From 1dc635947263c6cc34efb1f54222ffaec90fc494 Mon Sep 17 00:00:00 2001
From: Mirko Birbaumer <mirko.birbaumer@hslu.ch>
Date: Sat, 17 Sep 2022 13:33:18 +0000
Subject: [PATCH] new transformation figures

---
 ...book - Introduction Numpy and Pandas.ipynb |  101 +-
 .../combined_transformation.eps               | 1436 ++++++++++++++++-
 .../rotation_transformation.eps               | 1291 ++++++++++++++-
 .../scaling_transformation.eps                |  198 +--
 4 files changed, 2890 insertions(+), 136 deletions(-)

diff --git a/notebooks/Preliminaries_Numpy_Pandas/Jupyter Notebook - Introduction Numpy and Pandas.ipynb b/notebooks/Preliminaries_Numpy_Pandas/Jupyter Notebook - Introduction Numpy and Pandas.ipynb
index 1fc3c5d..727121b 100644
--- a/notebooks/Preliminaries_Numpy_Pandas/Jupyter Notebook - Introduction Numpy and Pandas.ipynb	
+++ b/notebooks/Preliminaries_Numpy_Pandas/Jupyter Notebook - Introduction Numpy and Pandas.ipynb	
@@ -1907,7 +1907,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 284,
+   "execution_count": 295,
    "metadata": {},
    "outputs": [
     {
@@ -1944,7 +1944,8 @@
     "ax.set_xticks(np.arange(-2.5, 3, 0.5))\n",
     "ax.set_yticks(np.arange(-2.5, 3, 0.5))\n",
     "plt.grid()\n",
-    "plt.show()"
+    "plt.show()\n",
+    "fig.savefig('identity_transformation.eps')"
    ]
   },
   {
@@ -1973,26 +1974,26 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 285,
+   "execution_count": 290,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "[<matplotlib.axis.YTick at 0x7f0702b4ce10>,\n",
-       " <matplotlib.axis.YTick at 0x7f0702b4c410>,\n",
-       " <matplotlib.axis.YTick at 0x7f0702b51b50>,\n",
-       " <matplotlib.axis.YTick at 0x7f0702b3da50>,\n",
-       " <matplotlib.axis.YTick at 0x7f0702b3da10>,\n",
-       " <matplotlib.axis.YTick at 0x7f0702ac7650>,\n",
-       " <matplotlib.axis.YTick at 0x7f0702ac7350>,\n",
-       " <matplotlib.axis.YTick at 0x7f0702ace0d0>,\n",
-       " <matplotlib.axis.YTick at 0x7f0702ace550>,\n",
-       " <matplotlib.axis.YTick at 0x7f0702aceb10>,\n",
-       " <matplotlib.axis.YTick at 0x7f0702b3d590>]"
+       "[<matplotlib.axis.YTick at 0x7f0708290e50>,\n",
+       " <matplotlib.axis.YTick at 0x7f070397eed0>,\n",
+       " <matplotlib.axis.YTick at 0x7f07087db3d0>,\n",
+       " <matplotlib.axis.YTick at 0x7f07082f4590>,\n",
+       " <matplotlib.axis.YTick at 0x7f0703969bd0>,\n",
+       " <matplotlib.axis.YTick at 0x7f0703b22c10>,\n",
+       " <matplotlib.axis.YTick at 0x7f0703b22ad0>,\n",
+       " <matplotlib.axis.YTick at 0x7f07038c62d0>,\n",
+       " <matplotlib.axis.YTick at 0x7f07038c6e90>,\n",
+       " <matplotlib.axis.YTick at 0x7f0703794450>,\n",
+       " <matplotlib.axis.YTick at 0x7f0703794690>]"
       ]
      },
-     "execution_count": 285,
+     "execution_count": 290,
      "metadata": {},
      "output_type": "execute_result"
     },
@@ -2038,9 +2039,36 @@
     "ax.set_yticks(np.arange(-2.5, 3, 0.5))"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "From the plot above it should be very clear that the $x$ and $y$ dimensions were simply scaled up by a factor of two while the third dimension responsible for the ASCII letter index was left unchanged. In fact, those familiar with matrix algebra will have noticed that for all of the affine transformations the value represented in the third dimension is always left un-altered as indicated by the all zeros and one lone value in the third dimension index of the last column.\n",
+    "\n",
+    "Now let us describe how to interpret the _rotation transformation_. We will start by solving the two trigonometric functions for the desired angle of rotation of $90$ degrees, then we simply plug them into the rotation transformation matrix listed in the previous table.\n",
+    "\n",
+    "$$\n",
+    "\\begin{align}\n",
+    "\\sin(90) & =1 \\\\\n",
+    "\\cos(90) & = 1\\\\\n",
+    "\\end{align}\n",
+    "$$\n",
+    "\n",
+    "\n",
+    "$$T_r=\n",
+    "\\left(\\begin{array}{rrr} \n",
+    "0 & 1 & 0 \\\\ \n",
+    "-1 & 0 & 0 \\\\ \n",
+    "0 & 0 & 1 \\\\ \n",
+    "\\end{array}\\right)\n",
+    "$$\n",
+    "\n",
+    "Now all we need to do is apply the same logic to transform and plot the points, like so:\n"
+   ]
+  },
   {
    "cell_type": "code",
-   "execution_count": 277,
+   "execution_count": 292,
    "metadata": {},
    "outputs": [
     {
@@ -2078,9 +2106,40 @@
     "plt.show()"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Hopefully you can tell from the plot that all points were rotated 90 degrees around an axis of rotation at the origin.\n",
+    "\n",
+    "The neat thing about affine transformations being essentially linear transformations is that you can _combine_ the transformations and apply them in one step. To demonstrate this we will apply the dot product (matrix multiplication) of our two transformation matrices, like:\n",
+    "$$\n",
+    "\\begin{align*}\n",
+    "T_{\\text{comb}}=\n",
+    "\\left(\\begin{array}{rrr} \n",
+    "2 & 0 & 0 \\\\ \n",
+    "0 & 2 & 0 \\\\ \n",
+    "0 & 0 & 1 \\\\ \n",
+    "\\end{array}\\right)\n",
+    "\\star\n",
+    "\\left(\\begin{array}{rrr} \n",
+    "0 & 1 & 0 \\\\ \n",
+    "-1 & 0 & 0 \\\\ \n",
+    "0 & 0 & 1 \\\\ \n",
+    "\\end{array}\\right) = \\left(\\begin{array}{rrr} \n",
+    "0 & 2 & 0 \\\\ \n",
+    "-2 & 0 & 0 \\\\ \n",
+    "0 & 0 & 1 \\\\ \n",
+    "\\end{array}\\right)\n",
+    "\\end{align*}\n",
+    "$$\n",
+    "\n",
+    "Now we can apply this combined transformation matrix to the points and replot them to show a combination of scaling by two and rotation by 90 degrees."
+   ]
+  },
   {
    "cell_type": "code",
-   "execution_count": 278,
+   "execution_count": 294,
    "metadata": {},
    "outputs": [
     {
@@ -2106,7 +2165,7 @@
     "xs_comb = []\n",
     "ys_comb = []\n",
     "for row in A:\n",
-    "    output_row = T @ row\n",
+    "    output_row = np.matmul(T, row)\n",
     "    x, y, i = row\n",
     "    x_comb, y_comb, i_comb = output_row\n",
     "    xs_comb.append(x_comb)\n",
@@ -2125,14 +2184,14 @@
     "ax.set_xticks(np.arange(-2.5, 3, 0.5))\n",
     "ax.set_yticks(np.arange(-2.5, 3, 0.5))\n",
     "plt.grid()\n",
-    "plt.show()\n"
+    "plt.show()"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## A real use case"
+    "## A Real Use Case"
    ]
   },
   {
@@ -2173,7 +2232,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## Some more useful Numpy methods"
+    "## Some More Useful Numpy Methods"
    ]
   },
   {
diff --git a/notebooks/Preliminaries_Numpy_Pandas/combined_transformation.eps b/notebooks/Preliminaries_Numpy_Pandas/combined_transformation.eps
index 174289e..ba24191 100644
--- a/notebooks/Preliminaries_Numpy_Pandas/combined_transformation.eps
+++ b/notebooks/Preliminaries_Numpy_Pandas/combined_transformation.eps
@@ -1,12 +1,12 @@
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+%%CreationDate: Sat Sep 17 13:32:56 2022
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diff --git a/notebooks/Preliminaries_Numpy_Pandas/rotation_transformation.eps b/notebooks/Preliminaries_Numpy_Pandas/rotation_transformation.eps
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