From 153ad7fe3f2b412f4927e1c5b6007c58cc7d07df Mon Sep 17 00:00:00 2001
From: marteresagh <marteresa28@gmail.com>
Date: Sat, 20 Apr 2019 18:47:16 +0200
Subject: [PATCH 01/10] graph_img

---
 docs/src/images/dependency_graph.png | Bin 0 -> 60304 bytes
 1 file changed, 0 insertions(+), 0 deletions(-)
 create mode 100644 docs/src/images/dependency_graph.png

diff --git a/docs/src/images/dependency_graph.png b/docs/src/images/dependency_graph.png
new file mode 100644
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From a68c2b4e94cde008a41c67672c9a75dd121d4752 Mon Sep 17 00:00:00 2001
From: marteresagh <marteresa28@gmail.com>
Date: Sat, 20 Apr 2019 18:48:41 +0200
Subject: [PATCH 02/10] documentation

---
 docs/src/arrangement.md | 99 ++++++++++++++++++++++++++++++++++++++---
 1 file changed, 93 insertions(+), 6 deletions(-)

diff --git a/docs/src/arrangement.md b/docs/src/arrangement.md
index 528460ef..41bd8266 100644
--- a/docs/src/arrangement.md
+++ b/docs/src/arrangement.md
@@ -9,12 +9,25 @@ This operation can be seen as a boolean union of two cellular complexes. Here an
 ![10 x 10 Cube](./images/cube10x10.jpg)
 > **Figure 1:** Arrangement of ``2000=2\times10\times10\times10`` cubes
 
+## Graph of recall functions
+A graph is an ordered pair ``G(V,E)`` comprising a set ``V`` of vertices, nodes or points together with a set ``E`` of edges, arcs or lines, which are ``2``-elements subsets of ``V``. In this case, ``V`` is the set of functions of this sub-module and let ``v1``, ``v2`` be elements of ``V``, then ``e=v1v2`` ``\in`` ``E`` if ``v1`` calls ``v2``.
+In the following graph the functions are arranged in colored boxes:
+![dependency graph](./images/dependency_graph.png)
+- yellow boxes is for `spatial_arrangement`
+- red boxes is for `planar_arrangement`
+- green boxes is for `minimal_cycles`
+- blue boxes is for `dimension_travel`
+
+Functions in darker boxes are declared only in the local scope of its father.
+
 ## API
 
-Every function strictly relative to the arrangement has been collected in the `Lar.Arrangement` sub-module but the two main functions are accessible directly from the `LinearAlgebraicRepresentation` namespace.
+Every functions strictly relative to the arrangement has been collected in the `Lar.Arrangement` sub-module but the two main functions are accessible directly from the `LinearAlgebraicRepresentation` namespace.
+
 !!! warning
     `Lar.Arrangement` is the only place in `LinearAlgebraicRepresentation` where `Point` matrices store points per row and not per column as described in the documentation of `Lar.Points`
 
+
 ```@docs
 Lar.spatial_arrangement
 Lar.planar_arrangement
@@ -47,14 +60,14 @@ the spatial index: it is a mapping ``\mathcal{I}(\sigma)`` from a cell
 where the ``box`` function provides the axis aligned bounding box (AABB) of a cell [fig. 2, c, 
 ``\sigma`` in red and ``\mathcal{I}(\sigma)`` in blue]. The spatial arrangement
 calculation is speeded up by storing the AABBs as dimensional wise intervals
-into an interval tree \cite{interval_trees}. 
+into an interval tree ``\cite{interval_trees}``. 
 Now for each cell ``\sigma`` we transform ``\sigma \cup \mathcal{I}(\sigma)`` 
-in a way that ``\sigma`` lays on the ``x_3=0`` plane [fig. 2, d] and we find the intersections 
-of the ``\mathcal{I}(\sigma)`` cells with ``x_3=0`` plane. So we have a "soup"
+in a way that ``\sigma`` lays on the ``x_3=0``  plane [fig. 2, d] and we find the intersections 
+of the ``\mathcal{I}(\sigma)`` cells with ``x_3=0``  plane. So we have a "soup"
 of 1-cells in ``\mathbb{E}^2`` [fig. 2, e], and we fragment each 1-cell 
 with every other cell obtaining a valid 1-skeleton [fig. 2, f].
 From this data it is possible to build the 2-cells using the ALGORITHM 1
-presented and explored by Paoluzzi et al. \cite{Paoluzzi}
+presented and explored by Paoluzzi et al. ``\cite{Paoluzzi}``
 [fig. 2, g, exploded]. The procedure to fragment 1-cells
 on a plane and return a 2-complex is called *planar arrangement*. When the planar arrangement 
 is complete, fragmented ``\sigma`` can be transformed back to its original position
@@ -97,4 +110,78 @@ cell 2 has cell 1 as an hole].
 !!! note
     A 2-cell with a non-intersecting shell can be trivially defined
     as a "face with holes"; the correct definition is that it cannot 
-    be shrunk to the dimension of a point.
\ No newline at end of file
+    be shrunk to the dimension of a point.
+
+## Main Interface
+
+### Dimension travel
+
+### Minimal cycles
+
+```@docs
+Lar.Arrangement.minimal_2cycles
+```
+
+```@docs
+Lar.Arrangement.minimal_3cycles
+```
+
+### Planar arrangement
+
+```@docs
+Lar.Arrangement.intersect_edges
+```
+
+```@docs
+Lar.Arrangement.frag_edge
+```
+
+```@docs
+Lar.Arrangement.merge_vertices!
+```
+
+```@docs
+Lar.Arrangement.biconnected_components
+```
+
+```@docs
+Lar.Arrangement.get_external_cycle
+```
+
+```@docs
+Lar.Arrangement.pre_containment_test
+```
+
+```@docs
+Lar.Arrangement.prune_containment_graph
+```
+
+```@docs
+Lar.Arrangement.transitive_reduction!
+```
+
+```@docs
+Lar.Arrangement.componentgraph
+```
+
+```@docs
+Lar.Arrangement.cell_merging
+```
+
+```@docs
+Lar.Arrangement.cleandecomposition
+```
+
+```@docs
+Lar.Arrangement.planar_arrangement_1
+```
+
+```@docs
+Lar.Arrangement.planar_arrangement_2
+```
+
+### Spatial arrangement
+
+
+
+

From f8715d8c1577ac39d393bc466233e79c848a5a4f Mon Sep 17 00:00:00 2001
From: marteresagh <marteresa28@gmail.com>
Date: Sat, 20 Apr 2019 18:50:15 +0200
Subject: [PATCH 03/10] Add docstrings

---
 src/arrangement/minimal_cycles.jl     |  13 +-
 src/arrangement/planar_arrangement.jl | 164 ++++++++++++++++++++++++++
 2 files changed, 176 insertions(+), 1 deletion(-)

diff --git a/src/arrangement/minimal_cycles.jl b/src/arrangement/minimal_cycles.jl
index 6ff94735..ce57cc0d 100644
--- a/src/arrangement/minimal_cycles.jl
+++ b/src/arrangement/minimal_cycles.jl
@@ -1,5 +1,11 @@
 Lar = LinearAlgebraicRepresentation
 
+"""
+	minimal_2cycles(V::Lar.Points, EV::Lar.ChainOp)
+
+Return all cycles of a 2D graph define by its vertices and edges.
+
+"""
 function minimal_2cycles(V::Lar.Points, EV::Lar.ChainOp)
 
     function edge_angle(v::Int, e::Int)
@@ -14,12 +20,17 @@ function minimal_2cycles(V::Lar.Points, EV::Lar.ChainOp)
         EV[i, j] = -1
     end
     VE = convert(Lar.ChainOp, SparseArrays.transpose(EV))
-
+@show V
+@show VE
     EF = Lar.Arrangement.minimal_cycles(edge_angle)(V, VE)
 
     return convert(Lar.ChainOp, SparseArrays.transpose(EF))
 end
 
+"""
+	minimal_3cycles(V::Lar.Points, EV::Lar.ChainOp, FE::Lar.ChainOp)
+
+"""
 function minimal_3cycles(V::Lar.Points, EV::Lar.ChainOp, FE::Lar.ChainOp)
 
     triangulated_faces = Array{Any, 1}(undef, FE.m)
diff --git a/src/arrangement/planar_arrangement.jl b/src/arrangement/planar_arrangement.jl
index ddcf7790..594b33ce 100644
--- a/src/arrangement/planar_arrangement.jl
+++ b/src/arrangement/planar_arrangement.jl
@@ -13,6 +13,42 @@ function frag_edge_channel(in_chan, out_chan, V, EV, bigPI)
     end
 end
 
+
+"""
+	frag_edge(V::Lar.Points, EV::Lar.ChainOp, edge_idx::Int, bigPI)
+
+Return vertices and edges after intersection.
+# Example
+```julia
+julia> V = [ 0 0; 1 1; 1 0; 0 1];
+
+julia> EV = Int8[ 1 1 0 0;
+                  0 0 1 1; 
+                  ];
+
+julia> EV = sparse(EV);
+
+julia> model = (convert(Lar.Points,V'),Lar.cop2lar(EV));
+
+julia> bigPI = Lar.spaceindex(model::Lar.LAR);
+
+julia> Lar.Arrangement.frag_edge(V, EV, 1, bigPI)
+([0.0 0.0; 1.0 1.0; 0.5 0.5], 
+  [1, 1]  =  1
+  [2, 2]  =  1
+  [1, 3]  =  1
+  [2, 3]  =  1)
+
+julia> Lar.Arrangement.frag_edge(V, EV, 2, bigPI)
+([1.0 0.0; 0.0 1.0; 0.5 0.5], 
+  [1, 1]  =  1
+  [2, 2]  =  1
+  [1, 3]  =  1
+  [2, 3]  =  1)
+
+
+```
+"""
 function frag_edge(V::Lar.Points, EV::Lar.ChainOp, edge_idx::Int, bigPI)
     alphas = Dict{Float64, Int}()
     edge = EV[edge_idx, :]
@@ -39,6 +75,27 @@ function frag_edge(V::Lar.Points, EV::Lar.ChainOp, edge_idx::Int, bigPI)
     return verts, ev
 end
 
+
+"""
+	intersect_edges(V::Lar.Points, edge1::Lar.Cell, edge2::Lar.Cell)
+
+Intersection of two edges. Return, if exist, points of intersection and parameter.
+
+# Example
+```julia
+julia> V=[0 0 ; 1 1; 1/2 0; 1/2 1];
+
+julia> EV = SparseArrays.sparse(Array{Int8, 2}([
+                            [1 1 0 0] #1->1,2
+                            [0 0 1 1] #2->3,4       
+                        ]));
+
+julia> Lar.Arrangement.intersect_edges(V, EV[1, :], EV[2, :])
+1-element Array{Tuple{Array{T,2} where T,Float64},1}:
+ ([0.5 0.5], 0.5)
+
+```
+"""
 function intersect_edges(V::Lar.Points, edge1::Lar.Cell, edge2::Lar.Cell)
     err = 10e-8
 
@@ -83,6 +140,36 @@ function intersect_edges(V::Lar.Points, edge1::Lar.Cell, edge2::Lar.Cell)
     return ret
 end
 
+"""
+	merge_vertices!(V::Lar.Points, EV::Lar.ChainOp, edge_map, err=1e-4)
+
+If two or more vertices are very close, return one vertex and right edges. 
+
+# Example
+```julia
+julia> p0 = 1e-2;
+
+julia> pm = 1-p0;
+
+julia> pp = 1+p0;
+
+julia> V = [ p0  p0; p0 -p0;
+                    pp pm; pp pp
+                  ];
+
+julia> EV = Int8[1 0 1 0 ;
+                 0 1 0 1 ;
+                 1 0 0 1 ;
+                 0 1 1 0 ];
+
+julia> EV = sparse(EV);
+
+julia> Lar.Arrangement.merge_vertices!(V, EV, [],1e-1)
+([0.01 0.01; 1.01 0.99], 
+  [1, 1]  =  1
+  [1, 2]  =  1)
+```
+"""
 function merge_vertices!(V::Lar.Points, EV::Lar.ChainOp, edge_map, err=1e-4)
     vertsnum = size(V, 1)
     edgenum = size(EV, 1)
@@ -144,6 +231,29 @@ function merge_vertices!(V::Lar.Points, EV::Lar.ChainOp, edge_map, err=1e-4)
     return Lar.Points(nV), nEV
 end
 
+"""
+	biconnected_components(EV::Lar.ChainOp)
+
+Find the biconnected components of a graph define by his edges. A biconnected component is a maximal biconnected subgraph. A biconnected graph has no **articulation vertices**. 
+
+# Example
+```julia
+julia> EV = Int8[1 1 0 0 0 0;
+               0 1 1 0 0 0;
+               1 0 1 0 0 0;
+               1 0 0 0 1 0;
+               0 0 0 1 1 0;
+               0 0 0 1 0 1;
+               0 0 0 0 1 1] ;
+
+julia> EV = sparse(EV);
+
+julia> bc = Lar.Arrangement.biconnected_components(EV)
+2-element Array{Array{Int64,1},1}:
+ [3, 2, 1]
+ [7, 6, 5]
+```
+"""
 function biconnected_components(EV::Lar.ChainOp)
     ps = Array{Tuple{Int, Int, Int}, 1}()
     es = Array{Tuple{Int, Int}, 1}()
@@ -240,6 +350,11 @@ function biconnected_components(EV::Lar.ChainOp)
     return bicon_comps
 end
 
+"""
+	get_external_cycle(V::Lar.Points, EV::Lar.ChainOp, FE::Lar.ChainOp)
+
+Get the face's index of external cell in FE. 
+"""
 function get_external_cycle(V::Lar.Points, EV::Lar.ChainOp, FE::Lar.ChainOp)
     FV = abs.(FE)*EV
     vs = sparsevec(mapslices(sum, abs.(EV), dims=1)').nzind
@@ -272,6 +387,13 @@ function get_external_cycle(V::Lar.Points, EV::Lar.ChainOp, FE::Lar.ChainOp)
         end
     end
 end
+
+"""
+	pre_containment_test(bboxes)
+
+Return containment graph. An element **(i,j)** is **1** if the **i-th** cell is contained in the **boundary box** of the **j-th** cell.
+   
+"""
 function pre_containment_test(bboxes)
     n = length(bboxes)
     containment_graph = spzeros(Int8, n, n)
@@ -286,6 +408,12 @@ function pre_containment_test(bboxes)
 
     return containment_graph
 end
+
+"""
+	prune_containment_graph(n, V, EVs, shells, graph)
+
+Check if the origin point of a cell is inside the face area of other cell in the graph.   
+"""
 function prune_containment_graph(n, V, EVs, shells, graph)
     
     for i in 1:n
@@ -309,6 +437,27 @@ function prune_containment_graph(n, V, EVs, shells, graph)
      end
      return graph
 end
+
+"""
+	transitive_reduction!(graph)
+
+Remove elements from containment graph that can be compute for transitivity.
+
+# Example
+```julia
+julia> graph = [0 1 1 1 ; 0 0 1 1 ; 0 0 0 1 ; 0 0 0 0 ];
+
+julia> Lar.Arrangement.transitive_reduction!(graph)
+
+julia> graph
+4×4 Array{Int64,2}:
+ 0  1  0  0
+ 0  0  1  0
+ 0  0  0  1
+ 0  0  0  0
+
+```
+"""
 function transitive_reduction!(graph)
     n = size(graph, 1)
     for j in 1:n
@@ -324,6 +473,12 @@ function transitive_reduction!(graph)
     end
 end
 
+"""
+	cell_merging(n, containment_graph, V, EVs, boundaries, shells, shell_bboxes)
+
+Merge all cells.
+
+"""
 function cell_merging(n, containment_graph, V, EVs, boundaries, shells, shell_bboxes)
     function bboxes(V::Lar.Points, indexes::Lar.ChainOp)
         boxes = Array{Tuple{Any, Any}}(undef, indexes.n)
@@ -379,7 +534,11 @@ function cell_merging(n, containment_graph, V, EVs, boundaries, shells, shell_bb
     return EV, FE
 end
 
+"""
+	componentgraph(V, copEV, bicon_comps)
 
+Return some properties of a graph, in order: `n`, `containment_graph`, `V`, `EVs`, `boundaries`, `shells`, `shell_bboxes`. 
+"""
 function componentgraph(V, copEV, bicon_comps)
 
 	# arrangement of isolated components
@@ -425,7 +584,12 @@ function componentgraph(V, copEV, bicon_comps)
 	return n, containment_graph, V, EVs, boundaries, shells, shell_bboxes
 end
 
+"""
+	cleandecomposition(V, copEV, sigma)
+
+Delete edges outside sigma area.
 
+"""
 function cleandecomposition(V, copEV, sigma)
     # Deletes edges outside sigma area
 	todel = []

From 24460f55ac1cbf4afb1e3cc044c188c9ec91f84a Mon Sep 17 00:00:00 2001
From: marteresagh <marteresa28@gmail.com>
Date: Sat, 20 Apr 2019 18:51:22 +0200
Subject: [PATCH 04/10] my test of planar arrangement

---
 test/planar_arr_mytest.jl | 176 ++++++++++++++++++++++++++++++++++++++
 1 file changed, 176 insertions(+)
 create mode 100644 test/planar_arr_mytest.jl

diff --git a/test/planar_arr_mytest.jl b/test/planar_arr_mytest.jl
new file mode 100644
index 00000000..1660fd24
--- /dev/null
+++ b/test/planar_arr_mytest.jl
@@ -0,0 +1,176 @@
+using Test
+using LinearAlgebraicRepresentation
+Lar = LinearAlgebraicRepresentation
+using Plasm,SparseArrays
+
+@testset "frag_edge" begin
+	V = [ 3 1; 6 8;2 -2;9 6;1 6;10 1]
+    
+	EV = Int8[ 1 1 0 0 0 0 ;
+        	   0 0 1 1 0 0 ;
+        	   0 0 0 0 1 1 ]
+	EV = sparse(EV)
+	edgenum = size(EV, 1)
+	rV = Lar.Points(zeros(0, 2))
+	rEV = SparseArrays.spzeros(Int8, 0, 0)
+	model = (convert(Lar.Points,V'),Lar.cop2lar(EV))
+	bigPI = Lar.spaceindex(model::Lar.LAR)
+	for i in 1:edgenum
+		v, ev = Lar.Arrangement.frag_edge(V, EV, i, bigPI)
+		rV, rEV = Lar.skel_merge(rV, rEV, v, ev) # block diagonal ...
+	end
+
+	@test Lar.cop2lar(rEV) == [[1,3],[2,3],[4,6],[5,6],[7,10],[9,10],[8,9]]
+end
+
+
+@testset "merge_vertices!" begin
+	p0 = 1e-2
+	pm = 1-p0
+	pp = 1+p0
+	V = [ p0  p0; p0 -p0; pp  pm; pp  pp]
+	EV = Int8[1 0 1 0 ;
+                  0 1 0 1 ;
+                  1 0 0 1 ;
+                  0 1 1 0 ]
+	EV = sparse(EV)
+	V, EV = Lar.Arrangement.merge_vertices!(V, EV, [],1e-1)
+	@test size(V,1)==2
+	@test size(EV,1)==1
+end
+
+
+@testset "biconnected_components" begin
+	@testset "Cycle graph" begin
+		EV = Int8[1 1 0 0;
+			0 1 1 0;
+			0 0 1 1;
+			1 0 0 1] 
+		EV = sparse(EV)
+
+		bc = Lar.Arrangement.biconnected_components(EV)
+		@test size(bc,1)==1
+	end
+
+	@testset "Graph with 2 biconnected components" begin
+		EV = Int8[1 1 0 0 0 0;
+			0 1 1 0 0 0;
+			1 0 1 0 0 0;
+			1 0 0 0 1 0;
+			0 0 0 1 1 0;
+			0 0 0 1 0 1;
+			0 0 0 0 1 1] ;
+		EV = sparse(EV);
+
+		bc = Lar.Arrangement.biconnected_components(EV)
+		@test size(bc,1)==2
+	end
+end
+
+
+@testset "get_external_cycle" begin
+	@testset "Example 1" begin
+
+		V = [ 0 0; 1 0;2 0;2 1;1 1;0 1]
+	    
+		EV = Int8[-1  1  0  0  0  0 ;
+			   0 -1  1  0  0  0 ;
+			   0  0 -1  1  0  0 ;
+			   0  0  0 -1  1  0 ;
+			   0  0  0  0 -1  1 ;
+			   1  0  0  0  0 -1 ;
+			   0  -1  0  0  1  0 
+			   ]
+		EV = sparse(EV)
+		
+		FE = Int8[ 1 1 1 1 1 1 0;
+			   1 0 0 0 1 1 -1;
+			   0 1 1 1 0 0 -1]
+		FE = sparse(FE)
+		@test Lar.Arrangement.get_external_cycle(V, EV, FE) == 1
+	end
+
+	@testset "Example 2" begin
+
+		V = [ 0 0; 2 0; 2 2; 0 2; 
+		      1 0; 3 0; 3 3; 1 2]
+	    
+		EV = Int8[-1  1  0  0  0  0  0  0;
+			   0 -1  1  0  0  0  0  0;
+			   0  0 -1  1  0  0  0  0;
+			  -1  0  0  1  0  0  0  0;
+			   0  0  0  0 -1  1  0  0;
+			   0  0  0  0  0 -1  1  0;
+			   0  0  0  0  0  0 -1  1;
+			   0  0  0  0 -1  0  0  1;
+			   ]
+		EV = sparse(EV)
+		
+		FE = Int8[-1 -1 -1  1  0  0  0  0;
+			   0  0  0  0 -1 -1 -1  1;
+			   0  0  0  1 -1 -1 -1  0]
+		FE = sparse(FE)
+		@test Lar.Arrangement.get_external_cycle(V, EV, FE) == 3
+	end
+
+end
+
+
+
+@testset "Containment graph" begin
+	@testset "pre_containment" begin
+
+		V = [ 0 0; 2 0; 0 2; 3/2 3/2; 1/2 3/2;1/2 1/2 ; 3/2 1/2]
+		    
+		EV = Int8[ -1  1  0  0  0  0  0;
+			   0  -1  1  0  0  0  0;
+			   -1  0  1  0  0  0  0]
+
+		EV1=Int8[  0  0  0  -1  1  0  0;
+			   0  0  0  0  -1  1  0;
+			   0  0  0  0  0  -1  1;
+			   0  0  0  -1  0  0  1
+			   ]
+		EVs = map(sparse, [EV, EV1])
+
+		shell1=Int8[-1 -1 1]
+		shell2=Int8[-1 -1 -1 1]
+		shells = map(sparsevec, [shell1, shell2])
+		     
+		shell_bboxes = []
+		n = 2
+		for i in 1:n
+		    vs_indexes = (abs.(EVs[i]')*abs.(shells[i])).nzind
+		    push!(shell_bboxes, Lar.bbox(V[vs_indexes, :]))
+		end
+		   
+		graph = Lar.Arrangement.pre_containment_test(shell_bboxes)
+		@test graph == [0 0;1 0];
+		   
+		graph = Lar.Arrangement.prune_containment_graph(n, V, EVs, shells, graph)
+		@test graph == [0 0;0 0]
+	end
+
+	@testset "transitive_reduction" begin
+
+		graph = [0 1 1 1 ; 0 0 1 1 ; 0 0 0 1 ; 0 0 0 0 ]
+	 	Lar.Arrangement.transitive_reduction!(graph)
+		@test graph == [0 1 0 0 ; 0 0 1 0; 0 0 0 1 ; 0 0 0 0 ]
+	end
+
+end
+
+
+@testset "Planar Arrangement" begin
+
+	V = [ 0 0; 1 0; 1 1; 0 1; 2 1]
+    
+	EV=[[1,2],[2,3],[3,4],[1,4],[3,5]]
+	
+	cop_EV = Lar.coboundary_0(EV::Lar.Cells)
+	cop_EW = convert(Lar.ChainOp, cop_EV)
+
+	V, copEV, copFE = Lar.Arrangement.planar_arrangement(V::Lar.Points, cop_EW::Lar.ChainOp)
+
+	@test Lar.cop2lar(copEV)==[[1,2],[2,3],[3,4],[1,4]]
+end

From 56bb1ca855fa88e4c0e6810d2282726167399c6c Mon Sep 17 00:00:00 2001
From: marteresagh <marteresa28@gmail.com>
Date: Sat, 20 Apr 2019 19:02:24 +0200
Subject: [PATCH 05/10] my link to the documentation

---
 README.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/README.md b/README.md
index b19e81aa..4c96c120 100644
--- a/README.md
+++ b/README.md
@@ -24,7 +24,7 @@ add("LinearAlgebraicRepresentation")
 
 ## Documentation
 
-Go to [the documentation page](https://cvdlab.github.io/LinearAlgebraicRepresentation.jl/)
+Go to [the documentation page](https://marteresagh.github.io/LinearAlgebraicRepresentation.jl/)
 
 ## Authors
 * [Giulio Martella](https://github.com/giuliom95)

From ef7ec03bbba9e1805090300b9d1d1ab77c865a1e Mon Sep 17 00:00:00 2001
From: marteresagh <marteresa28@gmail.com>
Date: Tue, 23 Apr 2019 09:06:35 +0200
Subject: [PATCH 06/10] update my test

---
 test/planar_arr_mytest.jl | 74 +++++++++++++++++++++++++++++++--------
 1 file changed, 60 insertions(+), 14 deletions(-)

diff --git a/test/planar_arr_mytest.jl b/test/planar_arr_mytest.jl
index 1660fd24..2deb6c57 100644
--- a/test/planar_arr_mytest.jl
+++ b/test/planar_arr_mytest.jl
@@ -3,6 +3,20 @@ using LinearAlgebraicRepresentation
 Lar = LinearAlgebraicRepresentation
 using Plasm,SparseArrays
 
+
+@testset "intersect edges" begin
+	V=[0 0 ; 1 1; 1/2 0; 1/2 1];
+	EV = SparseArrays.sparse(Array{Int8, 2}([
+		             [1 1 0 0] #1->1,2
+		             [0 0 1 1] #2->3,4       
+		         ]));
+
+	W=Lar.Arrangement.intersect_edges(V, EV[1, :], EV[2, :])
+	@test W[1][1]==[0.5 0.5]
+	@test W[1][2]==0.5
+ 
+end
+
 @testset "frag_edge" begin
 	V = [ 3 1; 6 8;2 -2;9 6;1 6;10 1]
     
@@ -17,7 +31,7 @@ using Plasm,SparseArrays
 	bigPI = Lar.spaceindex(model::Lar.LAR)
 	for i in 1:edgenum
 		v, ev = Lar.Arrangement.frag_edge(V, EV, i, bigPI)
-		rV, rEV = Lar.skel_merge(rV, rEV, v, ev) # block diagonal ...
+		rV, rEV = Lar.skel_merge(rV, rEV, v, ev) 
 	end
 
 	@test Lar.cop2lar(rEV) == [[1,3],[2,3],[4,6],[5,6],[7,10],[9,10],[8,9]]
@@ -25,18 +39,28 @@ end
 
 
 @testset "merge_vertices!" begin
-	p0 = 1e-2
-	pm = 1-p0
-	pp = 1+p0
-	V = [ p0  p0; p0 -p0; pp  pm; pp  pp]
-	EV = Int8[1 0 1 0 ;
-                  0 1 0 1 ;
-                  1 0 0 1 ;
-                  0 1 1 0 ]
-	EV = sparse(EV)
-	V, EV = Lar.Arrangement.merge_vertices!(V, EV, [],1e-1)
-	@test size(V,1)==2
-	@test size(EV,1)==1
+
+	@testset "close vertices" begin
+		V = [ 0.01 0.01; 0.01 -0.01; 1.01  0.99; 1.01  1.01]
+		EV = Int8[1 0 1 0 ;
+		          0 1 0 1 ;
+		          1 0 0 1 ;
+		          0 1 1 0 ]
+		EV = sparse(EV)
+		V, EV = Lar.Arrangement.merge_vertices!(V, EV, [],1e-1)
+		@test size(V,1)==2
+		@test size(EV,1)==1
+	end
+
+	@testset "double vertices" begin
+		V=[0 0; 0 0; 1 1; 1 1; 2 2 ;2 2]
+		EV=Int8[1 0 1 0;
+		    0 1 0 1]
+		EV = sparse(EV)
+		V, EV = Lar.Arrangement.merge_vertices!(V, EV, [])
+		@test size(V,1)==3
+		@test size(EV,1)==1
+	end
 end
 
 
@@ -118,7 +142,7 @@ end
 
 
 @testset "Containment graph" begin
-	@testset "pre_containment" begin
+	@testset "pre_containment and prune_containment" begin
 
 		V = [ 0 0; 2 0; 0 2; 3/2 3/2; 1/2 3/2;1/2 1/2 ; 3/2 1/2]
 		    
@@ -160,6 +184,28 @@ end
 
 end
 
+@testset "component graph" begin
+	
+	V = [ 0 0; 2 0; 0 1; 3/2 1/2; 3/2 1/2; 1/2 3/2; 3/2 3/2]
+	    
+	EV = Int8[ 1  1  0  0  0  0  0;
+		   0  1  1  0  0  0  0;
+		   1  0  1  0  0  0  0;
+		   0  0  0  1  1  0  0;
+		   0  0  0  0  1  0  1;
+		   0  0  0  0  0  1  1;
+		   0  0  0  1  0  1  0
+		   ]
+	EV = sparse(EV)
+		
+	bc = Lar.Arrangement.biconnected_components(EV)
+
+	n, containment_graph, V, EVs, boundaries, shells, shell_bboxes=Lar.Arrangement.componentgraph(V, EV, bc)
+
+	@test n==2
+	@test boundaries==[[1 1 -1 -1],[1 1 -1]]
+	@test shell_bboxes==[([0.5 0.5], [1.5 1.5]), ([0.0 0.0], [2.0 1.0])]
+end 
 
 @testset "Planar Arrangement" begin
 

From d1edb57535f08f38ae4b05fae9abacd8ef9922a8 Mon Sep 17 00:00:00 2001
From: marteresagh <marteresa28@gmail.com>
Date: Tue, 23 Apr 2019 09:07:33 +0200
Subject: [PATCH 07/10] update docstrings

---
 src/arrangement/planar_arrangement.jl | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/arrangement/planar_arrangement.jl b/src/arrangement/planar_arrangement.jl
index 594b33ce..47f0af98 100644
--- a/src/arrangement/planar_arrangement.jl
+++ b/src/arrangement/planar_arrangement.jl
@@ -234,7 +234,7 @@ end
 """
 	biconnected_components(EV::Lar.ChainOp)
 
-Find the biconnected components of a graph define by his edges. A biconnected component is a maximal biconnected subgraph. A biconnected graph has no **articulation vertices**. 
+Find the biconnected components of a graph define by its edges. A biconnected component is a maximal biconnected subgraph. A biconnected graph has no **articulation vertices**. 
 
 # Example
 ```julia
@@ -587,7 +587,7 @@ end
 """
 	cleandecomposition(V, copEV, sigma)
 
-Delete edges outside sigma area.
+Delete edges outside sigma area and remove dangling edges in 3D arrangement.
 
 """
 function cleandecomposition(V, copEV, sigma)
@@ -742,7 +742,7 @@ function planar_arrangement_2(V, copEV, bicon_comps,
 
 	# Topological Gift Wrapping
 	n, containment_graph, V, EVs, boundaries, shells, shell_bboxes = 
-		componentgraph(V, copEV, bicon_comps)
+		Lar.Arrangement.componentgraph(V, copEV, bicon_comps)
 	@show containment_graph
 	# only in the context of 3D arrangement
 	if sigma.n > 0

From 21841ebee82a4aad110081973ff37d5e35038b88 Mon Sep 17 00:00:00 2001
From: marteresagh <marteresa28@gmail.com>
Date: Tue, 23 Apr 2019 09:11:10 +0200
Subject: [PATCH 08/10] graph_img


From 843bd96d4e2ac850c80cbb5b6868a7e028996f0a Mon Sep 17 00:00:00 2001
From: marteresagh <marteresa28@gmail.com>
Date: Tue, 23 Apr 2019 09:11:57 +0200
Subject: [PATCH 09/10] documentation


From 00c84d9cdbaeda812d038e0561c67994ff5b04af Mon Sep 17 00:00:00 2001
From: marteresagh <marteresa28@gmail.com>
Date: Tue, 23 Apr 2019 10:59:15 +0200
Subject: [PATCH 10/10] update

---
 docs/src/arrangement.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/docs/src/arrangement.md b/docs/src/arrangement.md
index 41bd8266..ee9aeeeb 100644
--- a/docs/src/arrangement.md
+++ b/docs/src/arrangement.md
@@ -22,7 +22,7 @@ Functions in darker boxes are declared only in the local scope of its father.
 
 ## API
 
-Every functions strictly relative to the arrangement has been collected in the `Lar.Arrangement` sub-module but the two main functions are accessible directly from the `LinearAlgebraicRepresentation` namespace.
+Every function strictly relative to the arrangement has been collected in the `Lar.Arrangement` sub-module but the two main functions are accessible directly from the `LinearAlgebraicRepresentation` namespace.
 
 !!! warning
     `Lar.Arrangement` is the only place in `LinearAlgebraicRepresentation` where `Point` matrices store points per row and not per column as described in the documentation of `Lar.Points`