Introduction

coefficient_field(P::Union{Polyhedron{T}, Cone{T}, PolyhedralFan{T}, PolyhedralComplex{T}) where T<:scalar_types

Return the parent Field of the coefficients of P.

Examples

julia> c = cross_polytope(2)
Polytope in ambient dimension 2

julia> coefficient_field(c)
Rational field

point_vector(p = QQ, v::AbstractVector)

Return a PointVector resembling a point whose coordinates equal the entries of v. p specifies the Field or Type of its coefficients.

ray_vector(p = QQ, v::AbstractVector)

Return a RayVector resembling a ray from the origin through the point whose coordinates equal the entries of v. p specifies the Field or Type of its coefficients.

affine_halfspace(p = QQ, a, b)

Return the Oscar.AffineHalfspace H(a,b), which is given by a vector a and a value b such that $H(a,b) = \{ x | ax ≤ b \}.$ p specifies the Field or Type of its coefficients.

linear_halfspace(p = QQ, a, b)

Return the Oscar.LinearHalfspace H(a), which is given by a vector a such that $H(a,b) = \{ x | ax ≤ 0 \}.$ p specifies the Field or Type of its coefficients.

affine_hyperplane(p = QQ, a, b)

Return the Oscar.AffineHyperplane H(a,b), which is given by a vector a and a value b such that $H(a,b) = \{ x | ax = b \}.$ p specifies the Field or Type of its coefficients.

linear_hyperplane(p = QQ, a, b)

Return the Oscar.LinearHyperplane H(a), which is given by a vector a such that $H(a,b) = \{ x | ax = 0 \}.$ p specifies the Field or Type of its coefficients.

IncidenceMatrix

A matrix with boolean entries. Each row corresponds to a fixed element of a collection of mathematical objects and the same holds for the columns and a second (possibly equal) collection. A 1 at entry (i, j) is interpreted as an incidence between object i of the first collection and object j of the second one.

Examples

Note that the input of this example and the print of an IncidenceMatrix list the non-zero indices for each row.

julia> IM = incidence_matrix([[1,2,3],[4,5,6]])
2×6 IncidenceMatrix
[1, 2, 3]
[4, 5, 6]


julia> IM[1, 2]
true

julia> IM[2, 3]
false

julia> IM[:, 4]
2-element SparseVectorBool
[2]

incidence_matrix(r::Base.Integer, c::Base.Integer)

Return an IncidenceMatrix of size r x c whose entries are all false.

Examples

julia> IM = incidence_matrix(8, 5)
8×5 IncidenceMatrix
[]
[]
[]
[]
[]
[]
[]
[]

incidence_matrix(mat::Union{AbstractMatrix{Bool}, IncidenceMatrix})

Convert mat to an IncidenceMatrix.

Examples

julia> IM = incidence_matrix([true false true false true false; false true false true false true])
2×6 IncidenceMatrix
[1, 3, 5]
[2, 4, 6]

incidence_matrix(mat::AbstractMatrix)

Convert the 0/1 matrix mat to an IncidenceMatrix. Entries become true if the initial entry is 1 and false if the initial entry is 0.

Examples

julia> IM = incidence_matrix([1 0 1 0 1 0; 0 1 0 1 0 1])
2×6 IncidenceMatrix
[1, 3, 5]
[2, 4, 6]

incidence_matrix(r::Base.Integer, c::Base.Integer, incidenceRows::AbstractVector{<:AbstractVector{<:Base.Integer}})

Return an IncidenceMatrix of size r x c. The i-th element of incidenceRows lists the indices of the true entries of the i-th row.

Examples

julia> IM = incidence_matrix(3, 4, [[2, 3], [1]])
3×4 IncidenceMatrix
[2, 3]
[1]
[]

incidence_matrix(incidenceRows::AbstractVector{<:AbstractVector{<:Base.Integer}})

Return an IncidenceMatrix where the i-th element of incidenceRows lists the indices of the true entries of the i-th row. The dimensions of the result are the smallest possible row and column count that can be deduced from the input.

Examples

julia> IM = incidence_matrix([[2, 3], [1]])
2×3 IncidenceMatrix
[2, 3]
[1]

incidence_matrix(g::Graph{T}) where {T <: Union{Directed, Undirected}}

Return an unsigned (boolean) incidence matrix representing a graph g.

Examples

julia> g = Graph{Directed}(5);

julia> add_edge!(g, 1, 3);

julia> add_edge!(g, 3, 4);

julia> incidence_matrix(g)
5×2 IncidenceMatrix
[1]
[]
[1, 2]
[2]
[]

row(i::IncidenceMatrix, n::Int)

Return the indices where the n-th row of i is true, as a Set{Int}.

Examples

julia> IM = incidence_matrix([[1,2,3],[4,5,6]])
2×6 IncidenceMatrix
[1, 2, 3]
[4, 5, 6]


julia> row(IM, 2)
Set{Int64} with 3 elements:
  5
  4
  6

column(i::IncidenceMatrix, n::Int)

Return the indices where the n-th column of i is true, as a Set{Int}.

Examples

julia> IM = incidence_matrix([[1,2,3],[4,5,6]])
2×6 IncidenceMatrix
[1, 2, 3]
[4, 5, 6]


julia> column(IM, 5)
Set{Int64} with 1 element:
  2

visualize(P::Union{Polyhedron{T}, Cone{T}, PolyhedralFan{T}, PolyhedralComplex{T}, SubdivisionOfPoints{T}}; kwargs...) where T<:Union{FieldElem, Float64}

Visualize a polyhedral object of dimension at most four (in 3-space). In dimensions up to 3 a usual embedding is shown. Four-dimensional polytopes are visualized as a Schlegel diagram, which is a projection onto one of the facets; e.g., see Chapter 5 of [Zie95].

In higher dimensions there is no standard method; use projections to lower dimensions or try ideas from [GJRW10].

Extended help

Keyword Arguments

Colors

Colors can be given as

• a literal String, e.g. "green".
• a String of the format "r g b" where $r, g, b \in {0, \dots, 255}$ are integers corresponding to the R/G/B values of the color.
• a String of the format "r g b" where $r, g, b \in [0, 1]$ are decimal values corresponding to the R/G/B values of the color.

Possible arguments are:

• FacetColor: Filling color of the polygons.
• EdgeColor: Color of the boundary lines.
• PointColor/VertexColor: Color of the spheres or rectangles representing the points.

Scaling and other gradient properties

These arguments can be given as a floating point number:

• FacetTransparency: Transparency factor of the polygons between 0 (opaque) and 1 (completely translucent).
• EdgeThickness: Scaling factor for the thickness of the boundary lines.
• PointThickness/VertexThickness`: Scaling factor for the size of the spheres or rectangles representing the points.

Camera

These arguments can be given as a 3-element vector over floating point numbers:

• ViewPoint: Position of the camera.
• ViewDirection: Direction of the camera.

Appearance and Texts

These arguments can be given as a string:

• FacetStyle: If set to "hidden", the inner area of the polygons are not rendered at all.
• FacetLabels: If set to "hidden", the facet labels are not displayed (in the most cases this is the default behavior). TODO
• EdgeStyle: If set to "hidden", the boundary lines are not rendered.
• Name: The name of this visual object in the drawing.
• PointLabels/VertexLabels: If set to "hidden", no point labels are displayed.
• PointStyle/VertexStyle: If set to "hidden", neither point nor its label is rendered.
• LabelAlignment: Defines the alignment of the vertex labels: "left", "right" or "center".