{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "c4ad7169",
   "metadata": {},
   "source": [
    "## <center> Julia lang definition of   plain-knit yarns </center>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fa1a21b0",
   "metadata": {},
   "source": [
    "Recently, K Crane defined 3d curves to model yarns and fibers for simple knitting.\n",
    "\n",
    "We reproduce the plots  from K Crane's paper, [https://www.cs.cmu.edu/~kmcrane/Projects/Other/YarnCurve.pdf](https://www.cs.cmu.edu/~kmcrane/Projects/Other/YarnCurve.pdf),\n",
    "taking  the advantage of  Julia,  over C programming language,\n",
    "to write a more compact code, and moreover to plot  the   graphical objects directly, \n",
    "without involving an external  library. Notice that automatic derivation (via ForwardDiff.jl) of vector valued functions $t\\mapsto γ(t)$, $t\\mapsto\\dot{γ}(t)$, substantially reduces the lines of code, compared to the C version."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ffdc2e06",
   "metadata": {},
   "source": [
    "The yarns are represented as tubular surfaces, having as directrice a 3d curve, $t\\to\\gamma(t)\\in\\mathbb{R}^3$. To get the tube parameterizaion we implement the   A. Hanson's algorithm, presented in *Visualizing Quaternions*, Morgan Kaufmann, 2006:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "447f71f8",
   "metadata": {},
   "outputs": [],
   "source": [
    "import ForwardDiff.derivative\n",
    "import LinearAlgebra: norm, cross\n",
    "import Rotations.UnitQuaternion\n",
    "using  PlotlyJS "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "99ac8b3c",
   "metadata": {},
   "outputs": [],
   "source": [
    "dγ(t)=derivative(γ, t) # t->̇γ(t)\n",
    "ddγ(t)=derivative(dγ, t) #t->̈γ(t)\n",
    "\n",
    "#get the tubular surface  parameterization:\n",
    "function tube(γ::T, u::S, v::S; radius=0.1) where {S<:Real, T<:Function}\n",
    "    tang = dγ(u)   #tangent(γ, u)\n",
    "    ~iszero(tang) || error(\"null tangent vector!\")\n",
    "    unittan = tang/norm(tang)\n",
    "    θ = acos(unittan[1])/2\n",
    "    crossp= [0, -unittan[3], unittan[2]]  # cross product [1,0,0] x unittan\n",
    "    quvect = sin(θ)*crossp/norm(crossp) #3-vector to define the quaternion \n",
    "                                          #q=(cos(θ), sin(θ)*unitvect)\n",
    "    q = isapprox(θ, π/2) ? UnitQuaternion(0, 0, 1, 0) : isapprox(θ, 0) ? \n",
    "                   UnitQuaternion(1, 0 , 0, 0) : UnitQuaternion(cos(θ), quvect...)\n",
    "    _, n₁, n₂,   = eachcol(q)\n",
    "    return γ(u) + radius*cos(v) * n₁ + radius*sin(v) * n₂\n",
    "end \n",
    "\n",
    "#define  a fiber curve  parameterization (§3.2 Fiber Curves in K Crane's paper)\n",
    "function fiber_curve(γ::T₁; \n",
    "                     ω=4, r=0.5, Φ=0) where {T₁}  #, T₂, T₃<:Function\n",
    "    # curve is t->γ(t), dγ: t->̇γ(t), ddγ: t->̈γ(t)\n",
    "    θ(t) =  ω*t −2cos(t)+Φ\n",
    "    unittan(t) = dγ(t)/norm(dγ(t))\n",
    "    unitacc(t) = ddγ(t)/norm(ddγ(t))\n",
    "    n₂(t) = cross(unittan(t), unitacc(t))  #binormal\n",
    "    n₁(t) = cross(n₂(t), unittan(t))  #principal normal\n",
    "    t->γ(t) + r*cos(θ(t)) * n₁(t) + r*sin(θ(t)) * n₂(t)\n",
    "end "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a31165ef",
   "metadata": {},
   "source": [
    "Define the PlotlyJS type data for plotting yarns and fibers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e8840b00",
   "metadata": {},
   "outputs": [],
   "source": [
    "burgundy = [[0, \"#e26152\"], [1, \"#e26152\"]] #kind of burgundy color\n",
    "silver = [[0, \"#bebebe\"], [1.0, \"#bebebe\"]]  #silver color\n",
    "\n",
    "function surf(x::Matrix{T} , y::Matrix{T}, z::Matrix{T}; colorscale=burgundy) where T<:Real\n",
    "    surface(x=x, y=y, z=z, hoverinfo=\"skip\",\n",
    "            colorscale=colorscale, \n",
    "            showscale=false,\n",
    "            lighting=attr(ambient=0.45,\n",
    "                                    diffuse=0.35,\n",
    "                                    fresnel=0.5,        \n",
    "                                    specular=0.25,\n",
    "                                    roughness=0.45),\n",
    "            lightposition=attr(x=100,\n",
    "                               y=100,\n",
    "                               z=100 \n",
    "                   ))  \n",
    "end\n",
    "\n",
    "axes_off(fig)=\n",
    "    relayout!(fig, scene =attr(xaxis_visible=false, \n",
    "                               yaxis_visible=false,\n",
    "                               zaxis_visible=false))\n",
    "\n",
    "function plot_data(traces) \n",
    "    fig = Plot(traces, \n",
    "               Layout(width=450, height=450, \n",
    "                      #margin=attr(t=2, r=2, b=2, l=2),\n",
    "                      scene=attr(aspectmode=\"data\", \n",
    "                               camera_eye=attr(x=2.7, y=0, z=0.5))\n",
    "                    ))  \n",
    "    axes_off(fig)\n",
    "    fig\n",
    "end    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "075b6734",
   "metadata": {},
   "source": [
    "### 1. Yarn curves "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "eac06197",
   "metadata": {},
   "outputs": [],
   "source": [
    "#K. Crane's yarn curve rotated π/2 about y-axis, then π/2 about x-axis\n",
    "γ(t; a=1.5, h=4, d=1) = [d*cos(2t),   t+a*sin(2t),  h*cos(t)]\n",
    "u = range(0, 8π, length=600)\n",
    "v = range(0, 2π, length=100); \n",
    "\n",
    "x, y, z = [getindex.(tube.(γ,  u, v'; radius=4/5), i) for i∈1:3]\n",
    "fig1 = plot_data([surf(x, y, z .+ k*4.5) for k in 0:3])  \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a4f57bcf",
   "metadata": {},
   "source": [
    "![burgundy](https://raw.githubusercontent.com/empet/Datasets/master/Images/yarncurves1.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3343727",
   "metadata": {},
   "source": [
    "![silverish](https://raw.githubusercontent.com/empet/Datasets/master/Images/yarncurves3.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "466555fb",
   "metadata": {},
   "source": [
    "### 2. Twisted yarn fibers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "bca238da",
   "metadata": {},
   "outputs": [],
   "source": [
    "γ(t; a=1.5, h=4, d=1)= [d*cos(2t),   t+a*sin(2t),  h*cos(t)] #curve \n",
    "fcurves = [fiber_curve(γ; ω= 4,  Φ=k*π/2) for k ∈ 0:3];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0dd0e8cc",
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "u = range(0, 8π, length=600)\n",
    "v = range(0, 2π, length=100); \n",
    "fibers = GenericTrace{Dict{Symbol, Any}}[]\n",
    "for fcurve  ∈ fcurves\n",
    "    x, y, z = [getindex.(tube.(fcurve,  u, v'; radius=7/20), i)  for i∈ 1:3]\n",
    "    push!(fibers, surf(x, y, z))\n",
    "end    \n",
    "fig2 = plot_data(fibers)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b5f4cbed",
   "metadata": {},
   "source": [
    "Fiber curves for $\\omega=2, 4, 6$:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fccff3bd",
   "metadata": {},
   "source": [
    "![fibers](https://raw.githubusercontent.com/empet/Datasets/master/Images/fibers.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4c94109b",
   "metadata": {},
   "outputs": [],
   "source": [
    "fibers = GenericTrace{Dict{Symbol, Any}}[]\n",
    "for fcurve  ∈ fcurves\n",
    "    x, y, z = [getindex.(tube.(fcurve,  u, v'; radius=7/20),i)  for i∈ 1:3]\n",
    "    append!(fibers, [surf(x, y, z .+ j*4.5) for j∈0:3])#a fiber contains 4 twisted yarns(j=0:3)\n",
    "end    \n",
    "fig3 = plot_data(fibers)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f86d02a4",
   "metadata": {},
   "source": [
    "![knitted o4](https://raw.githubusercontent.com/empet/Datasets/master/Images/knittedomega4.webp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5df4ac0b",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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