edits
This commit is contained in:
jverzani
@@ -19,11 +19,47 @@ using SymPy
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```{julia}
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#| echo: false
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#| results: "hidden"
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import LinearAlgebra: norm
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import LinearAlgebra: norm, cross
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using SplitApplyCombine
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nothing
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```
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```{julia}
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#| echo: false
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# commands used for plotting from https://github.com/SigurdAngenent/WisconsinCalculus/blob/master/figures/221/09surf_of_rotation2.py
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#linear projection of R^3 onto R^2
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function _proj(X, v)
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# a is ⟂ to v and b is v × a
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vx, vy, vz = v
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a = [-vy, vx, 0]
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b = cross([vx,vy,vz], a)
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a′, b′ = a/norm(a), b/norm(b)
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return (a′ ⋅ X, b′ ⋅ X)
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end
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# project a curve in R3 onto R2
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pline(viewp, ps...) = [_proj(p, viewp) for p in ps]
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# determinant of Jacobian; area multiplier
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# det(J); used to identify folds
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function jac(X, u, v)
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return det(ForwardDiff.jacobian(xs -> collect(X(xs...)), [u,v]))
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end
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function _fold(F, t, θmin, θmax)
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λ = θ -> jac(F, t, θ) # F is projected surface, psurf
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iszero(λ(θmin)) && return θmin
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iszero(λ(θmax)) && return θmax
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return solve(ZeroProblem(λ, (θmin, θmax)))
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end
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nothing
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```
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---
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@@ -133,38 +169,13 @@ plt = let
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# plot surface of revolution around x axis between [0, 3]
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# best if r(t) descreases
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rad(x) = 2/(1 + exp(x))#2/(2.0+x)
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viewp = [2,-2,1]
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##
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unitize(x) = x / norm(x)
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"""Orthogonal projection along the vector viewp"""
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function make_Pmat(viewp)
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a = unitize( [-viewp[2], viewp[1], 0] )
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b = unitize( [-viewp[3]*viewp[1],
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-viewp[3]*viewp[2],
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viewp[1]^2 + viewp[2]^2] )
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collect(zip(a,b))
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end
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#linear projection of R^3 onto R^2
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function proj(X, viewp)
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Pmat = make_Pmat(viewp)
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x=sum([Pmat[i][1]*X[i] for i in 1:3])
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y=sum([Pmat[i][2]*X[i] for i in 1:3])
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(x, y) # a point
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end
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proj(X) = proj(X, viewp)
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# discrete determinant of Jacobian; area multiplier?
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function jac(X, u, v)
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ϵ = 0.000001
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A = map((p,q) -> (p-q)/ϵ, X(u+ϵ/2, v), X(u-ϵ/2, v))
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B = map((p,q) -> (p-q)/ϵ, X(u, v+ϵ/2), X(u, v-ϵ/2))
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return A[1]*B[2]-A[2]*B[1]
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end
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rad(x) = 2/(1 + exp(x))
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trange = (0,3)
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θrange = (0, 2pi)
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viewp = [2,-2, 1]
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##
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proj(X) = _proj(X, viewp)
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# surface of revolution
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surf(t, z) = [t, rad(t)*cos(z), rad(t)*sin(z)]
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@@ -172,80 +183,63 @@ plt = let
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# project the surface at (t, a=theta)
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psurf(t,z) = proj(surf(t,z))
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bisect(f, a, b) = find_zero(f, (a,b), Bisection())
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_fold(t, zmin, zmax) = bisect(z -> jac(psurf, t, z), zmin, zmax)
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# create shape holding project disc
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drawdiscF(t) = Shape(invert([psurf(t, 2*i*pi/100) for i in 1:101])...)
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# project a line between two points
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pline!(p,q; kwargs...) = plot!([proj(p),proj(q)];
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line_style..., kwargs...)
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α = 1.0
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line_style = (; line=(:black, 1))
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α = 1.0 # opacity
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line_style = (; line=(:black, 1))
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plot(; empty_style..., aspect_ratio=:equal)
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# by layering, we get x-axis as desired
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pline!([-1,0,0], [0,0,0])
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plot!(pline(viewp, [-1,0,0], [0,0,0]); line_style...)
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plot!(drawdiscF(0); fill =(:lightgray, α))
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pline!([0,0,0], [1,0,0])
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plot!(pline(viewp, [0,0,0], [1,0,0]); line_style...)
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plot!(drawdiscF(1); fill =(:black, α)) # black to lightgray gives thickness
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plot!(drawdiscF(1.1); fill=(:lightgray, α))
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pline!([1.1,0,0], [2,0,0])
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plot!(pline(viewp, [1.1,0,0], [2,0,0]); line_style...)
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plot!(drawdiscF(2); fill=(:lightgray, α))
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pline!([2,0,0], [3,0,0])
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plot!(pline(viewp, [2,0,0], [3,0,0]); line_style...)
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plot!(drawdiscF(3); fill=(:lightgray, α))
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pline!([3,0,0], [4,0,0]; arrow=true, side=:head)
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pline!([0,0,0], [0,0,1.25]; arrow=true, side=:head)
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plot!(pline(viewp, [3,0,0], [4,0,0]); line_style..., arrow=true, side=:head)
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plot!(pline(viewp, [0,0,0], [0,0,1.25]); line_style..., arrow=true, side=:head)
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tt = range(0, pi, 30)
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curve = [psurf(t, pi/2) for t in tt]
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tt = range(trange..., 30)
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curve = psurf.(tt, pi/2)
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plot!(curve; line=(:black, 2))
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f1 = [[t, _fold(t, 0, pi)] for t in tt]
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f1 = [(t, _fold(psurf, t, 0, pi)) for t in tt]
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curve = [psurf(f[1], f[2]) for f in f1]
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plot!(curve; line=(:black,))
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plot!(curve; line=(:black,1))
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f2 = [[t, _fold(t, pi, 2*pi)] for t in tt]
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f2 = [(t, _fold(psurf, t, pi, 2*pi)) for t in tt]
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curve = [psurf(f[1], f[2]) for f in f2]
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plot!(curve; line=(:black,))
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plot!(curve; line=(:black,1))
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tt= [0.025*i for i in 1:121]
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f1 = [[t, _fold(t, pi, 2*pi)] for t in tt]
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for f in f1
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plot!([psurf(f[1], f[2]-k*0.01*(4-f[1])) for k in 1:21];
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line=(:black,1))
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## find bottom edge (t,θ) again
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tt = range(0, 3, 120)
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f1 = [(t, _fold(psurf, t, pi, 2*pi)) for t in range(trange..., 100)]
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# shade bottom by adding bigger density of lines near bottom
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for (i,f) ∈ enumerate(f1)
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λ = iseven(i) ? 6 : 4 # adjust density by have some lines only extend to 6
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isnan(f[1]) || isnan(f[2]) && continue
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curve = [psurf(f[1], θ) for θ in range(f[2] - 0.2*(λ - f[1]), f[2], 20)]
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plot!(curve; line=(:black, 1))
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end
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tt= [0.05*i for i in 1:61]
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f1 = [[t, _fold(t, pi, 2*pi)] for t in tt]
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for f in f1
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plot!([psurf( f[1], f[2]-k*0.01*(6-f[1]) )
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for k in 1:21]; line=(:black, 1))
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end
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#=
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ts = 0:.1:2.95
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θs = range(pi + pi/4, 3pi/2, 25)
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for ti in ts
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plot!([psurf(ti,θ) for θ in θs]; line=(:black, 1))
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end
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θs = range(pi + pi/6, 3pi/2, 25)
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for ti in ts
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plot!([psurf(ti +0.05,θ) for θ in θs]; line=(:black, 1))
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end
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=#
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current()
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#ts = range(0, 3, 100)
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#θs = range(0, 2pi, 100)
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#contour(ts, θs, (t,z) -> jac(psurf,t,z), levels=[0])
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end
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plt
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```
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@@ -501,37 +495,13 @@ plt = let
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# plot surface of revolution around x axis between [0, 3]
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# best if r(t) descreases
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rad(x) = 2/(1 + exp(x))#2/(2.0+x)
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rad(x) = 2/(1 + exp(x))
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trange = (0, 3)
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θrange = (0, 2pi)
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viewp = [2,-2,1]
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##
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unitize(x) = x / norm(x)
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"""Orthogonal projection along the vector viewp"""
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function make_Pmat(viewp)
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a = unitize( [-viewp[2], viewp[1], 0] )
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b = unitize( [-viewp[3]*viewp[1],
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-viewp[3]*viewp[2],
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viewp[1]^2 + viewp[2]^2] )
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collect(zip(a,b))
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end
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#linear projection of R^3 onto R^2
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function proj(X, viewp)
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Pmat = make_Pmat(viewp)
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x=sum([Pmat[i][1]*X[i] for i in 1:3])
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y=sum([Pmat[i][2]*X[i] for i in 1:3])
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(x, y) # a point
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end
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proj(X) = proj(X, viewp)
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# discrete determinant of Jacobian; area multiplier?
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function jac(X, u, v)
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ϵ = 0.000001
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A = map((p,q) -> (p-q)/ϵ, X(u+ϵ/2, v), X(u-ϵ/2, v))
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B = map((p,q) -> (p-q)/ϵ, X(u, v+ϵ/2), X(u, v-ϵ/2))
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return A[1]*B[2]-A[2]*B[1]
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end
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proj(X) = _proj(X, viewp)
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# surface of revolution
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@@ -542,67 +512,64 @@ plt = let
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psurf(t,z) = proj(surf(t,z))
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psurf2(t, z) = proj(surf2(t,z))
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bisect(f, a, b) = find_zero(f, (a,b), Bisection())
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_fold(t, zmin, zmax) = bisect(z -> jac(psurf, t, z), zmin, zmax)
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# create shape holding project disc
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drawdiscF(t) = Shape(invert([psurf(t, 2*i*pi/100) for i in 1:101])...)
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drawdiscI(t) = Shape([psurf2(t, 2*i*pi/100) for i in 1:101])
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# project a line between two points
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pline!(p,q; kwargs...) = plot!([proj(p),proj(q)];
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line_style..., kwargs...)
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α = 1.0
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line_style = (; line=(:black, 1))
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plot(; empty_style..., aspect_ratio=:equal)
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# by layering, we get x-axis as desired
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pline!([-1,0,0], [0,0,0])
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plot!(pline(viewp, [-1,0,0], [0,0,0]); line_style...)
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plot!(drawdiscF(0); fill =(:lightgray, α))
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plot!(drawdiscI(0); fill=(:white, .5))
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pline!([0,0,0], [1,0,0])
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plot!(pline(viewp, [0,0,0], [1,0,0]); line_style...)
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plot!(drawdiscF(1); fill =(:black, α)) # black to lightgray gives thickness
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plot!(drawdiscI(1); fill=(:white, .5))
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plot!(drawdiscF(1.1); fill=(:lightgray, α))
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plot!(drawdiscI(1.1); fill=(:white, .5))
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pline!([1.1,0,0], [2,0,0])
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plot!(pline(viewp, [1.1,0,0], [2,0,0]); line_style...)
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plot!(drawdiscF(2); fill=(:lightgray, α))
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plot!(drawdiscI(2); fill=(:white, .5))
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pline!([2,0,0], [3,0,0])
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plot!(pline(viewp, [2,0,0], [3,0,0]); line_style...)
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plot!(drawdiscF(3); fill=(:lightgray, α))
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plot!(drawdiscI(3); fill=(:white, .5))
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pline!([3,0,0], [4,0,0]; arrow=true, side=:head)
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pline!([0,0,0], [0,0,1.25]; arrow=true, side=:head)
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plot!(pline(viewp, [3,0,0], [4,0,0]); line_style..., arrow=true, side=:head)
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plot!(pline(viewp, [0,0,0], [0,0,1.25]); line_style..., arrow=true, side=:head)
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tt = range(0, pi, 30)
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## bounding curves
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### main spine
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tt = range(trange..., 30)
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curve = [psurf(t, pi/2) for t in tt]
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plot!(curve; line=(:black, 2))
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f1 = [[t, _fold(t, 0, pi)] for t in tt]
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### the folds
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f1 = [(t, _fold(psurf, t, 0, pi)) for t in tt]
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curve = [psurf(f[1], f[2]) for f in f1]
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plot!(curve; line=(:black,))
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f2 = [[t, _fold(t, pi, 2*pi)] for t in tt]
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f2 = [(t, _fold(psurf, t, pi, 2*pi)) for t in tt]
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curve = [psurf(f[1], f[2]) for f in f2]
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plot!(curve; line=(:black,))
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## add shading
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### find bottom edge (t,θ) again
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f1 = [[t, _fold(psurf, t, pi, 2*pi)] for t in range(trange..., 120)]
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### shade bottom by adding bigger density of lines near bottom
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for (i,f) ∈ enumerate(f1)
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λ = iseven(i) ? 6 : 4 # adjust density by have some lines only extend to 6
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isnan(f[1]) || isnan(f[2]) && continue
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tt= [0.025*i for i in 1:121]
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f1 = [[t, _fold(t, pi, 2*pi)] for t in tt]
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for f in f1
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plot!([psurf(f[1], f[2]-k*0.01*(4-f[1])) for k in 1:21];
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line=(:black,1))
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end
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tt= [0.05*i for i in 1:61]
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f1 = [[t, _fold(t, pi, 2*pi)] for t in tt]
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for f in f1
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plot!([psurf( f[1], f[2]-k*0.01*(6-f[1]) )
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for k in 1:21]; line=(:black, 1))
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curve = [psurf(f[1], θ) for θ in range(f[2] - 0.2*(λ - f[1]), f[2], 20)]
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plot!(curve; line=(:black, 1))
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end
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current()
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@@ -686,44 +653,12 @@ Let $h$ be the distance from the apex to the base. Consider cones with the prope
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plt = let
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gr()
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rad(t) = 3/2 - t
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trange = (0, 3/2)
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θrange = (0, 2pi)
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viewp = [2,-1/1.5,1/2+.2]
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empty_style = (xaxis=([], false),
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yaxis=([], false),
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framestyle=:origin,
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legend=false)
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axis_style = (arrow=true, side=:head, line=(:gray, 1))
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unitize(x) = x / norm(x)
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"""Orthogonal projection along the vector viewp"""
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function make_Pmat(viewp)
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a = unitize( [-viewp[2], viewp[1], 0] )
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b = unitize( [-viewp[3]*viewp[1],
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-viewp[3]*viewp[2],
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viewp[1]^2 + viewp[2]^2] )
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collect(zip(a,b))
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end
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#linear projection of R^3 onto R^2
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function proj(X, viewp)
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Pmat = make_Pmat(viewp)
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x=sum([Pmat[i][1]*X[i] for i in 1:3])
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y=sum([Pmat[i][2]*X[i] for i in 1:3])
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(x, y) # a point
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end
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proj(X) = proj(X, viewp)
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drawdiscF(t) = Shape([psurf(t, 2*i*pi/100) for i in 1:101])
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# discrete determinant of Jacobian; area multiplier?
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function jac(X, u, v)
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ϵ = 0.000001
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A = map((p,q) -> (p-q)/ϵ, X(u+ϵ/2, v), X(u-ϵ/2, v))
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B = map((p,q) -> (p-q)/ϵ, X(u, v+ϵ/2), X(u, v-ϵ/2))
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return A[1]*B[2]-A[2]*B[1]
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end
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##
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proj(X) = _proj(X, viewp)
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# our surface
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R, r, rho = 1, 1/4, 1/4
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f(t) = (R-r) * cos(t) + rho * cos((R-r)/r * t)
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@@ -731,6 +666,17 @@ plt = let
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surf(t, θ) = (rad(t)*f(θ), rad(t)*g(θ), t)
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psurf(t,θ) = proj(surf(t,θ))
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empty_style = (xaxis=([], false),
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yaxis=([], false),
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framestyle=:origin,
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legend=false)
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axis_style = (arrow=true, side=:head, line=(:gray, 1))
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drawdiscF(t) = Shape([psurf(t, 2*i*pi/100) for i in 1:101])
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plot(; empty_style..., aspect_ratio=:equal)
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for (i,t) in enumerate(range(0, 3/2, 30))
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plot!(drawdiscF(t); fill=(:gray,1), line=(:black,1))
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@@ -943,38 +889,37 @@ plt = let
|
||||
rad2(t) = 1/2
|
||||
viewp = [2,-2,1]
|
||||
|
||||
##
|
||||
function _proj(X, v)
|
||||
# a is ⟂ to v and b is v × a
|
||||
vx, vy, vz = v
|
||||
a = [-vy, vx, 0]
|
||||
b = cross([vx,vy,vz], a)
|
||||
a′, b′ = a/norm(a), b/norm(b)
|
||||
return (a′ ⋅ X, b′ ⋅ X)
|
||||
end
|
||||
|
||||
# project a curve in R3 onto R2
|
||||
pline(viewp, ps...) = [_proj(p, viewp) for p in ps]
|
||||
|
||||
# determinant of Jacobian; area multiplier
|
||||
# det(J); used to identify folds
|
||||
function jac(X, u, v)
|
||||
return det(ForwardDiff.jacobian(xs -> collect(X(xs...)), [u,v]))
|
||||
end
|
||||
|
||||
function _fold(F, t, θmin, θmax)
|
||||
λ = θ -> jac(F, t, θ) # F is projected surface, psurf
|
||||
iszero(λ(θmin)) && return θmin
|
||||
iszero(λ(θmax)) && return θmax
|
||||
|
||||
return solve(ZeroProblem(λ, (θmin, θmax)))
|
||||
end
|
||||
|
||||
##
|
||||
unitize(x) = x / norm(x)
|
||||
|
||||
"""Orthogonal projection along the vector viewp"""
|
||||
function make_Pmat(viewp)
|
||||
a = unitize( [-viewp[2], viewp[1], 0] )
|
||||
b = unitize( [-viewp[3]*viewp[1],
|
||||
-viewp[3]*viewp[2],
|
||||
viewp[1]^2 + viewp[2]^2] )
|
||||
collect(zip(a,b))
|
||||
end
|
||||
|
||||
#linear projection of R^3 onto R^2
|
||||
function proj(X, viewp)
|
||||
Pmat = make_Pmat(viewp)
|
||||
x=sum([Pmat[i][1]*X[i] for i in 1:3])
|
||||
y=sum([Pmat[i][2]*X[i] for i in 1:3])
|
||||
(x, y) # a point
|
||||
end
|
||||
proj(X) = proj(X, viewp)
|
||||
|
||||
# discrete determinant of Jacobian; area multiplier?
|
||||
function jac(X, u, v)
|
||||
ϵ = 0.000001
|
||||
A = map((p,q) -> (p-q)/ϵ, X(u+ϵ/2, v), X(u-ϵ/2, v))
|
||||
B = map((p,q) -> (p-q)/ϵ, X(u, v+ϵ/2), X(u, v-ϵ/2))
|
||||
return A[1]*B[2]-A[2]*B[1]
|
||||
end
|
||||
|
||||
proj(X) = _proj(X, viewp)
|
||||
|
||||
# surface of revolution about the z axis
|
||||
|
||||
surf(t, z) = (rad(t)*cos(z), rad(t)*sin(z), t)
|
||||
surf2(t, z) = (rad2(t)*cos(z), rad2(t)*sin(z), t)
|
||||
|
||||
@@ -983,15 +928,12 @@ plt = let
|
||||
psurf2(t, z) = proj(surf2(t,z))
|
||||
|
||||
bisect(f, a, b) = find_zero(f, (a,b), Bisection())
|
||||
_fold(t, zmin, zmax) = bisect(z -> jac(psurf, t, z), zmin, zmax)
|
||||
_foldz(z, tmin, tmax) = bisect(t -> jac(psurf, t, z), tmin, tmax)
|
||||
|
||||
# create shape holding project disc
|
||||
drawdiscF(t) = Shape([psurf(t, 2*i*pi/100) for i in 1:101])
|
||||
drawdiscI(t) = Shape([psurf2(t, 2*i*pi/100) for i in 1:101])
|
||||
|
||||
# project a line between two points
|
||||
pline!(p,q; kwargs...) = plot!([proj(p),proj(q)];
|
||||
line_style..., kwargs...)
|
||||
|
||||
α = 1.0
|
||||
line_style = (; line=(:black, 1))
|
||||
|
||||
@@ -1014,17 +956,8 @@ plt = let
|
||||
plot!(drawdiscI(x₀); fill=(:white,1.0), line=(:black,1))
|
||||
|
||||
z0 = 3pi/2 - δ
|
||||
pline!(surf(t0, z0), surf(-t0, z0); line=(:black, 1))
|
||||
pline!(surf(t0, z0+pi), surf(-t0, z0+pi); line=(:black, 1))
|
||||
|
||||
|
||||
# boundary of sphere
|
||||
z0 = 3pi/2 - δ
|
||||
curve = [psurf(t, z0) for t in range(-t0, t0, 100)]
|
||||
plot!(curve; line=(:black,3))
|
||||
z0 = 3pi/2 - δ + pi
|
||||
curve = [psurf(t, z0) for t in range(-t0, t0, 100)]
|
||||
plot!(curve; line=(:black,3))
|
||||
plot!(pline(viewp, surf(t0, z0), surf(-t0, z0)); line=(:black, 1))
|
||||
plot!(pline(viewp, surf(t0, z0+pi), surf(-t0, z0+pi)); line=(:black, 1))
|
||||
|
||||
# caps
|
||||
curve = [psurf(t0, θ) for θ in range(0, 2pi, 100)]
|
||||
@@ -1033,6 +966,17 @@ plt = let
|
||||
plot!(curve, line=(:black, 2))
|
||||
|
||||
|
||||
## folds
|
||||
tθs = [(t, _fold(psurf, t, 0,pi)) for t in range(-t0, t0, 50)]
|
||||
curve = [psurf(t, θ) for (t,θ) ∈ tθs]
|
||||
plot!(curve, line=(:black, 3))
|
||||
|
||||
tθs = [(t, _fold(psurf, t, pi, 2pi)) for t in range(-t0, t0, 50)]
|
||||
curve = [psurf(t, θ) for (t,θ) ∈ tθs]
|
||||
plot!(curve, line=(:black, 3))
|
||||
|
||||
|
||||
|
||||
# Shade lines
|
||||
δ = pi/6
|
||||
Δₜ = (4pi/2 - (3pi/2 - δ))/(2*25)
|
||||
@@ -1046,7 +990,7 @@ plt = let
|
||||
end
|
||||
|
||||
#=
|
||||
f1 = [[t, _fold(t, 0, pi/2)] for t in range(-0.5, -0.1, 26)]
|
||||
f1 = [[t, _fold(psurf, t, 0, pi/2)] for t in range(-0.5, -0.1, 26)]
|
||||
for f in f1
|
||||
plot!([psurf( f[1], f[2]-k*0.01*(6-f[1]) )
|
||||
for k in 1:21]; line=(:black, 1))
|
||||
|
||||
Reference in New Issue
Block a user