pluto_affine_function.jl

### A Pluto.jl notebook ###
# v0.19.36

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

# ╔═╡ 50c3ca64-919a-4ae1-a778-4aa21096ca76
begin
	using PlutoUI
	using Plots
    using Symbolics
	using LaTeXStrings
	#using ForwardDiff
	#using Images
	#using TestImages
	#using ImageView
	#using LinearAlgebra
 	#using FFTW
 	#using DSP
end

# ╔═╡ 4527fef0-5851-4d55-bd88-b99870f859f3
begin
nbsp = html"&nbsp"
end

# ╔═╡ 76cb2ec2-fc76-4ad5-8d57-777395755f7f
md"""
## Affine function $\ $ $y=ax+b$  

"""

# ╔═╡ 092ba07e-4866-40c6-ade8-25d9c2218011
md"""
Slope 
$a$ = $(@bind a Slider(-10:0.1:10, show_value=true, default=1))  

Intercept 
$b$ = $(@bind b Slider(-10:0.1:10, show_value=true, default=0))
"""

# ╔═╡ adc0be40-10fd-4270-aedd-b0e1e8a7ce69


# ╔═╡ ab1c3d69-0499-4fad-b2a8-5360ed7339e1
begin
	x = range(-5, 5, length = 101)
	f= a*x+b*ones(length(x));
	c=0;
end

# ╔═╡ 8ed1312b-f5a3-4220-a865-9a12ed09a691
begin
    plot(f,grid=false,
		linewidth=2,
		ylim=(-20,20),
        title="\$y=ax+b\$",
        xticks = ([51,61], [0 1]),
		yticks = ([0],[0]),
		xlabel="\$x\$",
		ylabel="\$y\$",
	    label=:false)
	plot!([1,101],[0,0],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	plot!([51,51],[-19,19],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	plot!([61,61],[-19,19],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
	scatter!((51,b),markersize = 5,label="\$b\$", color=:red)
	scatter!((61,a+b),markersize = 5,label="\$a+b\$", color=:magenta)
end

# ╔═╡ 438abd15-e638-47dc-8499-a9ab3d41a73c


# ╔═╡ 63515570-50f8-45d1-9cdd-0b0d0f085a43


# ╔═╡ 31e3b9f8-465e-49b4-a6d3-4249dd14a27d
md"""
 $a$ = $(@bind a0 Slider(-10:0.1:10, show_value=true, default=1))  
 
 $p$ = $(@bind p0 Slider(-4:0.1:4, show_value=true, default=0)) 

 $q$ = $(@bind q0 Slider(-4:0.1:4, show_value=true, default=0))
"""

# ╔═╡ a780a761-5e4d-434e-8d11-e602d07b9d9b
begin
	x00 = range(-10, 10, length = 201)
	f00=a0*x00
	f01=a0*(x00-p0*ones(length(x00)))+(q0-0.1)*ones(length(x00))
	0;
end

# ╔═╡ da9be8a6-0d61-4e5b-957e-e28726a08ffc
begin
    plot([f00,f01],
		grid=false,
		linewidth=2,
		xlim=(-2,211),
		ylim=(-5.5,5.5),
        title="Parallel transport of \$y=ax\$",
        xticks = ([51 10*(p0+10)+1;], [0, "\$p\$"]),
		yticks = ([0 q0],[0, "\$q\$"]),
		xlabel="\$x\$",
		ylabel="\$y\$",
	    label=["\$y=ax\$" "\$y=a(x-p)+q\$"],
	    legendfontsize=12,
		labelfontsize=12,
		xtickfontsize=12,
		ytickfontsize=12,
		legend=:topleft,
		aspect_ratio=:10
	    )
	plot!([1,201],[0,0],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	plot!([101,101],[-5,5],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	plot!([10*(p0+10)+1,10*(p0+10)+1],[-5,5],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
	plot!([1,201],[q0,q0],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
	scatter!((10*(p0+10)+1,q0),markersize = 5,label="\$(p,q)\$", color=:magenta)
	scatter!((101,0),markersize = 5,label="\$(0,0)\$", color=:red)
end

# ╔═╡ 5268cbee-c311-4582-8699-d9f5be083baf


# ╔═╡ c7e2c310-4f45-4003-ad33-7acd179d12a6
begin
	t = range(-1, 5, length = 61)
	f1=t;
	f2=t-ones(length(t));
	z=0;
end

# ╔═╡ c07da875-b5d5-4e87-96ce-676f8436d3fe
begin
    plot([f1,f2],
		grid=false,
		linewidth=2,
		ylim=(-2.2,5),
        title="\$y=ax\$, \$y-q=a(x-p)\$",
        xticks = ([11 41], [0,"\$p\$"]),
		yticks = ([0 2],[0,"\$q\$"]),
		xlabel="\$x\$",
		ylabel="\$y\$",
	    label=["\$y=ax\$" "\$y-q=a(x-p)\$"])
	plot!([1,61],[0,0],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	plot!([11,11],[-2,5],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	plot!([41,41],[-2,5],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
	plot!([1,61],[2,2],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
	scatter!((11,0),markersize = 5,label="\$(0,0)\$", color=:red)
	scatter!((41,2),markersize = 5,label="\$(p,q)\$", color=:magenta)
end

# ╔═╡ 2e6986d8-8b3b-4fed-8fca-f7d9e7797c95


# ╔═╡ 54237fdd-2e60-4980-b905-ed336c1fe6cd
md"""
## Linear Systems

"""

# ╔═╡ 60dc6f18-6c57-4cdc-9638-b679a5bba41e
begin
	t1 = range(-1, 5, length = 61);
    h1=t1/2+ones(length(t1));
	h2=-2*t1+4*ones(length(t1));
	h3=t1/2+0.95*ones(length(t1));
	h4=t1/2+2*ones(length(t1));
	0;
end

# ╔═╡ 5e233fdf-66bb-44d4-b55b-c8ae7ce55fd7
begin
    plot([h1,h2],
		grid=false,
		linewidth=2,
		ylim=(-2.2,5),
        title="\$-x+2y=2\$, \$2x+y=4\$",
        xticks = ([11 23], [0,"\$6/5\$"]),
		yticks = ([0 8/5],[0,"\$8/5\$"]),
		xlabel="\$x\$",
		ylabel="\$y\$",
		legendfontsize=12,
	    label=["\$-x+2y=2\$" "\$2x+y=4\$"])
	plot!([1,61],[0,0],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	plot!([11,11],[-2,5],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	plot!([23,23],[-2,5],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
	plot!([1,61],[8/5,8/5],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
	scatter!((23,8/5),markersize = 5,label="\$(6/5,8/5)\$", color=:magenta)
	#savefig("linearsystem1.png")
end

# ╔═╡ 0a213e6e-67d5-4328-b431-b7bb73dfefcd
begin
    plot([h1,h3],
		grid=false,
		linewidth=2,
		ylim=(-2.2,5),
        title="\$-x+2y=2\$, \$-2x+4y=4\$",
        xticks = ([11], [0]),
		yticks = ([0],[0]),
		xlabel="\$x\$",
		ylabel="\$y\$",
		legendfontsize=12,
	    label=["\$-x+2y=2\$" "\$-2x+4y=4\$"])
	plot!([1,61],[0,0],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	plot!([11,11],[-2,5],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	#savefig("linearsystem2.png")
end

# ╔═╡ 0a2366a4-3d53-4518-99a1-2cceba8490d1
begin
    plot([h1,h4],
		grid=false,
		linewidth=2,
		ylim=(-2.2,5),
        title="\$-x+2y=2\$, \$-x+2y=4\$",
        xticks = ([11], [0]),
		yticks = ([0],[0]),
		xlabel="\$x\$",
		ylabel="\$y\$",
		legendfontsize=12,
	    label=["\$-x+2y=2\$" "\$-x+2y=4\$"])
	plot!([1,61],[0,0],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	plot!([11,11],[-2,5],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	#savefig("linearsystem3.png")
end

# ╔═╡ 23950250-c5fd-4956-b1ec-8642d9bea06e


# ╔═╡ 2feb4d84-a2d7-41eb-8131-fa80f027980d
md"""
## Quadratic Function and Square Root

"""

# ╔═╡ 023b92a4-d008-4177-bfae-0ede40645df9
begin
	t2 = range(0, 3.1, length = 1241);
    g1=t2;
	g2=t2.^2;
	g3=sqrt.(t2);
	0;
end

# ╔═╡ 1e3563da-8e9d-47c6-9622-77dae776c740
begin
    plot([g1,g2,g3],
		grid=false,
		linewidth=2,
		xlim=(-50,850),
		ylim=(-0.5,4.5),
        title="\$y=x\$, \$y=x^2\$, \$y=x^{1/2}\$",
        xticks = ([1 401 801;], [0,1,2]),
		yticks = ([0 1 2 3 4;],[0,1,2,3,4]),
		xlabel="\$x\$",
		ylabel="\$y\$",
	    label=["\$y=x\$" "\$y=x^2\$" "\$y=x^{1/2}\$"],
		legendfontsize=12,
		legend=:top,
		)
	plot!([-30,850],[0,0],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	plot!([1,1],[-0.3,4.5],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	scatter!((400,1),markersize = 5,label="\$(1,1)\$", color=:magenta)
	#savefig("quadratic.png")
end

# ╔═╡ bc3c5a7c-3c9d-413c-b464-4d204b2688e0
begin
    plot([g1,g2,g3],
		grid=false,
		linewidth=2,
		xlim=(-50,1250),
		ylim=(-0.2,3.1),
        title="\$y=x\$, \$y=x^2\$, \$y=x^{1/2}\$",
        xticks = ([1 401 801 1201;], [0,1,2,3]),
		yticks = ([0 1 2 3;],[0,1,2,3]),
		xlabel="\$x\$",
		ylabel="\$y\$",
	    label=["\$y=x\$" "\$y=x^2\$" "\$y=x^{1/2}\$"],
		legendfontsize=14,
		legend=:bottomright,
		aspect_ratio=400
		)
	plot!([-30,1250],[0,0],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	plot!([1,1],[-0.1,3.1],label=false,
		  arrow=3,linewidth=1,linecolor=:black)
	scatter!((400,1),markersize = 5,label="\$(1,1)\$", color=:magenta)
	#savefig("quadratic2.png")
end

# ╔═╡ 568cad58-ae37-4eba-837d-80c1f2905477
md"""
## Axial Symmetry

"""

# ╔═╡ 0a16fa90-05b9-4a16-8453-69697ce43719
md"""
Slope $(@bind a1 Slider(-20:0.1:20, show_value=true, default=1))  
"""

# ╔═╡ 4d0d945a-9398-41c2-87b6-a05d45b30668
begin
	y = range(-3, 10, length = 131)
	g=a1*y;
	D=1+a1^2/100;
	C=1-a1^2/100;
	P1=30+(C*41+2*a1/10*(-20))/D;
	Q1=(2*a1/10*41-C*(-20))/D;
    P2=30+(C*61+2*a1/10*(-20))/D;
	Q2=(2*a1/10*61-C*(-20))/D;
	P3=30+(C*51+2*a1/10*(-10))/D;
	Q3=(2*a1/10*51-C*(-10))/D;
	d=0;
end

# ╔═╡ 31a639c0-f3a1-4c59-9cbb-598987d240da
begin
    plot(g,grid=false,
		linewidth=2,
		ylim=(-71,71),
		xlim=(-31,101),
        title="Axial Symmetry",
		axis=false,
		xticks=false,
		yticks=false,
	    label=false,
	    aspect_ratio=1.0)
	plot!([71,91],[-20,-20],label=false,linewidth=3,linecolor=:green)
	plot!([91,81],[-20,-10],label=false,linewidth=3,linecolor=:blue)
	plot!([81,71],[-10,-20],label=false,linewidth=3,linecolor=:red)
	plot!([P1,P2],[Q1,Q2],label=false,linewidth=3,linecolor=:green)
	plot!([P2,P3],[Q2,Q3],label=false,linewidth=3,linecolor=:blue)
	plot!([P3,P1],[Q3,Q1],label=false,linewidth=3,linecolor=:red)
	plot!([71,P1],[-20,Q1],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
	plot!([91,P2],[-20,Q2],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
	plot!([81,P3],[-10,Q3],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
	#plot!([41,61],[-20,-20],label=false,linewidth=3,linecolor=:green)
	#plot!([61,51],[-20,-10],label=false,linewidth=3,linecolor=:blue)
	#plot!([51,41],[-10,-20],label=false,linewidth=3,linecolor=:red)
	#plot!([P1,P2],[Q1,Q2],label=false,linewidth=3,linecolor=:green)
	#plot!([P2,P3],[Q2,Q3],label=false,linewidth=3,linecolor=:blue)
	#plot!([P3,P1],[Q3,Q1],label=false,linewidth=3,linecolor=:red)
	#plot!([41,P1],[-20,Q1],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
	#plot!([61,P2],[-20,Q2],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
	#plot!([51,P3],[-10,Q3],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
end

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uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.29.0+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.52.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

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