Stereographic projection

A point light illuminates the grid points on the ground.

Load packages

using Luxor
using IntervalSets
using BasicBSpline
using BasicBSplineFitting
using StaticArrays
using ElasticSurfaceEmbedding

Compute the embedding shapes

Shape definition

ElasticSurfaceEmbedding.𝒑₍₀₎(u¹,u²) = SVector(2*u¹/(1+u¹^2+u²^2), 2*u²/(1+u¹^2+u²^2), (-1+u¹^2+u²^2)/(1+u¹^2+u²^2))
n = 10
D(i,n) = (-2.0..2.0, 2(i-1)/n..2i/n)

D (generic function with 1 method)

Strain estimation

show_strain(D(1,n))

┌ Info: Strain - domain: [-2.0, 2.0]×[0.0, 0.2]
└ Predicted: (min: -0.00653530699604614, max: 0.013070613992092282)

Main computation

steptree = StepTree()
for i in 1:10
    initial_state!(steptree, D(i,n))
    newton_onestep!(steptree, fixingmethod=:fix3points)
    newton_onestep!(steptree)
    refinement!(steptree, p₊=(0,1), k₊=suggest_knotvector(steptree))
    newton_onestep!(steptree)
    newton_onestep!(steptree)
    pin!(steptree)
end

Helper functions to export svg images

function create_bezierpath(C::BSplineManifold{1,(3,),Point})
    P = bsplinespaces(C)[1]
    k = knotvector(P)
    k′ = 3*unique(k) + k[[1,end]]
    P′ = BSplineSpace{3}(k′)
    C′ = refinement(C,P′)
    a′ = controlpoints(C′)
    n′ = dim(P′)
    m = (n′-1) ÷ 3
    bezierpath = BezierPath([BezierPathSegment(a′[3i-2], a′[3i-1], a′[3i], a′[3i+1]) for i in 1:m])
    return bezierpath
end
function svector2point(M::BSplineManifold, unitlength)
    P = bsplinespaces(M)
    a = controlpoints(M)
    a′ = [Point(p[1]*unitlength[1], -p[2]*unitlength[1]) for p in a]
    M′ = BSplineManifold(a′, P)
    return M′
end

svector2point (generic function with 1 method)

Settings for export

xlims=(-3,3)
ylims=(-1,1)
unitlength = (200, "mm")
r = 0.025

0.025

Export all embedded shapes with arcs

mkpath("stereographicprojection")
for i in 1:10
    M = svector2point(steptree.steps[6i].manifold, unitlength)
    D¹ = domain(bsplinespaces(M)[1])
    D² = domain(bsplinespaces(M)[2])
    u¹s = range(extrema(D¹)...,21)[2:end-1]
    u²₋ = minimum(D²)
    u²₊ = maximum(D²)

    width = (xlims[2] - xlims[1]) * unitlength[1]
    height = (ylims[2] - ylims[1]) * unitlength[1]

    filepath = joinpath("stereographicprojection", "embedding-$(i).svg")
    Drawing(width, height, filepath)
    origin()
    background("white")
    sethue("red")

    C = M(:,u²₋)
    path = create_bezierpath(C)
    drawbezierpath(path, :stroke)
    C = M(:,u²₊)
    path = create_bezierpath(C)
    drawbezierpath(path, :stroke)
    C = M(2,:)
    path = create_bezierpath(C)
    drawbezierpath(path, :stroke)
    C = M(-2,:)
    path = create_bezierpath(C)
    drawbezierpath(path, :stroke)

    for u¹ in u¹s
        k = KnotVector([0,0,0,0,0.25,0.5,0.75,1,1,1,1])
        P = BSplineSpace{3}(k)
        dim(P)

        a = fittingcontrolpoints(t -> M(u¹+r*cospi(t), u²₋+r*sinpi(t)), P)
        C = BSplineManifold(a,P)
        path = create_bezierpath(C)
        drawbezierpath(path, :stroke)

        a = fittingcontrolpoints(t -> M(u¹+r*cospi(t), u²₊-r*sinpi(t)), P)
        C = BSplineManifold(a,P)
        path = create_bezierpath(C)
        drawbezierpath(path, :stroke)
    end

    finish()
    preview()

    script = read(filepath, String)
    lines = split(script, "\n")
    lines[2] = replace(lines[2],"pt\""=>"mm\"")
    write(filepath, join(lines,"\n"))
end

The output files will be saved as embedding-$(i).svg. By modifying these files, we can place all of the shapes in yatsugiri-size (八ツ切, approximately 270×390 mm) paper like this:

Cutting and weaving these shape will result the sphere in the top image. Please check the following references for more information.

References

• 紙工作で立体射影をつくった話
• 立体射影製作キット
• Stereographic projection weaving kit
• Further adventures in stereographic projection

This page was generated using DemoCards.jl and Literate.jl.