Stereographic projection

A point light illuminates the grid points on the ground. You can buy the kit via my Booth site!

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

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