implement Align (fix JuliaGraphs#79)

Author: thchr

docs/src/index.md

@@ -104,6 +104,23 @@ nothing #hide
 ```
 ![spring animation](spring_animation.mp4)
 
+## Aligning Layouts
+
+Any two-dimensional layout can have its principal axis aligned along a desired angle (default, zero angle), by nesting an "inner" layout into an [`Align`](@ref) layout.
+For example, we may align the above `Spring` layout of the small cubical graph along the horizontal or vertical axes:
+
+```@docs
+Align
+```
+```@example layouts
+g = smallgraph(:cubical)
+layout = Spring(Ptype=Float32)
+f, ax, p = graphplot(g, layout=layout) # horizontal alignment (zero-angle by default)
+hidedecorations!(ax); hidespines!(ax); ax.aspect = DataAspect(); f #hide
+f, ax, p = graphplot(g, layout=Align(layout, pi/2)) # vertical alignment
+hidedecorations!(ax); hidespines!(ax); ax.aspect = DataAspect(); f #hide
+```
+
 ## Stress Majorization
 ```@docs
 Stress

src/NetworkLayout.jl

@@ -238,5 +238,6 @@ include("stress.jl")
 include("spectral.jl")
 include("shell.jl")
 include("squaregrid.jl")
+include("align.jl")
 
 end

src/align.jl

@@ -0,0 +1,47 @@
+export Align
+
+"""
+    Align(inner_layout :: AbstractLayout{2}, angle :: Real = zero(Float64))
+    
+Align the vertex positions of `inner_layout` so that the principal axis of the resulting
+layout makes an `angle` with the **x**-axis.
+
+Only supports two-dimensional inner layouts.
+"""
+struct Align{Ptype, L <: AbstractLayout{2, Ptype}} <: AbstractLayout{2, Ptype}
+    inner_layout :: L
+    angle :: Ptype
+    function Align(inner_layout::L, angle::Real) where {L <: AbstractLayout{2, Ptype}} where Ptype
+        new{Ptype, L}(inner_layout, convert(Ptype, angle))
+    end
+end
+Align(inner_layout::AbstractLayout{2, Ptype}) where Ptype = Align(inner_layout, zero(Ptype))
+
+function layout(algo::Align{Ptype, <:AbstractLayout{2, Ptype}}, adj_matrix::AbstractMatrix) where {Ptype}
+    # compute "inner" layout
+    rs = layout(algo.inner_layout, adj_matrix)
+
+    # align the "inner" layout to have its principal axis make `algo.angle` with x-axis
+    # step 1: compute covariance matrix for PCA analysis: 
+    #       C = ∑ᵢ (rᵢ - ⟨r⟩) (rᵢ - ⟨r⟩)ᵀ
+    # for vertex positions rᵢ, i = 1, …, N, and center of mass ⟨r⟩ = N⁻¹ ∑ᵢ rᵢ.
+    centerofmass = sum(rs) / length(rs)
+    C = zeros(SMatrix{2, 2, Ptype})
+    for r in rs
+        C += (r - centerofmass) * (r - centerofmass)'
+    end
+    vs = eigen(C).vectors
+
+    # step 2: pick principal axis (largest eigenvalue → last eigenvalue/vector)
+    axis = vs[:, end]
+    axis_angle = atan(axis[2], axis[1])
+
+    # step 3: rotate positions `rs` so that new axis is aligned with `algo.angle`
+    s, c = sincos(-axis_angle + algo.angle)
+    R = @SMatrix [c -s; s c] # [cos(θ) -sin(θ); sin(θ) cos(θ)]
+    for (i, r) in enumerate(rs)
+        rs[i] = Point2{Ptype}(R * r) :: Point2{Ptype}
+    end
+
+    return rs
+end