed directly (e.g. CartesianGrid) or based on other constructs such as connectivity lists and topological structures (e.g. SimpleMesh).

Overview

Meshes.Mesh — Type

Mesh{M,CRS,TP}

A mesh of geometries in a given manifold M with point coordinates specified in a coordinate reference system CRS. Unlike a general domain, a mesh has a well-defined topology TP.

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Meshes.Grid — Type

Grid{M,CRS,Dim}

A grid of geometries in a given manifold M with points coordinates specified in a coordinate reference system CRS, which is embedded in Dim dimensions.

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Meshes.RegularGrid — Type

RegularGrid(dims, origin, spacing)

A regular grid with dimensions dims, lower left corner at origin and cell spacing spacing. The three arguments must have the same length.

RegularGrid(dims, origin, spacing, offset)

A regular grid with dimensions dims, with lower left corner of element offset at origin and cell spacing spacing.

RegularGrid(start, finish, dims=dims)

Alternatively, construct a regular grid from a start point to a finish with dimensions dims.

RegularGrid(start, finish, spacing)

Alternatively, construct a regular grid from a start point to a finish point using a given spacing.

RegularGrid(dims)
RegularGrid(dim1, dim2, ...)

Finally, a regular grid can be constructed by only passing the dimensions dims as a tuple, or by passing each dimension dim1, dim2, ... separately. In this case, the origin and spacing default to (0,0,...) and (1,1,...).

Examples

Create a 3D grid with 100x100x50 hexahedrons:

julia> RegularGrid(100, 100, 50)

Create a 2D grid with 100 x 100 quadrangles and origin at (10.0, 20.0):

julia> RegularGrid((100, 100), (10.0, 20.0), (1.0, 1.0))

Create a 1D grid from -1 to 1 with 100 segments:

julia> RegularGrid((-1.0,), (1.0,), dims=(100,))

Connectivities

Meshes.Connectivity — Type

Connectivity{PL,N}

A connectivity list of N indices representing a Polytope of type PL. Indices are taken from a global vector of Point.

Connectivity objects are constructed with the connect function.

source

Meshes.connect — Function

connect(indices, [PL])

Connect a list of indices from a global vector of Point into a Polytope of type PL.

The type PL can be a Ngon in which case the length of the indices is used to identify the actual polytope type.

Finally, the type PL can be ommitted. In this case, the indices are assumed to be connected as a Ngon or as a Segment.

Examples

Connect indices into a Triangle:

connect((1,2,3), Triangle)

Connect indices into N-gons, a Triangle and a Quadrangle:

connect.([(1,2,3), (2,3,4,5)], Ngon)

Connect indices into N-gon or segment:

connect((1,2)) # Segment
connect((1,2,3)) # Triangle
connect((1,2,3,4)) # Quadrangle

source

Meshes.materialize — Function

materialize(connec, points)

Materialize a face using the connec list and a global vector of points.

source

Topology

Meshes.Topology — Type

Topology

A data structure for constructing topological relations in a Mesh.

References

Floriani, L. & Hui, A. 2007. Shape representations based on simplicial and cell complexes

source

Meshes.GridTopology — Type

GridTopology(dims, [periodic])

A data structure for grid topologies with dims elements. Optionally, specify which dimensions are periodic. Default to aperiodic dimensions.

Examples

julia> GridTopology((10,20)) # 10x20 elements in a grid
julia> GridTopology((10,20), (true,false)) # cylinder topology

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Meshes.HalfEdgeTopology — Type

HalfEdgeTopology(elements; sort=true)
HalfEdgeTopology(halfedges; nelems=nothing)

A data structure for orientable 2-manifolds based on half-edges constructed from a vector of connectivity elements or from a vector of pairs of halfedges.

The option sort can be used to sort the elements in adjacent-first order in case of inconsistent orientation (i.e. mix of clockwise and counter-clockwise).

The option nelems can be used to specify an approximate number of elements as a size hint.

Examples

Construct half-edge topology from a list of top-faces:

elements = connect.([(1,2,3),(3,2,4,5)])
topology = HalfEdgeTopology(elements)

Relations

Meshes.TopologicalRelation — Type

TopologicalRelation

A topological relation between faces of a Mesh implemented for a given Topology.

An object implementing this trait is a functor that can be evaluated at an integer index representing the face.

Examples

# create boundary relation mapping
# 2-faces to 0-faces (i.e. vertices)
∂ = Boundary{2,0}(topology)

# list of vertices for first face
∂(1)

References

Floriani, L. & Hui, A. 2007. Shape representations based on simplicial and cell complexes

source

Meshes.Boundary — Type

Boundary{P,Q}(topology)

The boundary relation from rank P to smaller rank Q for a given topology.

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Meshes.Coboundary — Type

Coboundary{P,Q}(topology)

The co-boundary relation from rank P to greater rank Q for a given topology.

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Meshes.Adjacency — Type

Adjacency{P}(topology)

The adjacency relation of rank P for a given topology.

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Consider the following examples with the Boundary and Coboundary relations defined for the HalfEdgeTopology:

# global vector of 2D points
points = [(0,0),(1,0),(0,1),(1,1),(0.25,0.5),(0.75,0.5)]

# connect the points into N-gon
connec = connect.([(1,2,6,5),(2,4,6),(4,3,5,6),(3,1,5)], Ngon)

# 2D mesh made of N-gon elements
mesh = SimpleMesh(points, connec)

4 SimpleMesh
  6 vertices
  ├─ Point(x: 0.0 m, y: 0.0 m)
  ├─ Point(x: 1.0 m, y: 0.0 m)
  ├─ Point(x: 0.0 m, y: 1.0 m)
  ├─ Point(x: 1.0 m, y: 1.0 m)
  ├─ Point(x: 0.25 m, y: 0.5 m)
  └─ Point(x: 0.75 m, y: 0.5 m)
  4 elements
  ├─ Quadrangle(1, 2, 6, 5)
  ├─ Triangle(2, 4, 6)
  ├─ Quadrangle(4, 3, 5, 6)
  └─ Triangle(3, 1, 5)

# convert topology to half-edge topology
topo = convert(HalfEdgeTopology, topology(mesh))

# boundary relation from faces (dim=2) to edges (dim=1)
∂₂₁ = Boundary{2,1}(topo)

# show boundary of first n-gon
∂₂₁(1)

(1, 2, 3, 4)

# co-boundary relation from edges (dim=1) to faces(dim=2)
𝒞₁₂ = Coboundary{1,2}(topo)

# show n-gons that share edge 3
𝒞₁₂(3)

(1, 3)

Matrices

Based on topological relations, we can extract matrices that are widely used in applications such as laplacematrix, and adjacencymatrix.

Laplace

Meshes.laplacematrix — Function

laplacematrix(mesh; kind=nothing)

The Laplace-Beltrami (a.k.a. Laplacian) matrix of the mesh. Optionally, specify the kind of discretization.

Available discretizations

:uniform - Lᵢⱼ = 1 / |𝒜(i)|, ∀j ∈ 𝒜(i)
:cotangent - Lᵢⱼ = cot(αᵢⱼ) + cot(βᵢⱼ), ∀j ∈ 𝒜(i)

where 𝒜(i) is the adjacency relation at vertex i.

References

Botsch et al. 2010. Polygon Mesh Processing.
Pinkall, U. & Polthier, K. 1993. Computing discrete minimal surfaces and their conjugates.

source

grid = CartesianGrid(10, 10)

laplacematrix(grid, kind = :uniform)

121×121 SparseArrays.SparseMatrixCSC{Float64, Int64} with 561 stored entries:
⎡⠻⣦⡀⠈⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⡀⠈⠻⣦⡀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠘⢢⡀⠈⠻⣦⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠘⠢⡀⠈⠻⣦⡀⠈⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠈⠢⡀⠈⠻⣦⡀⠈⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠈⠦⡀⠈⠛⣤⡀⠈⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠈⠻⣦⡀⠈⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠈⠱⣦⡀⠈⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠈⠻⣦⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠈⠻⣦⡀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠻⣦⡀⠈⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠈⠻⣦⡀⠈⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡀⠈⠻⢆⡀⠈⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡀⠈⠻⣦⡀⠈⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡀⠈⠛⣤⡀⠈⠲⡀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡀⠈⠻⣦⡀⠈⠢⡀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠈⠻⣦⡀⠈⠢⡄⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠻⣦⡀⠈⠣⡄⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠈⠻⣦⡀⠈⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠈⠻⣦⎦

points = [(0, 0), (1, 0), (0, 1), (1, 1), (0.5, 0.5)]
connec = connect.([(1, 2, 5), (2, 4, 5), (4, 3, 5), (3, 1, 5)])
mesh = SimpleMesh(points, connec)

laplacematrix(mesh, kind = :cotangent)

5×5 SparseArrays.SparseMatrixCSC{Float64, Int64} with 21 stored entries:
 -0.0  -1.0  -1.0    ⋅    2.0
 -1.0  -0.0    ⋅   -1.0   2.0
 -1.0    ⋅   -0.0  -1.0   2.0
   ⋅   -1.0  -1.0  -0.0   2.0
  2.0   2.0   2.0   2.0  -8.0

Measure

Meshes.measurematrix — Function

measurematrix(mesh)

The measure (or "mass") matrix of the mesh, i.e. a diagonal matrix with entries Mᵢᵢ = 2Aᵢ where Aᵢ is (one-third of) the sum of the areas of triangles sharing vertex i.

The discrete cotangent Laplace-Beltrami operator can be written as Δ = M⁻¹L. When solving systems of the form Δu = f, it is useful to write Lu = Mf and exploit the symmetry of L.

source

grid = CartesianGrid(10, 10)

measurematrix(grid)

121×121 LinearAlgebra.Diagonal{Unitful.Quantity{Float64, 𝐋^2, Unitful.FreeUnits{(m^2,), 𝐋^2, nothing}}, Vector{Unitful.Quantity{Float64, 𝐋^2, Unitful.FreeUnits{(m^2,), 𝐋^2, nothing}}}}:
 0.666667 m^2     ⋅            ⋅         …     ⋅            ⋅   
    ⋅          1.33333 m^2     ⋅               ⋅            ⋅   
    ⋅             ⋅         1.33333 m^2        ⋅            ⋅   
    ⋅             ⋅            ⋅               ⋅            ⋅   
    ⋅             ⋅            ⋅               ⋅            ⋅   
    ⋅             ⋅            ⋅         …     ⋅            ⋅   
    ⋅             ⋅            ⋅               ⋅            ⋅   
    ⋅             ⋅            ⋅               ⋅            ⋅   
    ⋅             ⋅            ⋅               ⋅            ⋅   
    ⋅             ⋅            ⋅               ⋅            ⋅   
 ⋮                                       ⋱               ⋮
    ⋅             ⋅            ⋅               ⋅            ⋅   
    ⋅             ⋅            ⋅               ⋅            ⋅   
    ⋅             ⋅            ⋅               ⋅            ⋅   
    ⋅             ⋅            ⋅         …     ⋅            ⋅   
    ⋅             ⋅            ⋅               ⋅            ⋅   
    ⋅             ⋅            ⋅               ⋅            ⋅   
    ⋅             ⋅            ⋅               ⋅            ⋅   
    ⋅             ⋅            ⋅            1.33333 m^2     ⋅   
    ⋅             ⋅            ⋅         …     ⋅         0.666667 m^2

points = [(0, 0), (1, 0), (0, 1), (1, 1), (0.5, 0.5)]
connec = connect.([(1, 2, 5), (2, 4, 5), (4, 3, 5), (3, 1, 5)])
mesh = SimpleMesh(points, connec)

measurematrix(mesh)

5×5 LinearAlgebra.Diagonal{Unitful.Quantity{Float64, 𝐋^2, Unitful.FreeUnits{(m^2,), 𝐋^2, nothing}}, Vector{Unitful.Quantity{Float64, 𝐋^2, Unitful.FreeUnits{(m^2,), 𝐋^2, nothing}}}}:
 0.333333 m^2     ⋅             ⋅             ⋅             ⋅   
    ⋅          0.333333 m^2     ⋅             ⋅             ⋅   
    ⋅             ⋅          0.333333 m^2     ⋅             ⋅   
    ⋅             ⋅             ⋅          0.333333 m^2     ⋅   
    ⋅             ⋅             ⋅             ⋅          0.666667 m^2

Adjacency

Meshes.adjacencymatrix — Function

adjacencymatrix(mesh; rank=paramdim(mesh))

The adjacency matrix of the mesh using the adjacency relation of given rank for the underlying topology.

source

grid = CartesianGrid(10, 10)

adjacencymatrix(grid)

100×100 SparseArrays.SparseMatrixCSC{Int64, Int64} with 360 stored entries:
⎡⠪⡦⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠈⠪⡦⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠑⢄⠀⠀⠪⡦⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠑⢄⠀⠈⠪⡦⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠑⢄⠀⠀⠪⡦⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠪⡦⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠪⡦⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠪⡦⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠪⡦⡀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠪⡦⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠺⡢⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠈⠺⡢⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠺⡢⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠺⡢⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠺⡢⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠺⡢⠀⠀⠑⢄⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠺⡢⡀⠀⠑⢄⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠺⡢⠀⠀⠑⢄⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠺⡢⡀⠀⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠺⡢⎦

adjacencymatrix(grid, rank = 0)

121×121 SparseArrays.SparseMatrixCSC{Int64, Int64} with 440 stored entries:
⎡⠺⣢⡀⠈⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⡀⠈⠺⡢⡀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠘⢢⡀⠈⠺⣢⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠘⠢⡀⠈⠺⡢⡀⠈⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠈⠢⡀⠈⠫⡦⡀⠈⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠈⠦⡀⠈⠋⡤⡀⠈⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠈⠫⡦⡀⠈⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠈⠡⡦⡀⠈⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠈⠫⡦⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠈⠪⣦⡀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠺⡢⡀⠈⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠈⠺⣢⡀⠈⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡀⠈⠺⢂⡀⠈⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡀⠈⠺⣢⡀⠈⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡀⠈⠚⣠⡀⠈⠲⡀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡀⠈⠺⣢⡀⠈⠢⡀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠈⠪⡦⡀⠈⠢⡄⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠫⡦⡀⠈⠣⡄⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠈⠪⡦⡀⠈⎥
⎣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠈⠫⡦⎦