{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## This notebook both plots an equation $r= f(\\theta)$ in polar coordinates and computes its curvature.\n",
    "\n",
    "Start by importing the required libraries:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *  #python symbolic math package.\n",
    "import numpy as np   #numerical mathmatics library.\n",
    "import matplotlib.pyplot as plt    #plotting library.\n",
    "init_printing()            #to get good looking output."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To make things a bit easier input with will use $t=\\theta$ \n",
    "for the variable.  The code below declares $t$ to be a\n",
    "symbol, gives the domain, $[a,b]$, of the function $f$\n",
    "and defines $f(t)$.  The definition of $f$ should be\n",
    "edited as needed.  Then, as we will need these values\n",
    "to plot the graph, define give the formulas for $x$ and $y$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = symbols('t')\n",
    "a = 0\n",
    "b = 2*pi\n",
    "f = 1 - 2 * cos(t) #default.  Edit as needed. \n",
    "x = f*cos(t)\n",
    "y = f*sin(t)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Divide the domain into pieces of length $h$ to be used in plotting\n",
    "and compute the $x$ and $y$ values at these subdivsions of the domain\n",
    "and store in them in lists."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = .01   #edit if need be.\n",
    "t_dom = np.arange(a,b,h)\n",
    "x_vals = [x.subs(t,s) for s in t_dom]\n",
    "y_vals = [y.subs(t,s) for s in t_dom]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the curve."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left( -3.1562430086752933, \\  0.2812426920810328, \\  -1.936172771830307, \\  1.9361434472925099\\right)$"
      ],
      "text/plain": [
       "(-3.1562430086752933, 0.2812426920810328, -1.936172771830307, 1.93614344729250\n",
       "99)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x_vals,y_vals)\n",
    "plt.axis('equal')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the speed $v$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(5 - 4 \\cos{\\left(t \\right)}\\right)^{0.5}$"
      ],
      "text/plain": [
       "              0.5\n",
       "(5 - 4⋅cos(t))   "
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = diff(f,t)\n",
    "v = (f**2 + df**2)**(1/2)\n",
    "v.simplify()  # simplfy v if possible."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the curvature."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{3 \\left(3 - 2 \\cos{\\left(t \\right)}\\right)}{\\left(5 - 4 \\cos{\\left(t \\right)}\\right)^{1.5}}$"
      ],
      "text/plain": [
       "                               -1.5\n",
       "3⋅(3 - 2⋅cos(t))⋅(5 - 4⋅cos(t))    "
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ddf = diff(df,t)\n",
    "numerator = simplify(f*f + 2*df*df - f*ddf)\n",
    "numerator.simplify()  # simplity some more\n",
    "kappa = numerator/v**3\n",
    "kappa.simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the curvature."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fbd884fe438>]"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "kappa_vals = [kappa.subs(t,s) for s in t_dom]\n",
    "plt.plot(t_dom,kappa_vals)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}
