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"Guiding Future STEM Leaders through Innovative Research Training" ~ thinkingbeyond.education
Project: stephanie's main branch
Path: ThinkingBeyond Activities / BeyondAI-2024-Mentee-Projects / shaana-karuna / Continuity_Video_Final.ipynb~1
Views: 1165Image: ubuntu2204
{
"nbformat": 4,
"nbformat_minor": 0,
"metadata": {
"colab": {
"provenance": [],
"authorship_tag": "ABX9TyMie/9tJVpOnaI2FhwUvsTH",
"include_colab_link": true
},
"kernelspec": {
"name": "python3",
"display_name": "Python 3"
},
"language_info": {
"name": "python"
}
},
"cells": [
{
"cell_type": "markdown",
"metadata": {
"id": "view-in-github",
"colab_type": "text"
},
"source": [
"<a href=\"https://colab.research.google.com/github/ThinkingBeyond/BeyondAI-2024/blob/main/shaana-karuna/Continuity_Video_Final.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
]
},
{
"cell_type": "markdown",
"source": [
"#Summary of code\n",
"\n",
"This code creates a video explaining the epsilon-delta definition of continuity. It is divided into 2 parts."
],
"metadata": {
"id": "JEyIrK1HZxNe"
}
},
{
"cell_type": "markdown",
"source": [
"#Setting up Manim\n"
],
"metadata": {
"id": "Uxfsc247Z94w"
}
},
{
"cell_type": "markdown",
"source": [
"This section installs Manim and IPython."
],
"metadata": {
"id": "x51Xk38NeE6u"
}
},
{
"cell_type": "code",
"source": [
"!sudo apt update\n",
"!sudo apt install libcairo2-dev ffmpeg \\\n",
" texlive texlive-latex-extra texlive-fonts-extra \\\n",
" texlive-latex-recommended texlive-science \\\n",
" tipa libpango1.0-dev\n",
"!pip install manim==0.18.1\n",
"!pip install IPython==8.21.0"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/"
},
"id": "rqUI4fVf5vPD",
"outputId": "7cd772be-17ab-488e-894e-c0589f4e7156",
"collapsed": true
},
"execution_count": 2,
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"\u001b[33m\r0% [Working]\u001b[0m\r \rHit:1 https://cloud.r-project.org/bin/linux/ubuntu jammy-cran40/ InRelease\n",
"\u001b[33m\r0% [Connecting to archive.ubuntu.com (185.125.190.82)] [Connecting to security.\u001b[0m\r \rHit:2 https://developer.download.nvidia.com/compute/cuda/repos/ubuntu2204/x86_64 InRelease\n",
"\u001b[33m\r0% [Connecting to archive.ubuntu.com (185.125.190.82)] [Waiting for headers] [C\u001b[0m\r \rHit:3 http://security.ubuntu.com/ubuntu jammy-security InRelease\n",
"Hit:4 http://archive.ubuntu.com/ubuntu jammy InRelease\n",
"Hit:5 https://r2u.stat.illinois.edu/ubuntu jammy InRelease\n",
"Hit:6 http://archive.ubuntu.com/ubuntu jammy-updates InRelease\n",
"Hit:7 http://archive.ubuntu.com/ubuntu jammy-backports InRelease\n",
"Hit:8 https://ppa.launchpadcontent.net/deadsnakes/ppa/ubuntu jammy InRelease\n",
"Hit:9 https://ppa.launchpadcontent.net/graphics-drivers/ppa/ubuntu jammy InRelease\n",
"0% [Waiting for headers]\u001b[0m^C\n",
"Reading package lists... Done\n",
"Building dependency tree... Done\n",
"Reading state information... Done\n",
"libcairo2-dev is already the newest version (1.16.0-5ubuntu2).\n",
"texlive is already the newest version (2021.20220204-1).\n",
"texlive-fonts-extra is already the newest version (2021.20220204-1).\n",
"texlive-latex-extra is already the newest version (2021.20220204-1).\n",
"texlive-latex-recommended is already the newest version (2021.20220204-1).\n",
"texlive-science is already the newest version (2021.20220204-1).\n",
"tipa is already the newest version (2:1.3-21).\n",
"libpango1.0-dev is already the newest version (1.50.6+ds-2ubuntu1).\n",
"ffmpeg is already the newest version (7:4.4.2-0ubuntu0.22.04.1).\n",
"0 upgraded, 0 newly installed, 0 to remove and 51 not upgraded.\n",
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]
}
]
},
{
"cell_type": "code",
"source": [
"from manim import *"
],
"metadata": {
"id": "QXeDXoIW5z2g"
},
"execution_count": 1,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"#Setting up colours"
],
"metadata": {
"id": "9eQlgBLMaAuy"
}
},
{
"cell_type": "markdown",
"source": [
"This section uses specific hex codes to define colours to be used later."
],
"metadata": {
"id": "Lxg3kDxUhK1b"
}
},
{
"cell_type": "code",
"source": [
"GREENISHWHITE = ManimColor.from_hex(\"#e6ebd5\")\n",
"NEWGREEN = ManimColor.from_hex(\"#93c280\")\n",
"NEWDARKGREEN = ManimColor.from_hex(\"#17282a\")\n",
"NEWMEDIUMGREEN = ManimColor.from_hex(\"#0a5936\")"
],
"metadata": {
"id": "GRjN8R2GDa2o"
},
"execution_count": 5,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"#Part 1\n",
"\n",
"The first part of the video introduces the concept of a continuous function as a 'function you can draw without lifting your pen off the page'. It uses an animation of a straight line bending into different continuous functions to illustrate this idea.\n",
"\n",
"Then, it adds more formality to this concept by introducing the idea that as 2 inputs approach each other, so should the outputs at these 2 points."
],
"metadata": {
"id": "b7D-jBggaFo5"
}
},
{
"cell_type": "code",
"source": [
"class Continuity(Scene):\n",
" def construct(self):\n",
"\n",
"#Set the background colour.\n",
" self.camera.background_color = NEWDARKGREEN\n",
"\n",
"#Define the axes and set the colour of the axes.\n",
" axes = Axes(\n",
" x_range=[-5, 6, 1],\n",
" y_range=[-3, 3, 1],\n",
" axis_config={\"color\": GREENISHWHITE}\n",
" )\n",
"\n",
"#Add numbers to the axes.\n",
" numbered_axes = axes.add_coordinates()\n",
"\n",
"#Define a straight line on the axes.\n",
" line = axes.plot_line_graph(\n",
" x_values=[-5, 5],\n",
" y_values=[1, 1],\n",
" add_vertex_dots=False,\n",
" line_color=NEWGREEN,\n",
" )\n",
"\n",
"#Define a quadratic on the axes, with the same start and end points as the line for smooth transformation.\n",
" quadratic = axes.plot(\n",
" lambda x: (x**2 - 25)/12+1,\n",
" x_range=[-5, 5],\n",
" color=NEWGREEN\n",
" )\n",
"\n",
"#Define a sine curve on the axes, with the same start and end points as the line for smooth transformation.\n",
" sin = axes.plot(\n",
" lambda x: np.sin(5*x/6),\n",
" x_range=[-5, 5],\n",
" color=NEWGREEN\n",
" )\n",
"\n",
"#Define cubic on the axes, with the same start and end points as the line for smooth transformation.\n",
" cubic = axes.plot(\n",
" lambda x: x*(x-5)*(x+5)/35+1,\n",
" x_range=[-5, 5],\n",
" color=NEWGREEN\n",
" )\n",
"\n",
"#Define and position the point y.\n",
" point_coords = (-1.5, 1.975)\n",
" point_location = axes.coords_to_point(*point_coords)\n",
" y = Dot(point_location, color=NEWMEDIUMGREEN)\n",
"\n",
"#Define and position text describing a continuous function.\n",
" text = Tex(\"A continuous function is often described as 'a function that you can draw without lifting your pen off the page'.\", font_size=20)\n",
" text.move_to([0,3.5,0])\n",
"\n",
"##Sets up a variable tracking the position of x (dot) and a function used to update its position.\n",
"#The starting position of dot is at x = -3.\n",
" x_tracker = ValueTracker(-3)\n",
"\n",
"#Defines a function for updating the position of dot. Updates the position depending on the tracking variable x_tracker and the y value at this point.\n",
" def update_dot(dot):\n",
" x1 = x_tracker.get_value()\n",
" y1 = x1*(x1-5)*(x1+5)/35+1\n",
" dot.move_to(axes.coords_to_point(x1, y1))\n",
"\n",
"#Defines dot (x) and assignes it the previously defined update function update_dot.\n",
"\n",
" dot = Dot(color=GREENISHWHITE)\n",
" dot.add_updater(update_dot)\n",
"\n",
"##Sets up a variable tracking the position of x2 (dot2) and a function used to update its position.\n",
"#The starting position of dot2 is at x = -0.9.\n",
" x2_tracker = ValueTracker(-0.9)\n",
"\n",
"#Defines a function for updating the position of dot2. Updates the position depending on the tracking variable x2_tracker and the y value at this point.\n",
" def update_dot2(dot2):\n",
" x2 = x2_tracker.get_value()\n",
" y2 = x2*(x2-5)*(x2+5)/35+1\n",
" dot2.move_to(axes.coords_to_point(x2, y2))\n",
"\n",
"#Defines dot2 (x2) and assignes it the previously defined update function update_dot2.\n",
" dot2 = Dot(color=GREENISHWHITE)\n",
" dot2.add_updater(update_dot2)\n",
"\n",
"#Defines fx, a horizontal line through dot to illustrate the output at x approaching the output at y. This line is always drawn at the y value of dot.\n",
" fx = always_redraw(lambda: DashedLine(\n",
" start=axes.c2p(-5, dot.get_y()),\n",
" end=axes.c2p(5, dot.get_y()),\n",
" color=GREENISHWHITE,\n",
" ))\n",
"\n",
"#Defines fx, a horizontal line through dot to illustrate the output at x2 approaching the output at y. This line is always drawn at the y value of dot2.\n",
" fx2 = always_redraw(lambda: DashedLine(\n",
" start=axes.c2p(-5, dot2.get_y()),\n",
" end=axes.c2p(5, dot2.get_y()),\n",
" color=GREENISHWHITE,\n",
" ))\n",
"\n",
"#Defines fy, a horizontal line through dot to illustrate the output at y.\n",
" fy = DashedLine(start=axes.c2p(-5, 1.975),\n",
" end=axes.c2p(5, 1.975),\n",
" color=GREEN)\n",
"\n",
"## Start the animation\n",
"\n",
"#Writes 'A continuous function is often described as 'a function that you can draw without lifting your pen off the page'.', then waits 1 second.\n",
" self.play(Write(text))\n",
" self.wait(1)\n",
"\n",
"#Adds the axes and line. Then transforms the line to the quadratic, then the sine curve, then the cubic, then fades out the text.\n",
" self.play(Create(numbered_axes))\n",
" self.play(Create(line))\n",
" self.play(Transform(line, quadratic))\n",
" self.play(Transform(line, sin))\n",
" self.play(Transform(line, cubic))\n",
" self.play(FadeOut(text))\n",
"\n",
"#Changes the text, then writes it and waits 1 second.\n",
" text = Tex(\"But how can we define a continuous function more precisely?\", font_size=20)\n",
" text.move_to([0,3.5,0])\n",
" self.play(Write(text))\n",
" self.wait(1)\n",
"\n",
"#Fades out the text, changes it then writes it, adds y to the scene then waits 1 second.\n",
" self.play(FadeOut(text))\n",
" text = Tex(\"Consider a fixed point on the function, $(y, f(y))$.\",font_size=20)\n",
" text.move_to([0,3.5,0])\n",
" self.play(Write(text))\n",
" self.wait(0.5)\n",
" self.play(Create(y))\n",
" self.wait(1)\n",
"\n",
"#Fades out the text, changes it then writes it, adds x to the scene and waits 1 second.\n",
" self.play(FadeOut(text))\n",
" text = Tex(\"Now consider another variable point on the function, $(x,f(x))$.\",font_size=20)\n",
" text.move_to([0,3.5,0])\n",
" self.play(Write(text))\n",
" self.wait(0.5)\n",
" dot.move_to(axes.coords_to_point(-3, 2.37))\n",
" self.add(dot)\n",
" self.wait(1)\n",
"\n",
"#Fades out the text, changes the text and rewrites it.\n",
"#Adds fx and fy to the scene and animates x to approach y (and therefore fx to approach fy) using x_tracker, and waits 2 seconds.\n",
" self.play(FadeOut(text))\n",
" text = Tex(\"The function is continuous at the point $y$, if as x approaches y, f(x) approaches $f(y)$.\",font_size=20)\n",
" text.move_to([0,3.5,0])\n",
" self.play(Write(text))\n",
" self.add(fy,fx)\n",
" self.play(x_tracker.animate.set_value(-1.5), run_time=3, rate_func=linear)\n",
" self.wait(2)\n",
"\n",
"#Fades out the text, dot (x) and fx, replaces the text and rewrites it.\n",
"#Resets the value of x_tracker and adds dot and fx back.\n",
"#Also adds dot2 (x2) and fx2.\n",
"#Animates dot and dot2 to approach y.\n",
"#Waits 2 seconds, then fades out text then removes fx, fx2, dot and dot2.\n",
"\n",
" self.play(FadeOut(text, dot, fx))\n",
" text = Tex(\"This has to work on both sides of $y$.\",font_size=20)\n",
" text.move_to([0,3.5,0])\n",
" self.play(Write(text))\n",
" x_tracker = ValueTracker(-3)\n",
" self.add(dot, dot2, fx, fx2)\n",
" self.play(x_tracker.animate.set_value(-1.5), x2_tracker.animate.set_value(-1.5), run_time=3, rate_func=linear)\n",
" self.wait(2)\n",
" self.play(FadeOut(text))\n",
" self.remove(fx, fx2, dot, dot2)\n",
" self.wait()\n",
"\n"
],
"metadata": {
"id": "rsq3mb8xbwi-"
},
"execution_count": 12,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"Finally, the following command is used to render the scene and save it in the subfolder 'jupyter' in the folder 'media'.\n"
],
"metadata": {
"id": "4sD46Q5Vhyjq"
}
},
{
"cell_type": "code",
"source": [
"%%manim -qm -v WARNING Continuity"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 204
},
"id": "UUND1SyWhnJK",
"outputId": "cf48ae9c-725f-4fc7-8af3-dcd2f332d4ad"
},
"execution_count": 13,
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": [
"Manim Community \u001b[32mv0.\u001b[0m\u001b[32m18.1\u001b[0m\n",
"\n"
],
"text/html": [
"<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">Manim Community <span style=\"color: #008000; text-decoration-color: #008000\">v0.18.1</span>\n",
"\n",
"</pre>\n"
]
},
"metadata": {}
},
{
"output_type": "stream",
"name": "stderr",
"text": []
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"<IPython.core.display.Video object>"
],
"text/html": [
"<video src=\"media/jupyter/Continuity@2024-12-18@16-44-26.mp4\" controls autoplay loop style=\"max-width: 60%;\" >\n",
" Your browser does not support the <code>video</code> element.\n",
" </video>"
]
},
"metadata": {}
}
]
},
{
"cell_type": "markdown",
"source": [
"#Part 2\n",
"\n",
"The second and final part of the animation links what has been previously shown to the formal definition of continuity, and applies this definition to another example."
],
"metadata": {
"id": "hTrqa6GoaSV5"
}
},
{
"cell_type": "code",
"source": [
"class Continuity2(Scene):\n",
" def construct(self):\n",
"\n",
"#Set the background colour.\n",
" self.camera.background_color = NEWDARKGREEN\n",
" axes = Axes(\n",
" x_range=[-5, 6, 1],\n",
" y_range=[-3, 3, 1],\n",
" axis_config={\"color\": GREENISHWHITE}\n",
" )\n",
"\n",
"#Add numbers to the axes.\n",
" numbered_axes = axes.add_coordinates()\n",
"\n",
"#Define a cubic on the axes.\n",
" cubic = axes.plot(\n",
" lambda x: x*(x-5)*(x+5)/35+1,\n",
" x_range=[-5, 5],\n",
" color=NEWGREEN\n",
" )\n",
"\n",
"#Define a sine curve on the axes.\n",
" sin = axes.plot(\n",
" lambda x: np.sin(5*x/6),\n",
" x_range=[-5, 5],\n",
" color=NEWGREEN\n",
" )\n",
"\n",
"#Define and position text.\n",
" text = Tex(\"We have now seen that if we want $|f(x)-f(y)|<\\epsilon$ for at a point $y$, we can find a $\\delta$ such that if $|x-y|<\\delta$ then this is true.\", font_size=20)\n",
" text.move_to([0,3.5,0])\n",
"\n",
"#Defines fy, a horizontal line through dot to illustrate the output at y.\n",
" fy = DashedLine(start=axes.c2p(-5, 1.975),\n",
" end=axes.c2p(5, 1.975),\n",
" color=GREEN)\n",
"\n",
"#Defines fx1, a horizontal line.\n",
" fx1 = DashedLine(start=axes.c2p(-5, 2.25),\n",
" end=axes.c2p(5, 2.25),\n",
" color=GREENISHWHITE)\n",
"\n",
"#Defines fx2, a horizontal line. fx2 and fx1 are equidistant from fy and are on the opposite sides of fy.\n",
" fx2 = DashedLine(start=axes.c2p(-5, 1.75),\n",
" end=axes.c2p(5, 1.75),\n",
" color=GREENISHWHITE)\n",
"\n",
"#Defines y.\n",
" point_coords = (-1.5, 1.975)\n",
" point_location = axes.coords_to_point(*point_coords)\n",
" y = Dot(point_location, color=NEWMEDIUMGREEN)\n",
"\n",
"## Animation starts here\n",
"#Axes, the cubic, y and dy are added to the scene. Creates fx1 and fx2 and then writes the text, then waits 2 seconds.\n",
" self.add(axes, cubic, y, fy)\n",
" self.play(Create(fx1))\n",
" self.play(Create(fx2))\n",
" self.play(Write(text))\n",
" self.wait(2)\n",
"\n",
"#Fades out the text then changes and rewrites it. Waits 0.5 seconds, then fades fx1, fx2, y and fy, waits 0.1 seconds then transforms the cubic into a sin curve.\n",
" self.play(FadeOut(text))\n",
" text = Tex(\"This is the epsilon-delta definition of a continuous function, as used in our proof. Let's look at another function to see that this definition works.\", font_size=20)\n",
" text.move_to([0,3.5,0])\n",
" self.play(Write(text))\n",
" self.wait(0.5)\n",
" self.play(FadeOut(fx1, fx2, y, fy))\n",
" self.wait(0.1)\n",
" self.play(Transform(cubic, sin))\n",
"\n",
"\n",
"#Redefines fx1 and fx2 to change their positions.\n",
" fx1 = DashedLine(start=axes.c2p(-5, 0.75),\n",
" end=axes.c2p(5, 0.75),\n",
" color=GREENISHWHITE)\n",
"\n",
" fx2 = DashedLine(start=axes.c2p(-5, 0.25),\n",
" end=axes.c2p(5, 0.25),\n",
" color=GREENISHWHITE)\n",
"\n",
"#Redefines y and fy to change their positions.\n",
" point_coords = (0.65, np.sin(5*0.65/6))\n",
" point_location = axes.coords_to_point(*point_coords)\n",
" y = Dot(point_location, color=NEWMEDIUMGREEN)\n",
"\n",
" fy = DashedLine(start=axes.c2p(-5, np.sin(5*0.65/6)),\n",
" end=axes.c2p(5, np.sin(5*0.65/6)),\n",
" color=GREEN)\n",
"\n",
"#Defines a brace to show where 'epsilon' is measured and defines a label for the brace.\n",
" brace = BraceBetweenPoints(axes.coords_to_point(5, np.sin(5*0.65/6)), axes.coords_to_point(5, 0.75))\n",
" label = brace.get_text(r\"$\\epsilon$\")\n",
"\n",
"#Waits 0.5 seconds, then creates y, fy, fx1, fx2, and the brace and its label.\n",
" self.wait(0.5)\n",
" self.play(Create(y))\n",
" self.play(Create(fy))\n",
" self.play(Create(fx1))\n",
" self.play(Create(fx2))\n",
" self.play(Create(brace), Write(label))\n",
"\n",
"#Fades out the text then replaces it.\n",
" self.play(FadeOut(text))\n",
" text = Tex(\"If we want $|f(x) - f(y)| < \\epsilon$, where $y$ is the green dot, it is clear to see that we can choose a $\\delta$ such that when $|x-y| < \\delta$, this is true.\", font_size=20)\n",
" text.move_to([0,3.5,0])\n",
"\n",
"#Defines fx3, fx4 and fy2, vertical lines illustrating the position of the x values at all 3 positions focused on in the animation.\n",
" fx3 = DashedLine(start=axes.c2p(1, -3),\n",
" end=axes.c2p(1,3),\n",
" color=GREENISHWHITE)\n",
"\n",
" fx4 = DashedLine(start=axes.c2p(0.25, -3),\n",
" end=axes.c2p(0.25, 3),\n",
" color=GREENISHWHITE)\n",
"\n",
" fy2 = DashedLine(start=axes.c2p(0.65, -3),\n",
" end=axes.c2p(0.65, 3),\n",
" color=GREEN)\n",
"\n",
"#Defines a second brace, brace2 to show where 'delta' is measured and defines a label for the brace.\n",
" brace2 = BraceBetweenPoints(axes.coords_to_point(0.65, -3), axes.coords_to_point(1, -3))\n",
" label2 = Tex(\"$\\delta$\", font_size=30)\n",
" label2.next_to(brace2, direction=DOWN)\n",
"\n",
"#Writes the text and creates fx3, fy2, fx4 and the second brace and its label.\n",
" self.play(Write(text))\n",
" self.play(Create(fx3))\n",
" self.play(Create(fy2))\n",
" self.play(Create(fx4))\n",
" self.play(Create(brace2), Write(label2))\n",
" self.wait()"
],
"metadata": {
"id": "7sf4Pu8wi4kv"
},
"execution_count": 19,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"The following command is used to render the scene and save it in the subfolder 'jupyter' in the folder 'media'."
],
"metadata": {
"id": "f4yiUMsCqZjg"
}
},
{
"cell_type": "code",
"source": [
"%%manim -qm -v WARNING Continuity2"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/",
"height": 204
},
"id": "yVqHCuMrqZ8U",
"outputId": "762bf1dd-01a5-450e-91db-95598b7d14de"
},
"execution_count": 20,
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": [
"Manim Community \u001b[32mv0.\u001b[0m\u001b[32m18.1\u001b[0m\n",
"\n"
],
"text/html": [
"<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">Manim Community <span style=\"color: #008000; text-decoration-color: #008000\">v0.18.1</span>\n",
"\n",
"</pre>\n"
]
},
"metadata": {}
},
{
"output_type": "stream",
"name": "stderr",
"text": []
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"<IPython.core.display.Video object>"
],
"text/html": [
"<video src=\"media/jupyter/Continuity2@2024-12-18@17-10-00.mp4\" controls autoplay loop style=\"max-width: 60%;\" >\n",
" Your browser does not support the <code>video</code> element.\n",
" </video>"
]
},
"metadata": {}
}
]
}
]
}