From 8a4f02853768e1b3cb1ec702dbeb7346a40da161 Mon Sep 17 00:00:00 2001
From: Eric Wieser <wieser.eric@gmail.com>
Date: Fri, 17 Dec 2021 01:29:52 +0000
Subject: [PATCH] Update euler-angles.ipynb

---
 docs/tutorials/euler-angles.ipynb | 43 +++++++++++++++++++++++--------
 1 file changed, 32 insertions(+), 11 deletions(-)

diff --git a/docs/tutorials/euler-angles.ipynb b/docs/tutorials/euler-angles.ipynb
index bb14667f..35748a99 100644
--- a/docs/tutorials/euler-angles.ipynb
+++ b/docs/tutorials/euler-angles.ipynb
@@ -50,8 +50,7 @@
     "2. rotate about the rotated $e_1$-axis, call it  $e_1^{'}$\n",
     "3. rotate about the twice rotated axis of $e_3$-axis, call it  $e_3^{''}$\n",
     "\n",
-    "So the elemental rotations are about  $e_3,  e_{1}^{'},  e_3^{''}$-axes, respectively.",
-    "\n",
+    "So the elemental rotations are about  $e_3,  e_{1}^{'},  e_3^{''}$-axes, respectively.\n",
     "![](../_static/Euler2a.gif)\n",
     "\n",
     "(taken from wikipedia)"
@@ -217,7 +216,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "The components of this frame **are**  the rotation matrix, so we just enter the frame components into a matrix."
+    "The components of this frame **are**  the rotation matrix, so we just enter the frame components as columns into a matrix."
    ]
   },
   {
@@ -226,11 +225,13 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "from numpy import array \n",
-    "\n",
-    "M = [float(b|a) for b in B for a in A] # you need float() due to bug in clifford\n",
-    "M = array(M).reshape(3,3)\n",
+    "import numpy as np\n",
     "\n",
+    "# b|a is a multivector, `[()]` selects the scalar part\n",
+    "M = np.array([\n",
+    "    [(b|a)[()] for a in A]\n",
+    "    for b in B\n",
+    "])  \n",
     "M"
    ]
   },
@@ -238,7 +239,27 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Thats a rotation  matrix."
+    "That's a rotation matrix. We can compare transforming the vector $e_1 + 2e_2$ via the rotor and via the matrix:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "X = e1+2*e2\n",
+    "R*X*~R"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "Xv = np.array([1, 2, 0])\n",
+    "M @ Xv"
    ]
   },
   {
@@ -252,9 +273,9 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "In 3 Dimenions, there is a simple formula which can be used to directly transform a rotations matrix into a rotor.\n",
+    "In 3 Dimensions, there is a simple formula which can be used to directly transform a rotations matrix into a rotor.\n",
     "For arbitrary dimensions you have to use a different algorithm (see `clifford.tools.orthoMat2Versor()` ([docs](../api/generated/clifford.tools.orthoMat2Versor.rst))).\n",
-    "Anyway, in 3 dimensions there is a closed form solution, as described in Sec. 4.3.3 of \"Geometric Algebra for Physicists\".\n",
+    "Anyway, in 3 dimensions there is a closed form solution, as described in Sec. 4.3.3 of <cite data-cite=\"doran-ga4ph\">Geometric Algebra for Physicists</cite>.\n",
     "Given a rotor $R$ which transforms an  orthonormal frame $A={a_k}$  into $B={b_k}$ as such,\n",
     "\n",
     "$$b_k = Ra_k\\tilde{R}$$\n",
@@ -324,7 +345,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.5.4"
+   "version": "3.7.10"
   }
  },
  "nbformat": 4,