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Copy pathGrowthEllipse.jl
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GrowthEllipse.jl
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### A Pluto.jl notebook ###
# v0.20.0
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ 2317834e-c15f-11ef-2f73-4b33c7344e99
using PlutoUI, Plots
# ╔═╡ e3766ed5-e08f-4e12-9a97-b00d406e6ddb
p = @bind p Slider(LinRange(0,1,101); default=.5, show_value=true)
# ╔═╡ 12c35e44-f916-433e-84a6-9b93b2433339
let
# full ellipse
θ = LinRange(0, 2π, 100); q = 1-p
x = √p *cos.(θ); y = √q *sin.(θ)
plot(x,y,aspectratio=1,legend=false,lw=5)
θ = LinRange(acos(√p), acos(-√p), 100)
x = √p *cos.(θ); y = -√q *sin.(θ)
plot!(x,y,color=:red,lw=5)
s = 1
plot!([0,1,0,-1,0],[1,0,-1,0,1],color=:black)
scatter!([p p -p -p],[q -q q -q], color=:blue, ms=7)
end
# ╔═╡ 74d11d69-0e80-4b42-9fb4-5ae5564a72bd
let
# full ellipse
L = 5 # → ∞
p = 1/L
θ = LinRange(0, 2π, 100); q = 1-p
x = √p *cos.(θ); y = √q *sin.(θ)
plot()
#plot(x,y,aspectratio=1,legend=false,lw=5)
θ = LinRange(acos(√p), acos(-√p), 100)
x = √p *cos.(θ); y = -√q *sin.(θ)
plot!(x*L, L*(1 .+y) ,color=:red,lw=5)
s = 1
plot!(x->(x^2+1)/2, color=:green, lw=5)
#plot!([0,1,0,-1,0],[1,0,-1,0,1],color=:black)
#scatter!([p p -p -p],[q -q q -q], color=:blue, ms=7, ylims=(-1.1,-.5))
end
# ╔═╡ fb40b027-5c33-4810-a47f-3adcf2c7fd3a
md"""
The ellipse in the unit square may be described as
$\frac{x^2}p + \frac{y^2}q = 1,$
rotated 45°, reduced by $\sqrt{2}$ and center translated
to $(1/2,1/2)$ so as to fit in the unit square.
If $x = \sqrt{p} \cos \theta$, $y = \sqrt{q} \sin \theta$ then the ellipse
in the unit square coordinates are
$X = (1-x-y)/2$
$Y=(1+x-y)/2.$
The bottom left "red" arc corresponds to $\theta \in [\cos^{-1} (\sqrt p), \cos^{-1} (-\sqrt p)]$.
"""
# ╔═╡ b3868430-1ac0-479a-9bcc-43c7398620c0
let
# full ellipse
θ = LinRange(0, 2π, 100); q = 1-p
x = √p *cos.(θ); y = √q *sin.(θ)
plot( (-x-y.+1)/2 , (x-y.+1)/2,
xlims=(-.1,1.1),ylims=(-.1,1.1),lw=5, aspectratio=1, legend=nothing)
# ellipse arc
θ = LinRange(acos(√p), acos(-√p), 100)
x = √p *cos.(θ); y = √q *sin.(θ)
plot!( 1/2 .*(1 .- x-y), 1/2 .*(1 .+ x-y),
xlims=(-.1,1.1),ylims=(-.1,1.1),lw=5)
plot!([0,1,1,0,0],[1,1,0,0,1], lw=1, color=:black)
# "red" axis
z=p/(2+2√q); plot!([z,1-z],[z,1-z], color=:red); scatter!([z,1-z],[z,1-z], color=:red, ms=7)
# "blue" axis
z1 = (1+√p)/2; z2=(1-√p)/2; scatter!([z1,z2], [z2,z1],color=:blue,ms=7);
plot!([z1,z2], [z2,z1],color=:blue)
# "black" bounding box points
scatter!([0,p,1,q],[p,0,q,1],color=:black,ms=7)
#title
title!("p=$p")
end
# ╔═╡ b1c6f95a-102b-4772-a53f-7af97b78cf2a
md"""
Plot many ellipses
"""
# ╔═╡ 0dee19c0-bc6d-4f97-ac81-40416d2451bb
@bind num_ellipses Slider(2:100; default=5, show_value=true)
# ╔═╡ 979e9fd1-d391-4d00-ab3a-050aafde692e
let
θ = LinRange(0, 2π, 100)
plot([0,1,1,0,0],[1,1,0,0,1], lw=5, color=:black)
for p = LinRange(1/num_ellipses,.9,num_ellipses)
q = 1-p
x = √p *cos.(θ); y = √q *sin.(θ)
plot!( 1/2 .*(1 .- x-y), 1/2 .*(1 .+ x-y),
xlims=(-.1,1.1),ylims=(-.1,1.1),lw=2, aspectratio=1, legend=nothing)
end
plot!()
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
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PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
[compat]
Plots = "~1.40.9"
PlutoUI = "~0.7.60"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
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