# Plot

Polyhedra contains utilities to visualize either a 2-dimensional or a 3-dimensional polyhedron, see Polyhedron for how to construct a polyhedron, e.g. from its H- or V-representation.

## 2D plotting with Plots

A 2-dimensional polytope, i.e. bounded polyhedron, can be visualized with Plots. Suppose for instance that we want to visualize the polyhedron having the following H-representation:

using Polyhedra
h = HalfSpace([1, 1], 1) ∩ HalfSpace([-1, 0], 0) ∩ HalfSpace([0, -1], 0)
H-representation Polyhedra.Intersection{Int64, Vector{Int64}, Int64}:
3-element iterator of HalfSpace{Int64, Vector{Int64}}:
HalfSpace([1, 1], 1)
HalfSpace([-1, 0], 0)
HalfSpace([0, -1], 0)

The H-representation cannot be given to Plots directly, it first need to be transformed into a polyhedron:

p = polyhedron(h)
Polyhedron DefaultPolyhedron{Rational{BigInt}, Polyhedra.Intersection{Rational{BigInt}, Vector{Rational{BigInt}}, Int64}, Polyhedra.Hull{Rational{BigInt}, Vector{Rational{BigInt}}, Int64}}:
3-element iterator of HalfSpace{Rational{BigInt}, Vector{Rational{BigInt}}}:
HalfSpace(Rational{BigInt}[1//1, 1//1], 1//1)
HalfSpace(Rational{BigInt}[-1//1, 0//1], 0//1)
HalfSpace(Rational{BigInt}[0//1, -1//1], 0//1)

The polyhedron can be given to Plots as follows. We use ratio=:equal so that the horizontal and vertical axis have the same scale.

using Plots
plot(p, ratio=:equal)

See Polyhedral Function and 3D Plotting a projection of the 4D permutahedron for example notebooks.

## 3D plotting with Plots

A 3-dimensional polyhedron can be visualized with either MeshCat or Makie. Unbounded polyhedron are supported by truncating the polyhedron into a polytope and not triangularizing the faces in the directions of unbounded rays.

Suppose for instance that we want to visualize the polyhedron having the following H-representation:

julia> using Polyhedra

julia> v = convexhull([0, 0, 0]) + conichull([1, 0, 0], [0, 1, 0], [0, 0, 1])
V-representation Polyhedra.Hull{Int64, Vector{Int64}, Int64}:
1-element iterator of Vector{Int64}:
[0, 0, 0],
3-element iterator of Ray{Int64, Vector{Int64}}:
Ray([1, 0, 0])
Ray([0, 1, 0])
Ray([0, 0, 1])

The V-representation cannot be given to MeshCat or Makie directly, it first need to be transformed into a polyhedron:

julia> p = polyhedron(v)
Polyhedron DefaultPolyhedron{Rational{BigInt}, Polyhedra.Intersection{Rational{BigInt}, Vector{Rational{BigInt}}, Int64}, Polyhedra.Hull{Rational{BigInt}, Vector{Rational{BigInt}}, Int64}}:
1-element iterator of Vector{Rational{BigInt}}:
Rational{BigInt}[0//1, 0//1, 0//1],
3-element iterator of Ray{Rational{BigInt}, Vector{Rational{BigInt}}}:
Ray(Rational{BigInt}[1//1, 0//1, 0//1])
Ray(Rational{BigInt}[0//1, 1//1, 0//1])
Ray(Rational{BigInt}[0//1, 0//1, 1//1])

Then, we need to create a mess from the polyhedron:

julia> m = Polyhedra.Mesh(p)
Polyhedra.Mesh{3, Rational{BigInt}, DefaultPolyhedron{Rational{BigInt}, Polyhedra.Intersection{Rational{BigInt}, Vector{Rational{BigInt}}, Int64}, Polyhedra.Hull{Rational{BigInt}, Vector{Rational{BigInt}}, Int64}}}(convexhull([0//1, 0//1, 0//1]) + convexhull(Ray(Rational{BigInt}[1//1, 0//1, 0//1]), Ray(Rational{BigInt}[0//1, 1//1, 0//1]), Ray(Rational{BigInt}[0//1, 0//1, 1//1])), nothing, nothing, nothing)
Polyhedra.MeshType
struct Mesh{N, T, PT <: Polyhedron{T}} <: GeometryBasics.GeometryPrimitive{N, T}
polyhedron::PT
coordinates::Union{Nothing, Vector{GeometryBasics.Point{3, T}}}
faces::Union{Nothing, Vector{GeometryBasics.TriangleFace{Int}}}
normals::Union{Nothing, Vector{GeometryBasics.Point{3, T}}}
end

Mesh wrapper type that inherits from GeometryPrimitive to be used for plotting a polyhedron. Note that Mesh(p) is type unstable but one can use Mesh{3}(p) instead if it is known that p is defined in a 3-dimensional space.

source

The polyhedron can be plotted with MeshCat as follows

julia> using MeshCat

julia> vis = Visualizer()

julia> setobject!(vis, m)

julia> open(vis)

To plot it in a notebook, replace open(vis) with IJuliaCell(vis).

To plot it with Makie instead, you can use for instance mesh or wireframe.

julia> import Makie

julia> Makie.mesh(m, color=:blue)

julia> Makie.wireframe(m)

See 3D Plotting a projection of the 4D permutahedron for an example notebook.