jiyometrik/water-droplet.ipynb

## water-droplet.ipynb
{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "# Plotting Challenge: Water Droplet\n",
        "\n",
        "Let's plot the shape of a **water droplet** using parametric differential\n",
        "equations!\n",
        "\n",
        "A set of two [**parametric equations**](https://en.wikipedia.org/wiki/Parametric_equation)\n",
        "are a type of equation which employs an independent variable: a **parameter**\n",
        "(in this case, $\\phi$). Dependent variables are defined as continuous functions\n",
        "of the parameter: $x\\left(\\phi\\right)$ and $y\\left(\\phi\\right)$. The\n",
        "$x$-coordinates of all the points in the graph are outputs of $x\\left(\\phi\\right)$,\n",
        "and the $y$-coordinates of all the points in the graph are outputs of $y\\left(\\phi\\right)$.\n",
        "\n",
        "However, this task comes with an additional challenge. We're not just dealing\n",
        "with parametric equations $x\\left(\\phi\\right)$ and $y\\left(\\phi\\right)$, we're\n",
        "going to have to deal with [differential equations](https://en.wikipedia.org/wiki/Differential_equation)\n",
        "$dx/d\\phi$ and $dy/d\\phi$, functions defined with respect to $\\phi$ that relate to\n",
        "the derivatives of $x\\left(\\phi\\right)$ and $y\\left(\\phi\\right)$. These two equations\n",
        "are defined with the functions $x\\left(\\phi\\right)$ and $y\\left(\\phi\\right)$ already in them."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 1,
      "metadata": {},
      "outputs": [],
      "source": [
        "%matplotlib inline\n",
        "\n",
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "import scipy as sp\n",
        "\n",
        "plt.rcParams[\"figure.autolayout\"] = True\n",
        "plt.rcParams[\"figure.dpi\"] = 750\n",
        "plt.rcParams[\"font.family\"] = \"serif\"\n",
        "plt.rcParams[\"text.usetex\"] = True\n",
        "plt.rcParams[\n",
        "    \"text.latex.preamble\"\n",
        "] = \"\"\"\n",
        "\\\\usepackage{siunitx}\n",
        "\\\\usepackage{fourier}\n",
        "\"\"\"\n",
        "plt.style.use(\"seaborn-v0_8-dark\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Equations\n",
        "\n",
        "Let's take a look at the equations we're given:\n",
        "\n",
        "$$\n",
        "\\begin{align*}\n",
        "\\frac{dx}{d\\phi} &= \\frac{x \\cos \\phi}{R^2xy + xp - \\sin \\phi} \\\\\n",
        "\\frac{dy}{d\\phi} &= \\frac{x \\sin \\phi}{R^2xy + xp - \\sin \\phi} \\\\\n",
        "\\end{align*}\n",
        "$$\n",
        "\n",
        "Notice that $x$ is present in both of these differential equations, and that the\n",
        "denominators are the same! $R$ and $p$ are experimental constants, which we can\n",
        "helpfully take as\n",
        "\n",
        "$$\n",
        "\\begin{align*}\n",
        "R &= 0.01 \\\\\n",
        "p &= 0.00\n",
        "\\end{align*}\n",
        "$$\n",
        "\n",
        "However, we can't just let our differential equations run amok! We're going to\n",
        "need to set some **boundaries**:\n",
        "\n",
        "$$\n",
        "\\begin{cases}\n",
        "\\phi = 0 &\\to x = 0, y = 0 \\\\\n",
        "\\phi = \\pi/2 &\\to x = 1\n",
        "\\end{cases}\n",
        "$$\n",
        "\n",
        "Lastly, we're going to need to set some restrictions on our **domain**: what\n",
        "values $\\phi$ can span. These will be:\n",
        "\n",
        "$$\n",
        "\\phi \\in \\left[0, 99 \\deg \\right]\n",
        "$$"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {},
      "outputs": [],
      "source": [
        "# Define our constants.\n",
        "R_, p_ = 0.01, 0.00"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 3,
      "metadata": {},
      "outputs": [],
      "source": [
        "# NOTE: Define our differential equations.\n",
        "def F(z, phi) -> list:\n",
        "    x, y = z\n",
        "    denominator = R_**2 * x * y + x * p_ - np.sin(phi)\n",
        "    dx = (x * np.cos(phi)) / denominator\n",
        "    dy = (x * np.sin(phi)) / denominator\n",
        "    return [dx, dy]"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Great! Now that we've established our functions, we can set our boundary\n",
        "conditions: $x\\left(0\\right) = 0$ and $y\\left(0\\right) = 0$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {},
      "outputs": [],
      "source": [
        "# Set our initial conditions.\n",
        "# NOTE: We cannot set them to 0.0, otherwise we get a division by zero error.\n",
        "z_initial = [1e-5, 1e-5] # x(0), y(0)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Now, let's get ready to compute our $\\phi$ values. Let's establish the number\n",
        "of distinct data points we want, and our domain for $\\phi$."
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 5,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([0.00000000e+00, 8.64370165e-04, 1.72874033e-03, ...,\n",
              "       1.72614722e+00, 1.72701159e+00, 1.72787596e+00])"
            ]
          },
          "execution_count": 5,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "n = 2 * (10**3)\n",
        "phi_vals = np.linspace(0, np.deg2rad(99), n)\n",
        "\n",
        "phi_vals"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        " Now we can finally evaluate the outputs that $x\\left(\\phi\\right)$ and\n",
        " $y\\left(\\phi\\right)$ give us."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Solving differential equations\n",
        "\n",
        "We can make use of the spectacular [`scipy.integrate.solve_ivp`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html#scipy.integrate.solve_ivp)\n",
        "method to solve differential equations. This will help us extract the true\n",
        "definitions of $x\\left(\\phi\\right)$ and $y\\left(\\phi\\right)$!\n",
        "\n",
        "After that, we can finally plug in our $n$ values of $\\phi$, and see some output!"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 6,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/plain": [
              "array([[1.00000000e-05, 1.00000000e-05],\n",
              "       [1.12277367e+03, 9.05694979e-02],\n",
              "       [1.19498324e+03, 1.80604222e-01],\n",
              "       ...,\n",
              "       [1.84229386e+03, 1.55508452e+02],\n",
              "       [1.84228493e+03, 1.55565313e+02],\n",
              "       [1.84227595e+03, 1.55622144e+02]])"
            ]
          },
          "execution_count": 6,
          "metadata": {},
          "output_type": "execute_result"
        }
      ],
      "source": [
        "MIN_TOLERANCE = 1e-9\n",
        "solutions = sp.integrate.odeint(\n",
        "    F,\n",
        "    z_initial,\n",
        "    phi_vals,\n",
        "    atol=MIN_TOLERANCE,\n",
        "    rtol=MIN_TOLERANCE,\n",
        ")\n",
        "\n",
        "solutions"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/plain": [
              "<matplotlib.legend.Legend at 0x7f85cb5e9ba0>"
            ]
          },
          "execution_count": 7,
          "metadata": {},
          "output_type": "execute_result"
        },
        {
          "data": {
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ASRkzWnvwg5EvvfRSvPLKKztej9nZ2XXPlcvlbW1mLMrcB4D8PP3003314Ym1ijSPKlJbAOhO/ThnsF8BANrXj3OGtYoyzlpnACBL/TZfsF8BAPIzmHcFAID0tXojf7t/7EZEyz9sG41GXLhwYVtl7JQi9VeR2gJA9owbrc3MzLQ85tixY6ncxaFSqSQ6Lq3+2uiuFEnrAEB/MWdoX1HGWdcegKSMGVvrlg2SERuvK2x3M2NR5j4A5OfQoUN5VyE3RZpHFaktAHSnfpsz2K8AAJ3ptznDg4oyzlpnACBL/TRfsF8BAPIl1AkACqZarUa9Xt/ymDSShiuVSss7LiW5S1TeitRfRWoLANkzbiST5A4c3/72t1Mpq1QqJdpk8NZbb6VS3kZ3pUhjsycAxWLO0JkijLOuPQBJGTNaW7u+UCqVctsg2Wg01q11lMvlmJyc3PZ5H9Rrcx8AyEOR5lFFagsAdAv7FQCAThRhnLXOAADpsV8BAPIl1AkACibJ3Zm2+8du0vPU6/VEGwvyVKT+KlJbAMiecSOZy5cvb/nzUqmU6p0cRkdHWx5Tq9U2fPNhu0qlUurnBKD3mTOkoxfHWdcegKSMGa1dunTp3tdTU1O51WOjD16m8aGHB/Xi3AcA8lCkeVSR2gIA3cJ+BQAgDb04zlpnAID02K8AAPkS6gQABZJkwTjNpOEkGwKSLKjnpUj9VaS2AJA940ZyrTYtvvDCC6mW1+quT6tabd5sZaNNlknLBqB/mDN0pgjjrGsPQFLGjGTeeeediFjePJjWhww68WDfZHXXy16b+wBAHoo0jypSWwCgm9ivAAC0qwjjrHUGAEiX/QoAkK/hvCsAAKQnyWLxs88+m1p5SRbDq9VqNBqNrkw5LlJ/FaktAGTPuJFcpVKJ2dnZTX8+Pj6eankjIyOJjqvX69sqZ25ubt1zjz322LbOCUDxmDN0pgjjrGsPQFLGjGRee+21iMh382C1Wl23npDGXS+LMPcBgDwUaR5VpLYAQDexXwEAaFcRxlnrDACQLvsVACBfg3lXAABIT6s7EkREPPfcc6mWmeTOBEnqlYci9VeR2gJA9owbyU1NTW36BkapVErUrnY8/vjjiY7b6K4S7XBXCgCSMGfoTBHGWdcegKSMGcmMj4/H+Ph4qneOblcWd72MKMbcBwDyUKR5VJHaAgDdxH4FAKBdRRhnrTMAQLrsVwCAfAl1AoCCaDQaie6AlPYb+UnO9+abb6ZaZhqK1F9FagsA2TNutKdUKsX58+c3rP+3vvWt1MsbHR1NdNx273y5kaQbNAHoD+YM6eqlcda1ByApY0bvqNVq6z5skMZdLzfTS3MfAMhDkeZRRWoLAHQb+xUAgDT00jhrnQEAisd+BQD63XDeFQAA0pEk+T+LpOEkKc3vvPNO6uVuV5H6q0htASB7xo32lUqlOHfuXNRqtXjzzTejXq/Hc889l8rdIR40NzeX6LirV6+mXk7SDZoA9Adzhs71+jjr2gOQlDGjd0xPT9/3fVp3vYzo/bkPAOShSPOoIrUFALqR/QoAQDt6fZy1zgAAxWO/AgD9TqgTABREkuT/p59+OvVykyyKr94xIYsF9E4Vqb+K1BYAsmfc6NzY2FiiN++3o9FoZHr+rcrp1n4HIB/mDJ3r9XHWtQcgKWNGb6hWq+s+BHHmzJnUzt/rcx8AyEOR5lFFagsAdDP7FQCAJHp9nLXOAADFYr8CAEQM5l0BACAdSZL/s/ijNOk5k9w1YScVqb+K1BYAsmfc6G5J72i53Wu00V0pvIEBwFrmDJ3r9XHWtQcgKWNGbzh16tR93x87dizVD4H2+twHAPJQpHlUkdoCAP3OfgUA6H29Ps5aZwCAYrFfAQCEOgFAITQajUR3Sfra176WSflJ/tit1WqZlN2JIvVXkdoCQPaMG92vXq8nOu7QoUPbKmej10GpVNrWOQEoDnOG7enlcda1ByApY0ZvmJ2dXbfW8N3vfjfVMnp57gMAeSjSPKpIbQEA7FcAgCLo5XHWOgMAFIv9CgCwTKgTABRA0jfTs0oaTnLepHXcCUXqryK1BYDsGTe635UrVxIdt91r9OAdNt2RAoC1zBm2p5fHWdcegKSMGd2v0WjE9PT0fc+dPn069Q2MvTz3AYA8FGkeVaS2AAD2KwBAEfTyOGudAQCKw34FAPiMUCcAKICki8NZJQ0n+YP38uXLmZTdiSL1V5HaAkD2jBvdrdFoJL5GlUplW2XNz8/f9703MABYy5xhe3p5nHXtAUjKmNH9pqen77srZblcjsnJydTL6eW5DwDkoUjzqCK1BQD6nf0KAFAMvTzOWmcAgOKwXwEAPiPUCQAKIO+7EoztasIjAABTS0lEQVSMjLQ8Zu0f4nkrUn8VqS0AZM+40d2SvuFfLpe3vTHBXSkA2Io5w/b08jjr2gOQlDGju9VqtZidnb3vuTNnzmRSVi/PfQAgD0WaRxWpLQDQ7+xXAIBi6OVx1joDABSD/QoAcL/hvCsAAGxf0gXsrBw6dCjRcfV6vSv+OC5SfxWpLQBkz7jR3f7sz/4s0XHHjh3bdlnt3pWiXq9HtVqNv/zLv4z5+fmo1+v3Xk+lUinK5XKMjo5GuVyOSqUSlUolsztiAZA9c4bt6eVx1rUHICljRnc7efLkfd9PTk7G2NhYJmX18twHAPJQpHlUkdoCAP3OfgUAKIZeHmetMwBAMdivAAD3E+oEAAWQZAE7yz8gky5Kd8sCdpH6q0htASB7xo3u1Wg04uLFi4mO/fa3v73t8ubm5u77fqP+rtfrMTMzE9VqdcvXTqPRiFqtdu/71TtrlMvlOHHiRExOTm67vgDsLHOG7enlcda1ByApY0b3ev311++7PqVSKaampjIrr5fnPgCQhyLNo4rUFgDoZ/YrAEBx9PI4a50BAHqf/QoAsN5g3hUAALYvyQL26OhoZuUnXRzP++4Jq4rUX0VqCwDZM250r7feeivRcZOTk6lsCGg0Gvd9v/actVotXn755Xj++edjdna24+tRr9fj1KlT8fWvfz1mZma2VV8AdpY5w/b08jjr2gOQlDGjO9VqtTh79ux9z01NTWX6IYdenvsAQB6KNI8qUlsAoJ/ZrwAAxdHL46x1BgDobfYrAMDGhDoBQAE8mCq8kSz/AM5ycTwLReqvIrUFgOwZN7rX9PR0ouPSulPFZnelePnll2NiYiLxXTiTaDQaMT09Hc8///x9d68AoHuZM2xPL4+zrj0ASRkzutPJkyfv+35sbCzzu0X28twHAPJQpHlUkdoCAP3MfgUAKI5eHmetMwBAb7NfAQA2Npx3BQCAndENi8wPph93syL1V5HaAkD2jBs77/XXX0/U5jNnzqS2KeHB8ur1erz44osb1uPYsWPxta99LUqlUpTL5RgdHY16vR71ej3eeuutxG9K1Ov1mJiYiNOnT2f+Bg0A2TNn2FzRx1nXHoCkjBk76/XXX193Z8nXXnst83KLPvcBgDwUaR5VpLYAQBHZrwAAxVL0cdY6AwB0J/sVAGBzQp0AoADyXhjO8o4HWShSfxWpLQBkz7jRfer1epw9e7blcZOTkzE+Pp5ZPSYmJu77vlKpbFnm2NhYREScOHHi3l0nZmdnE5V16tSpaDQaceLEie1VGoDMmDOkq5fGWdcegKSMGd2lVqutW184duzYvXnFTuqluQ8A5KFI86gitQUA+pH9CgBQfL00zlpnAIDeZL+CNQYAtjaYdwUAgP5x9erVvKvQU4rUX0VqCwDZ66dx4+TJky2PqVQqcfr06dTK3GrzQ6lUinPnzsW5c+cSb8oslUpx+vTpePvtt6NcLif6nenp6ahWq4mOBYDNdOOcwTi7M7rx2gPQnfplzNhofeG73/1u5uWa+wBAcRVpHlWktgBAmuxXAIBiMc7uDOsMAHA/+xWKPfcBYPuEOgFAnxgZGcm7CjE/P593FRIrUn8VqS0AZM+4sXNef/31qNVqWx5TqVTi3LlzqZY7Nze34fPHjh2LH/zgB1GpVDo6b7lcjvPnzye+q8bx48ejXq93VBYA+TNn2Fg/jLOuPQBJGTN2xqlTp9aN+y+99NKO3FG6H+Y+AJCHIs2jitQWACgS+xWsJQBQPP0wzlpnAIDuYr+CNQYAWhPqBAA9bqtU4Z0yOjqadxUSK1J/FaktAGTPuNFdLly4EGfPnt3ymGPHjqW+QTJi49fCSy+9FG+88ca230AplUpx/vz5xHenSHLnTwB2ljnD9vTyOOvaA5CUMaN71Gq1mJ2dXff8d77znR0pv5fnPgCQhyLNo4rUFgDoN/YrLLOWAEDR9PI4a50BAHqP/QqfscYAwFaEOgEAAADsoGq12nLhfvUNhZ3w0ksvxSuvvJLqOc+cOZPouFqtFtVqNdWyAaCbGGcBgCxttL4wOTm5I3e93Ii5DwAAAHQ3+xU+Yy0BgKIzzgIAWbJf4TPmPgBsRagTAPS4vP7QXWtubi7vKiRWpP4qUlsAyJ5xozvUarU4fvz4lsecOXMm9TcU1hobG4vz589HpVLJ5M2L1TJeeumlRMdOT0+nXj4AnTNn2J5eHmddewCSMmZ0h5mZmajX6+uen5qa2rE69PLcBwDyUKR5VJHaAgD9wn6F9awlAFAkvTzOWmcAgN5iv8J61hgA2IxQJwDoE/Pz83lXIUZGRvKuQmJF6q8itQWA7Bk3slOr1eLFF1/c9OflcjnefvvtGB8fz7wuY2Njce7cuUw3Y37nO99JdJw7UwD0JnOGzRV9nHXtAUjKmJGder2+4YbAPO56WfS5DwDkoUjzqCK1BQB6mf0KG7OWAEDRFH2ctc4AAPmzX2Fj1hgA2IxQJwBgxxw6dCjvKvSUIvVXkdoCQPaKOG6sbpBsNBob/vzYsWPx9ttvR7lc3uGaZadUKsWxY8cSHXvhwoWMawNAERVxzpBUv4+z/XztAWhPUceMU6dObfj8iRMndrgmO6Pf5z4AkIcizaOK1BYA6IT9CluzlgAA7en3cdY6AwD9zn6FzRVx7gPA9gl1AoAC2OkU4wdt9mZ/typSfxWpLQBkz7iRj1qtFhMTE5u2//Tp0/HGG2/scK12xre//e1Ex7krBUB3MWfoDVmMs649AEkZM/Jz4cKFDcf3Y8eOFerDlw+yxgBAURRpHlWktgBAUdmv0Jq1BABon/0KANCf7FfYmjUGADYi1AkA+sTc3FzeVch9ob0dReqvIrUFgOwZN9JVrVZjYmJiw5+Vy+V4++23Y3JycodrtXMqlUqi4+r1uk0RAD3GnCF/eY2zrj0ASRkzsjE9Pb3h80k3EfYqawwA9JMizaOK1BYA6DX2K1hLAICs2K8AAP3JfoWtWWMAYCNCnQCgAEZHR1sek+UfhN2wON6OIvVXkdoCQPaMGztrdnY2jh8/vuHPJicn4+233y70HSlWjY2NJTrOnSkAuoc5Q+9Ie5x17QFIypiRj9nZ2ajX6+ueL5VKiTcR9jJrDAAUQZHmUUVqCwAUjf0Ky6wlAEB27FcAgP5iv4I1BgA6I9QJAAogyZvrWS4yJ10c75ZNAEXqryK1BYDsGTd2zszMTJw6dWrDn505cyZOnz69wzXKz7PPPpvouI3e5AEgH+YMvSPtcda1ByApY0Y+Nrvr5QsvvLDDNcmHNQYAiqBI86gitQUAisR+hc9YSwCA7NivAAD9xX4FawwAdGY47woAANuXZGG4G+5K0C0L2EXqryK1BYDsGTd2xqlTp2J2dnbd8+VyOc6dO9fz7WvX448/nui4q1evZlsRABIzZ+gdaY+zrj0ASRkzdt7MzMymfTo5ObnDtcmHNQYAiqBI86gitQUAisJ+hftZSwCA7NivAAD9w34FawwAdE6oEwAUQN4Lw++9916i4/Ku56q865FmfxWpLQBkL+//H/fDuPHyyy/HxYsX1z1/7NixeOONN3KoUf5GR0cTHeeuFADdI++xuB/mDGlJe5zNu09de4Dekff/i/txzPje97636c/GxsZ2sCb5scYAQBHkPT+xXwEAist+hfWsJQBAduxXAID+Yb+CNQYAOjeYdwUAgO1LujCc1Z0J5ufnWx5TKpUyKbsTReqvIrUFgOwZN7J1/PjxDTdIvvTSS327QTIi+TVN8voAYGeYM/SOtMdZ1x6ApIwZO2uru15WKpUdrk1+rDEAUARFmkcVqS0A0OvsV9iYtQQAyI79CgDQH+xXWGaNAYBOCXUCgAJIuoCdVdJvkvM+88wzmZTdiSL1V5HaAkD2jBvZOX78eFSr1XXPnz59Ol555ZUcatQ9kt6VYm5uLuOaAJCUOUPvSHucde0BSMqYsbO2uuvl+Pj4DtYkX9YYACiCIs2jitQWAOhl9itszloCAGTHfgUA6A/2KyyzxgBAp4Q6AUAB9MICdtI67oQi9VeR2gJA9owb2ZiYmNhwg+SZM2dicnIyhxp1l6R3pXjssccyrgkASZkz9I60x1nXHoCkjBk758KFC1veddqdL9ezxgBANyvSPKpIbQGAXmW/wtasJQBAduxXAIDis1/hM9YYAOiUUCcAKIBSqZToD8NLly5lUn6SBeyxsbFMyu5EkfqrSG0BIHvGjfRNTExErVZb9/y5c+f66s4TW9nqjZy1Dh06lG1FAEjMnKF3pD3OuvYAJGXM2Dlb3fUyor8+QGCNAYAiKNI8qkhtAYBeZL9Ca9YSACA79isAQPHZr/AZawwAdEqoEwAUxLPPPtvymCzuSpD0nN2WvFyk/ipSWwDInnEjPcePH990g2QvtSNrc3NziY4bGRnJuCYAtMOcoTdkMc669gAkZczIXr1e33DtYVW/fXjAGgMARVGkeVSR2gIAvcR+hWSsJQBAduxXAIBis1/hftYYAOiUUCcAKIhvfvObLY+5cuVK6uUmWcAulUpdl7xcpP4qUlsAyJ5xIx0vv/xyVKvVdc+fOXPGG/cdclcKgO5izlAs7Yyzrj0ASRkzsvdnf/ZnW/78mWee2aGa9BZrDAB0uyLNo4rUFgDoFfYrpM9aAgBkx34FAOhN9it0xhoDAA8S6gQABZHkzfgs7kqwVeLyqiR3TNhpReqvIrUFgOwZN7bv9ddfj4sXL657fmpqKsbHx3OoUXdzFyuA3mTO0BuyGGddewCSMmZk78///M+3/Hm/3fnSGgMARVGkeVSR2gIAvcB+hfZYSwCA7NivAADFZr/C/awxANApoU4AUBBJk/+TLDi3Y6M7Pj3oueeeS7XMNBSpv4rUFgCyZ9zYntnZ2Th79uy6548dOxYnTpzIoUbtqdfr8fLLL8fXv/71eP755+PChQuZl5n0tdRvb+wAdDtzhvYVZZx17QFIypiRrVqtFo1GY8tj8rwjdFHmPgCQhyLNo4rUFgDodvYrtM9aAgD9oijjrHUGAOgO9iusZ40BgE4JdQKAAkmS5Pvmm2+mWmaSBewXXngh1TLTUqT+KlJbAMiecaMztVotTp06te75crkcb7zxRg41as/s7Gw8//zzcfHixWg0GlGv1+PkyZOZ3LlqrcuXL7c8xh0pALqTOUNyRRtnXXsAkjJmZCdJO/PaJFm0uQ8A5KFI86gitQUAupX9Cp2xlgBAPyjaOGudAQDyZ7/CetYYAOiUUCcAKJDJycmWx7zzzjuplZckYbhSqUSpVEqtzDQVqb+K1BYAsmfcaF+j0YiTJ09u+LMzZ87scG3i3p0ljh8/3vIuGBHLb6xstMEzImJmZibt6t0n6fUHoPuYMyRTxHHWtQcgKWNGdt56662Wx+SxSbKIcx8AyEOR5lFFagsAdCP7FTpnLQGAoiviOGudAQDyZ7/CetYYAOiUUCcAKJCxsbGWf/wl+QMyqSSpyydOnEitvIhINTW5SP1VpLYAkL1+GDfSvtPCq6++uuE5p6amYmxsLNWyWmk0GvfuLFGtVhNtFtjszYuI5btVJNlo2YkLFy4kOi7JRgwAdp45QzJFHGf74doDkI5+GDOyvpvjZtLstzQVce4DAEnZr7CxIrUFANJgv8L9rCUAwMbsV9iYdQYA+Iz9Cvcr4twHgOIT6gQABZNkwXh2djaVslqdp1wup5IwXKvVYmJiIp566ql4/vnn7z2+/vrr2z53kfqrSG0BIHtFHDeymjNUq9W4ePHiuudLpVIub9a/+uqr975O0m+NRqPlGzppXesHvfnmmy2PmZycdBcrgC5mzrC1Io+zRbz2AGSjiGNGlu9LJC2/lTz+li7y3AcANmO/QjJFagsAdMJ+hY1ZSwCA+9mvkIx1BgD6mf0KGyvy3AeAYhPqBAAFU6lUolwub3nMzMzMtsupVqst/xBOY9PA66+/HhMTE+sWBOr1epw9ezYmJia2laJcpP4qUlsAyF7Rxo0s5wwnT57c8PnXXnuto/Ntx4ULF+7bsJnWZoHvfe97qZxnrdU7dLZi3gDQ3cwZtq9Xx9miXXsAslO0MSOP+cKDLl++3PKY0dHRTOvQqV6d+wDARuxXSK5IbQGAdtmvsD3WEgDoF/YrJGedAYB+Zb/C9vTq3AeAYhPqBAAFNDU1teXP6/V6otTkrbRaBC+VSjE5ObmtMqrVapw9e3bLY2q12n13f+pEUforolhtASB7RRk3spwzzMzMbPjGR6lUivHx8bbPtx0XLlxYt2EzySbJUqnU8q4PjUYjlU0Oa/3pn/5py2OOHTvWcvMFAPkzZ9i6XkUeZ4ty7QHIXlHGjJ16X6KVbr3zZdHnPgCwlv0K7StSWwAgKfsVtmYtAQCW2a/QPusMAPQb+xVal1nkuQ8AxSXUCQAKaHx8vOUb5tPT0x2fv16vR7Va3fKY73//+x2ff9WpU6cSHXfx4sVtLcgXpb8iitUWALJXlHEjqzlDo9HYtP3PPPNM1Gq1zP5Vq9WoVqtx4cKFmJmZieeff37DO3COjY0lasu3vvWtlsdMT09ve5PDqkaj0fJNpXK5HG+88UYq5QGQLXOGrRV5nC3KtQcge0UZM3bqfYlWuvnOl0We+wCQn6tXr+ZdhXXsV2hfkdoCQHfqpzmD/Qqds5YAQD/NGYo8zlpnACBL/TRfaJf9Cp+xxgBAGobzrgAAkI0zZ87EN77xjQ3v1BSxnN48Ozvb0Z0Djh8/vuXPp6amEr9hv5lGoxH1ej3x8ZcvX95Wmb3eX2sVqS0AZK/Xx40s5wxb3VWhWq3GxMRE4nKzkOSul6u+853vtHxDISLixRdfjB/84AfbvnvGiy++2PKYM2fObKsMAHaWOcPmij7O9vq1B2Dn9PqYsdPvS7SqS7cq+twHgHzMz8+nckxa7FfoXJHaAkD36ac5g/0KnbOWAEA/zRmKPs5aZwAgK/00X+ikLt2q6HMfAIppMO8KAADZKJVKLe8McOrUqbb+4E/yO5VKJU6cONHWOTfSbr22m6Dc6/21VpHaAkD2en3cyHLO8Od//uftVmdHjY+PJz62VCrFuXPnWh7XaDQ2vMNmO15++eWW/Xz69GkbHgB6jDnD5oo+zvb6tQdg5/T6mLHT70tsZW5uLrNzb1fR5z4A5KNaraZyTFrsV+hckdoCQPfppzmD/QqdsZYAQER/zRmKPs5aZwAgK/00X2iX/QrLrDEAkBahTgBQYGNjYzE1NbXlMcePH0+coDw7Oxuzs7Ob/rxUKvV0unCR+qtIbQEge8aN9S5cuNDVd5mIaO/Ol6vHnz59uuVxq3f1bLf9jUYjJiYm4uLFi1sed+bMmY7ujAVA/swZNlf0cda1ByApY0Y6emFNoshzHwB21uzsbKIPK9Tr9S3nBb2uSPOoIrUFgO7RT3MG+xWsJQDQuX6aM6wq+jhrnQGAtPXjfKEdvbAmUeS5DwDFM9BsNpt5VwIAyNbs7GycOnVq05+XSqV47bXXNr17UqPRiFdffXXLP0bL5XKcP38+SqXStuu76qmnnkp87OnTp1P7Q7hX+2sjRWoLANnr1XEjiznD8ePHd/TuGu0qlUrx7rvvdvS71Wo1Tp48megNipdeeim+853vbHm9Go1GzM7Oxve+970tz1kqleL8+fNRLpc7qjcA3cOcYXNFH2d79do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            "text/plain": [
              "<Figure size 4800x3600 with 1 Axes>"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "X_PHI, Y_PHI = \"x\\left(\\\\varphi\\\\right)\", \"y\\left(\\\\varphi\\\\right)\"\n",
        "x_vals, y_vals = solutions[:, 0], solutions[:, 1]\n",
        "plt.plot(x_vals, y_vals, label=f\"${Y_PHI}$ against ${X_PHI}$\")\n",
        "plt.title(\n",
        "    \"Plot of water droplet governed by parametric equations \"\n",
        "    + f\"$\\left[ {X_PHI}, {Y_PHI} \\\\right]$\"\n",
        ")\n",
        "plt.xlabel(f\"${X_PHI}$\")\n",
        "plt.ylabel(f\"${Y_PHI}$\")\n",
        "plt.ticklabel_format(\n",
        "    style=\"sci\", axis=\"both\", scilimits=(0, 0), useMathText=True\n",
        ")\n",
        "plt.legend(loc=\"best\")"
      ]
    }
  ],
  "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": "3.10.12"
    }
  },
  "nbformat": 4,
  "nbformat_minor": 2
}