{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Plotting smooth curves\n",
    "\n",
    "<hr/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "nbsphinx": "hidden",
    "tags": []
   },
   "outputs": [],
   "source": [
    "#| code-fold: true\n",
    "\n",
    "# Colab setup ------------------\n",
    "import os, sys, subprocess\n",
    "if \"google.colab\" in sys.modules:\n",
    "    cmd = \"pip install --upgrade watermark\"\n",
    "    process = subprocess.Popen(cmd.split(), stdout=subprocess.PIPE, stderr=subprocess.PIPE)\n",
    "    stdout, stderr = process.communicate()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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       "        }\n",
       "    </style>\n",
       "    <div>\n",
       "        <a href=\"https://bokeh.org\" target=\"_blank\" class=\"bk-notebook-logo\"></a>\n",
       "        <span id=\"a8f5feeb-6b53-495c-85e4-ed9477aabadd\">Loading BokehJS ...</span>\n",
       "    </div>\n"
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     "output_type": "display_data"
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If loading BokehJS from CDN, this \\n\"+\n     \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n     \"</p>\\n\"+\n     \"<ul>\\n\"+\n     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n     \"</ul>\\n\"+\n     \"<code>\\n\"+\n     \"from bokeh.resources import INLINE\\n\"+\n     \"output_notebook(resources=INLINE)\\n\"+\n     \"</code>\\n\"+\n     \"</div>\"}};\n\n  function display_loaded(error = null) {\n    const el = document.getElementById(\"a8f5feeb-6b53-495c-85e4-ed9477aabadd\");\n    if (el != null) {\n      const html = (() => {\n        if (typeof root.Bokeh === \"undefined\") {\n          if (error == null) {\n            return \"BokehJS is loading ...\";\n          } else {\n            return \"BokehJS failed to load.\";\n          }\n        } else {\n          const prefix = `BokehJS ${root.Bokeh.version}`;\n          if (error == null) {\n            return `${prefix} successfully loaded.`;\n          } else {\n            return `${prefix} <b>encountered errors</b> while loading and may not function as expected.`;\n          }\n        }\n      })();\n      el.innerHTML = html;\n\n      if (error != null) {\n        const wrapper = document.createElement(\"div\");\n        wrapper.style.overflow = \"auto\";\n        wrapper.style.height = \"5em\";\n        wrapper.style.resize = \"vertical\";\n        const content = document.createElement(\"div\");\n        content.style.fontFamily = \"monospace\";\n        content.style.whiteSpace = \"pre-wrap\";\n        content.style.backgroundColor = \"rgb(255, 221, 221)\";\n        content.textContent = error.stack ?? 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      "application/vnd.bokehjs_load.v0+json": ""
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import scipy.special\n",
    "\n",
    "import bokeh.plotting\n",
    "import bokeh.io\n",
    "bokeh.io.output_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr/>\n",
    "\n",
    "Sometimes you want to plot smooth functions, as opposed to measured data like we have done so far. To do this, you can use Numpy and/or Scipy to generate arrays of values of smooth functions.\n",
    "\n",
    "We will plot the [Airy disk](https://en.wikipedia.org/wiki/Airy_disk), which we encounter in biology when doing microscopy as the diffraction pattern of light passing through a pinhole. Here is a picture of the diffraction pattern from a laser (with the main peak overexposed).\n",
    "\n",
    "<div style=\"width: 300px; margin: auto;\">\n",
    "\n",
    "![Airy disk](airy_disk.png)\n",
    "    \n",
    "</div>\n",
    "\n",
    "The equation for the radial light intensity of an Airy disk is\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{I(x)}{I_0} = 4 \\left(\\frac{J_1(x)}{x}\\right)^2,\n",
    "\\end{align}\n",
    "\n",
    "where $I_0$ is the maximum intensity (the intensity at the center of the image) and $x$ is the radial distance from the center. Here, $J_1(x)$ is the first order Bessel function of the first kind. Yeesh. How do we plot *that*?\n",
    "\n",
    "Fortunately, SciPy has lots of special functions available. Specifically, `scipy.special.j1()` computes exactly what we are after! We pass in a NumPy array that has the values of $x$ we want to plot and then compute the $y$-values using the expression for the normalized intensity.\n",
    "\n",
    "To plot a smooth curve, we use the `np.linspace()` function with lots of points. We then connect the points with straight lines, which to the eye look like a smooth curve.  Let's try it. We'll use 400 points, which I find is a good rule of thumb for not-too-quickly-oscillating functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The x-values we want\n",
    "x = np.linspace(-15, 15, 400)\n",
    "\n",
    "# The normalized intensity\n",
    "norm_I = 4 * (scipy.special.j1(x) / x) ** 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we have the values we want to plot, we could construct a Pandas `DataFrame` to pass in as the `source` to `p.line()`. We do not need to take this extra step, though. If we instead leave `source` unspecified, and pass in NumPy arrays for `x` and `y`, Bokeh will directly use those in constructing the plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
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 const render_items = [{\"docid\":\"2adada6f-2227-4e00-ae22-0f09103b9a19\",\"roots\":{\"p1003\":\"e38b2437-8db0-4f0b-a648-9bc1c7fa1900\"},\"root_ids\":[\"p1003\"]}];\n  void root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n  }\n  if (root.Bokeh !== undefined) {\n    embed_document(root);\n  } else {\n    let attempts = 0;\n    const timer = setInterval(function(root) {\n      if (root.Bokeh !== undefined) {\n        clearInterval(timer);\n        embed_document(root);\n      } else {\n        attempts++;\n        if (attempts > 100) {\n          clearInterval(timer);\n          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n        }\n      }\n    }, 10, root)\n  }\n})(window);",
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "p1003"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = bokeh.plotting.figure(\n",
    "    height=250, \n",
    "    width=550, \n",
    "    x_axis_label=\"x\", \n",
    "    y_axis_label=\"I(x)/I₀\",\n",
    ")\n",
    "\n",
    "p.line(\n",
    "    x=x, \n",
    "    y=norm_I, \n",
    "    line_width=2,\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We could also plot dots (which doesn't make sense here, but we'll show it just to see who the line joining works to make a plot of a smooth function)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "  <div id=\"ff081dd1-e6fd-48b3-9197-5f350cbc5514\" data-root-id=\"p1003\" style=\"display: contents;\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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dices\",\"id\":\"p1056\"}}},\"glyph\":{\"type\":\"object\",\"name\":\"Scatter\",\"id\":\"p1051\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"size\":{\"type\":\"value\",\"value\":2},\"line_color\":{\"type\":\"value\",\"value\":\"orange\"},\"fill_color\":{\"type\":\"value\",\"value\":\"orange\"},\"hatch_color\":{\"type\":\"value\",\"value\":\"orange\"}}},\"nonselection_glyph\":{\"type\":\"object\",\"name\":\"Scatter\",\"id\":\"p1052\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"size\":{\"type\":\"value\",\"value\":2},\"line_color\":{\"type\":\"value\",\"value\":\"orange\"},\"line_alpha\":{\"type\":\"value\",\"value\":0.1},\"fill_color\":{\"type\":\"value\",\"value\":\"orange\"},\"fill_alpha\":{\"type\":\"value\",\"value\":0.1},\"hatch_color\":{\"type\":\"value\",\"value\":\"orange\"},\"hatch_alpha\":{\"type\":\"value\",\"value\":0.1}}},\"muted_glyph\":{\"type\":\"object\",\"name\":\"Scatter\",\"id\":\"p1053\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"size\":{\"type\":\"value\",\"value\":2},\"line_color\":{\"type\":\"value\",\"value\":\"orange\"},\"line_alpha\":{\"type\":\"value\",\"value\":0.2},\"fill_color\":{\"type\":\"value\",\"value\":\"orange\"},\"fill_alpha\":{\"type\":\"value\",\"value\":0.2},\"hatch_color\":{\"type\":\"value\",\"value\":\"orange\"},\"hatch_alpha\":{\"type\":\"value\",\"value\":0.2}}}}}],\"toolbar\":{\"type\":\"object\",\"name\":\"Toolbar\",\"id\":\"p1011\",\"attributes\":{\"tools\":[{\"type\":\"object\",\"name\":\"PanTool\",\"id\":\"p1024\"},{\"type\":\"object\",\"name\":\"WheelZoomTool\",\"id\":\"p1025\",\"attributes\":{\"renderers\":\"auto\"}},{\"type\":\"object\",\"name\":\"BoxZoomTool\",\"id\":\"p1026\",\"attributes\":{\"dimensions\":\"both\",\"overlay\":{\"type\":\"object\",\"name\":\"BoxAnnotation\",\"id\":\"p1027\",\"attributes\":{\"syncable\":false,\"line_color\":\"black\",\"line_alpha\":1.0,\"line_width\":2,\"line_dash\":[4,4],\"fill_color\":\"lightgrey\",\"fill_alpha\":0.5,\"level\":\"overlay\",\"visible\":false,\"left\":{\"type\":\"number\",\"value\":\"nan\"},\"right\":{\"type\":\"number\",\"value\":\"nan\"},\"top\":{\"type\":\"number\",\"value\":\"nan\"},\"bottom\":{\"type\":\"number\",\"value\":\"nan\"},\"left_units\":\"canvas\",\"right_units\":\"canvas\",\"top_units\":\"canvas\",\"bottom_units\":\"canvas\",\"handles\":{\"type\":\"object\",\"name\":\"BoxInteractionHandles\",\"id\":\"p1033\",\"attributes\":{\"all\":{\"type\":\"object\",\"name\":\"AreaVisuals\",\"id\":\"p1032\",\"attributes\":{\"fill_color\":\"white\",\"hover_fill_color\":\"lightgray\"}}}}}}}},{\"type\":\"object\",\"name\":\"SaveTool\",\"id\":\"p1034\"},{\"type\":\"object\",\"name\":\"ResetTool\",\"id\":\"p1035\"},{\"type\":\"object\",\"name\":\"HelpTool\",\"id\":\"p1036\"}]}},\"left\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p1019\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p1020\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p1021\"},\"axis_label\":\"I(x)/I\\u2080\",\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1022\"}}}],\"below\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p1014\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p1015\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p1016\"},\"axis_label\":\"x\",\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1017\"}}}],\"center\":[{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1018\",\"attributes\":{\"axis\":{\"id\":\"p1014\"}}},{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1023\",\"attributes\":{\"dimension\":1,\"axis\":{\"id\":\"p1019\"}}}]}}]}};\n  const render_items = [{\"docid\":\"c52c3e40-218c-4595-b7d9-a1f482c8e9e7\",\"roots\":{\"p1003\":\"ff081dd1-e6fd-48b3-9197-5f350cbc5514\"},\"root_ids\":[\"p1003\"]}];\n  void root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n  }\n  if (root.Bokeh !== undefined) {\n    embed_document(root);\n  } else {\n    let attempts = 0;\n    const timer = setInterval(function(root) {\n      if (root.Bokeh !== undefined) {\n        clearInterval(timer);\n        embed_document(root);\n      } else {\n        attempts++;\n        if (attempts > 100) {\n          clearInterval(timer);\n          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n        }\n      }\n    }, 10, root)\n  }\n})(window);",
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "p1003"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p.scatter(\n",
    "    x=x, \n",
    "    y=norm_I, \n",
    "    size=2, \n",
    "    color=\"orange\"\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is one detail I swept under the rug here. What happens if we compute the function for $x = 0$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/var/folders/8h/qwnxpqcx6vldhxr71n1582d00000gn/T/ipykernel_74974/557033242.py:1: RuntimeWarning: invalid value encountered in scalar divide\n",
      "  4 * (scipy.special.j1(0) / 0) ** 2\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "np.float64(nan)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "4 * (scipy.special.j1(0) / 0) ** 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We get a `RuntimeWarning` because we divided by zero. We know that\n",
    "\n",
    "\\begin{align}\n",
    "\\lim_{x\\to 0} \\frac{J_1(x)}{x} = \\frac{1}{2},\n",
    "\\end{align}\n",
    "\n",
    "so we could write a new function that checks if $x = 0$ and returns the appropriate limit for $x = 0$. In the `x` array I constructed for the plot, we hopped over zero, so it was never evaluated. If we were being careful, we could write our own Airy function that deals with this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def airy_disk(x):\n",
    "    \"\"\"Compute the Airy disk.\"\"\"\n",
    "    # Set up output array\n",
    "    res = np.empty_like(x)\n",
    "\n",
    "    # Where x is very close to zero (use np.isclose)\n",
    "    near_zero = np.isclose(x, 0)\n",
    "\n",
    "    # Compute values where x is close to zero\n",
    "    res[near_zero] = 1.0\n",
    "\n",
    "    # Everywhere else\n",
    "    x_vals = x[~near_zero]\n",
    "    res[~near_zero] = 4 * (scipy.special.j1(x_vals) / x_vals) ** 2\n",
    "\n",
    "    return res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing environment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python implementation: CPython\n",
      "Python version       : 3.13.7\n",
      "IPython version      : 9.5.0\n",
      "\n",
      "numpy     : 2.2.6\n",
      "scipy     : 1.16.1\n",
      "bokeh     : 3.7.3\n",
      "jupyterlab: 4.4.7\n",
      "\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -v -p numpy,scipy,bokeh,jupyterlab"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "default",
   "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": "3.13.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}