Planar_Transformations.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Transformations in a Plane"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As discussed in the previous section, any linear transformation $T:\\mathbb{R}^2\\to\\mathbb{R}^2$ can be represented as the multiplication of a $2\\times 2$ matrix and a coordinate vector.  For now, the coordinates are with respect to the standard basis $\\{E_1,E_2\\}$ as they were before.  The columns of the matrix are then the images of the the basis vectors.\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "A = \\left[ \\begin{array}{c|c} T(E_1) & T(E_2) \\end{array} \\right]\n",
    "\\end{equation}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 1:  Horizontal Stretch\n",
    "\n",
    "As a first example let us consider the transformation defined by the following images.\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "T\\left(\\left[ \\begin{array}{r} 1 \\\\ 0  \\end{array}\\right]\\right)= \\left[ \\begin{array}{r} 2 \\\\ 0  \\end{array}\\right] \\hspace{1in}  T\\left(\\left[ \\begin{array}{r} 0 \\\\ 1  \\end{array}\\right]\\right)= \\left[ \\begin{array}{r} 0 \\\\ 1  \\end{array}\\right]  \n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "The matrix corresponding to the transformation is then\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "A = \\left[ \\begin{array}{cc} 2 & 0 \\\\ 0 & 1 \\end{array}\\right]\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "The image of a general vector with components $c_1$ and $c_2$ is given by a linear combination \n",
    "of the images of the basis vectors.\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "T\\left(\\left[ \\begin{array}{r} c_1 \\\\ c_2  \\end{array}\\right]\\right) = T\\left(c_1\\left[ \\begin{array}{r} 1 \\\\ 0  \\end{array}\\right] + c_2 \\left[ \\begin{array}{r} 0 \\\\ 1  \\end{array}\\right]\\right) = c_1T\\left(\\left[ \\begin{array}{r} 1 \\\\ 0  \\end{array}\\right]\\right) + c_2 T\\left(\\left[ \\begin{array}{r} 0 \\\\ 1  \\end{array}\\right]\\right)  = c_1\\left[ \\begin{array}{r} 2 \\\\ 0  \\end{array}\\right] + c_2\\left[ \\begin{array}{r} 0 \\\\ 1  \\end{array}\\right]  = \\left[ \\begin{array}{cc} 2 & 0 \\\\ 0 & 1 \\end{array}\\right] \\left[\\begin{array}{c} c_1 \\\\ c_2 \\end{array}\\right]\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "To get a visual understanding of the transformation, we produce a plot that displays multiple input vectors, and their corresponding images.  Since we will multiply each vector by the same matrix, we can organize our calculations by constructing a matrix with columns that match the input vectors.  This matrix is the array labeled $\\texttt{coords}$ in the code below.  We use array slicing to label the first row $\\texttt{x}$, and the second row $\\texttt{y}$ for the purposes of plotting. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline  \n",
    "from math import pi, sin, cos\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "coords = np.array([[0,0],[0.5,0.5],[0.5,1.5],[0,1],[0,0]])\n",
    "coords = coords.transpose()\n",
    "coords\n",
    "x = coords[0,:]\n",
    "y = coords[1,:]\n",
    "\n",
    "A = np.array([[2,0],[0,1]])\n",
    "A_coords = A@coords"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The columns of the array $\\texttt{A_coords}$ are the images of the vectors that make up the columns of $\\texttt{coords}$.  Let's plot the coordinates and see the effect of the transformation.  Again, we need to access the first and second rows by slicing.  The red points show the original coordinates, and the new coordinates are shown in blue."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x_LT1 = A_coords[0,:]\n",
    "y_LT1 = A_coords[1,:]\n",
    "\n",
    "# Create the figure and axes objects\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "# Plot the points.  x and y are original vectors, x_LT1 and y_LT1 are images\n",
    "ax.plot(x,y,'ro')\n",
    "ax.plot(x_LT1,y_LT1,'bo')\n",
    "\n",
    "# Connect the points by lines\n",
    "ax.plot(x,y,'r',ls=\"--\")\n",
    "ax.plot(x_LT1,y_LT1,'b')\n",
    "\n",
    "# Edit some settings \n",
    "ax.axvline(x=0,color=\"k\",ls=\":\")\n",
    "ax.axhline(y=0,color=\"k\",ls=\":\")\n",
    "ax.grid(True)\n",
    "ax.axis([-2,2,-1,2])\n",
    "ax.set_aspect('equal')\n",
    "ax.set_title(\"Horizontal Stretch\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We observe that the result of the transformation is that the polygon has been stretched in the horizontal direction.  \n",
    "\n",
    "### Example 2:  Reflection\n",
    "\n",
    "Now that we have a set of vectors stored in the $\\texttt{coords}$ matrix, let's look at another transformation that is defined by the matrix $B$.\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "B = \\left[ \\begin{array}{cc} -1 & 0 \\\\ 0 & 1 \\end{array}\\right]\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "We make a new array $B$, but reuse the plotting code from the previous example."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "B = np.array([[-1,0],[0,1]])\n",
    "B_coords = B@coords\n",
    "\n",
    "x_LT2 = B_coords[0,:]\n",
    "y_LT2 = B_coords[1,:]\n",
    "\n",
    "# Create the figure and axes objects\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "# Plot the points.  x and y are original vectors, x_LT1 and y_LT1 are images\n",
    "ax.plot(x,y,'ro')\n",
    "ax.plot(x_LT2,y_LT2,'bo')\n",
    "\n",
    "# Connect the points by lines\n",
    "ax.plot(x,y,'r',ls=\"--\")\n",
    "ax.plot(x_LT2,y_LT2,'b')\n",
    "\n",
    "# Edit some settings \n",
    "ax.axvline(x=0,color=\"k\",ls=\":\")\n",
    "ax.axhline(y=0,color=\"k\",ls=\":\")\n",
    "ax.grid(True)\n",
    "ax.axis([-2,2,-1,2])\n",
    "ax.set_aspect('equal')\n",
    "ax.set_title(\"Reflection\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that this transformation reflects points around the vertical axis.\n",
    "\n",
    "### Example 3:  Rotation\n",
    "\n",
    "To rotate vectors in the plane, we choose an angle $\\theta$ and write down the matrix that represents the rotation counterclockwise by an angle $\\theta$.  Basic trigonometry can be used to calculate the columns in this case.   \n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "R = \\left[ \\begin{array}{cc} \\cos\\theta & -\\sin\\theta \\\\ \\sin\\theta & \\cos\\theta \\end{array}\\right]\n",
    "\\end{equation}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "theta = pi/6\n",
    "R = np.array([[cos(theta),-sin(theta)],[sin(theta),cos(theta)]])\n",
    "R_coords = R@coords\n",
    "\n",
    "x_LT3 = R_coords[0,:]\n",
    "y_LT3 = R_coords[1,:]\n",
    "\n",
    "# Create the figure and axes objects\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "# Plot the points.  x and y are original vectors, x_LT1 and y_LT1 are images\n",
    "ax.plot(x,y,'ro')\n",
    "ax.plot(x_LT3,y_LT3,'bo')\n",
    "\n",
    "# Connect the points by lines\n",
    "ax.plot(x,y,'r',ls=\"--\")\n",
    "ax.plot(x_LT3,y_LT3,'b')\n",
    "\n",
    "# Edit some settings \n",
    "ax.axvline(x=0,color=\"k\",ls=\":\")\n",
    "ax.axhline(y=0,color=\"k\",ls=\":\")\n",
    "ax.grid(True)\n",
    "ax.axis([-2,2,-1,2])\n",
    "ax.set_aspect('equal')\n",
    "ax.set_title(\"Rotation\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 4: Shear\n",
    "\n",
    "In the study of mechanics, shear forces occur when one force acts on some part of a body while a second force acts on another part of the body but in the opposite direction. To visualize this, imagine a deck of playing cards resting on the table, and then while resting your hand on top of the deck, sliding your hand parallel to the table. \n",
    "\n",
    "A **shear** transformation is any transformation that shifts points in a given direction by an amount proportional to their original (signed) distance from the line that is parallel to the movement of direction and that passes through the origin. For example, a horizontal shear applied to a vector would add to the vector's $x$-coordinate a value scaled by the value of its $y$-coordinate (or subtract, if the $y$-coordinate was negative). A vertical shear is expressed as a matrix in the form of the first matrix below, and a horizontal shear with the second where $k \\in \\mathbb{R}$ is called the **shearing factor**.\n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\left[ \\begin{array}{r} 1 & 0 \\\\ k & 1  \\end{array}\\right] \\hspace{1in}  \n",
    "\\left[ \\begin{array}{r} 1 & k \\\\ 0 & 1  \\end{array}\\right]  \n",
    "\\end{equation}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "S = np.array([[1,2],[0,1]])\n",
    "S_coords = S@coords\n",
    "\n",
    "x_LT4 = S_coords[0,:]\n",
    "y_LT4 = S_coords[1,:]\n",
    "\n",
    "\n",
    "# Create the figure and axes objects\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "# Plot the points.  x and y are original vectors, x_LT1 and y_LT1 are images\n",
    "ax.plot(x,y,'ro')\n",
    "ax.plot(x_LT4,y_LT4,'bo')\n",
    "\n",
    "# Connect the points by lines\n",
    "ax.plot(x,y,'r',ls=\"--\")\n",
    "ax.plot(x_LT4,y_LT4,'b')\n",
    "\n",
    "# Edit some settings \n",
    "ax.axvline(x=0,color=\"k\",ls=\":\")\n",
    "ax.axhline(y=0,color=\"k\",ls=\":\")\n",
    "ax.grid(True)\n",
    "ax.axis([-2,4,-1,2])\n",
    "ax.set_aspect('equal')\n",
    "ax.set_title(\"Shear\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example 5:  Composition of transformations\n",
    "\n",
    "One of the powerful aspects of representing linear transformations as matrices is that we can form compositions by matrix multiplication.  For example,  in order to find the matrix that represents the composition $B\\circ R$, the rotation from Example 3, followed by the reflection of Example 2,\n",
    "we can simply multiply the matrices that represent the individual transformations.  \n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "[B][R] = \\left[ \\begin{array}{cc} -1 & 0 \\\\ 0 & 1 \\end{array}\\right]\n",
    "\\left[ \\begin{array}{cc} \\cos\\theta & -\\sin\\theta \\\\ \\sin\\theta & \\cos\\theta \\end{array}\\right]\n",
    "=\\left[ \\begin{array}{cc} -\\cos\\theta & \\sin\\theta \\\\ \\sin\\theta & \\cos\\theta \\end{array}\\right]\n",
    "\\end{equation}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "C = np.array([[-cos(theta),sin(theta)],[sin(theta),cos(theta)]])\n",
    "C_coords = C@coords\n",
    "\n",
    "x_LT5 = C_coords[0,:]\n",
    "y_LT5 = C_coords[1,:]\n",
    "\n",
    "\n",
    "# Create the figure and axes objects\n",
    "fig, ax = plt.subplots()\n",
    "\n",
    "# Plot the points.  x and y are original vectors, x_LT1 and y_LT1 are images\n",
    "ax.plot(x,y,'ro')\n",
    "ax.plot(x_LT5,y_LT5,'bo')\n",
    "\n",
    "# Connect the points by lines\n",
    "ax.plot(x,y,'r',ls=\"--\")\n",
    "ax.plot(x_LT5,y_LT5,'b')\n",
    "\n",
    "# Edit some settings \n",
    "ax.axvline(x=0,color=\"k\",ls=\":\")\n",
    "ax.axhline(y=0,color=\"k\",ls=\":\")\n",
    "ax.grid(True)\n",
    "ax.axis([-2,2,-1,2])\n",
    "ax.set_aspect('equal')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 1:** \n",
    "\n",
    "($a$) Find a matrix that represents the reflection about the $x_1$-axis. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Code solution here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "($b$) Multiply the matrix by $\\texttt{coords}$ and plot the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Code solution here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 2:** \n",
    "\n",
    "($a$) Find a matrix that represents the reflection about the line $x_1=x_2$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Code solution here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "($b$) Multiply the matrix by $\\texttt{coords}$ and plot the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Code solution here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 3:** \n",
    "\n",
    "($a$) Find a matrix that represents the rotation clockwise by an angle $\\theta$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Code solution here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "($b$) Let $\\theta = 90^{\\circ}$.  Multiply the matrix by $\\texttt{coords}$ and plot the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Code solution here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 4:** \n",
    "\n",
    "($a$) Find a matrix that represents a vertical shear followed by the rotation in Example 3. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Code solution here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "($b$) Multiply the matrix by $\\texttt{coords}$ and plot the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Code solution here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 5:** Create a new matrix of coordinates and apply one of the transformations in the Examples.  Plot the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Code solution here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise 6:** \n",
    "\n",
    "($a$) Construct a matrix that represents a horizontal and a vertical stretch by a factor of 2 . "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Code solution here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "($b$) Create a new matrix of coordinates.  Apply this transformation and plot the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Code solution here"
   ]
  }
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