add example for worm cutting

examples/worm_cutting_tool/cutting_tool_worm_assembly.ipynb

@@ -0,0 +1,218 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "5e24589a-461d-46b5-8141-37f948dcf4dc",
+   "metadata": {},
+   "source": [
+    "# cutting tool for a worm gear\n",
+    "\n",
+    "1. idea 1:  \n",
+    "\n",
+    "<img src=\"../../docs/computing a profile_from_a_given_rack.jpg\">"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 35,
+   "id": "7eacf041-aa83-49e2-9cbe-066f177197f6",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import sympy as sp\n",
+    "import numpy as np"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 119,
+   "id": "980417d0-c79d-4501-a7cc-9725b3bbea83",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def symbolic_transformation(angle, axis, translation):\n",
+    "    \"\"\"\n",
+    "        see http://en.wikipedia.org/wiki/SO%284%29#The_Euler.E2.80.93Rodrigues_formula_for_3D_rotations\n",
+    "        sympy enabled transformation\n",
+    "        angle: angle of rotation\n",
+    "        axis: the axis of the rotation\n",
+    "        translation: translation of transformation\n",
+    "    \"\"\"\n",
+    "    assert len(axis) == 3\n",
+    "    a = sp.cos(angle / 2)\n",
+    "    axis_normalized = axis / sp.sqrt(axis.dot(axis))\n",
+    "    (b, c, d) = -axis_normalized * sp.sin(angle / 2)\n",
+    "    mat = sp.Matrix(\n",
+    "        [\n",
+    "            [\n",
+    "                a**2 + b**2 - c**2 - d**2,\n",
+    "                2 * (b * c - a * d),\n",
+    "                2 * (b * d + a * c),\n",
+    "                translation[0],\n",
+    "            ],\n",
+    "            [\n",
+    "                2 * (b * c + a * d),\n",
+    "                a**2 + c**2 - b**2 - d**2,\n",
+    "                2 * (c * d - a * b),\n",
+    "                translation[1],\n",
+    "            ],\n",
+    "            [\n",
+    "                2 * (b * d - a * c),\n",
+    "                2 * (c * d + a * b),\n",
+    "                a**2 + d**2 - b**2 - c**2,\n",
+    "                translation[2],\n",
+    "            ],\n",
+    "            [0.0, 0.0, 0.0, 1.0],\n",
+    "        ]\n",
+    "    )\n",
+    "    return sp.nsimplify(mat,tolerance=1e-10,rational=True)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 219,
+   "id": "5852aa56-e66b-4f3c-a50d-0cb4cb21abd2",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "T1 = symbolic_transformation(np.pi / 2.,\n",
+    "                             np.array([1., 0., 0.]),\n",
+    "                             np.array([12.5,0., 1.15]))\n",
+    "T2 = symbolic_transformation(-t / 7.5,\n",
+    "                             np.array([0., 0., 1.]),\n",
+    "                             np.array([0., 0., 0.]))\n",
+    "T3 = symbolic_transformation(0.,\n",
+    "                             np.array([1., 0., 0.]),\n",
+    "                             np.array([0., 0., t]))\n",
+    "\n",
+    "T = T2.inverse() @ T1.inverse() @ T3"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 220,
+   "id": "7c9837b8-caf7-4447-bf0d-eba9085197a5",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([[ 0.        ,  0.        , -0.13333333,  0.15333333],\n",
+       "       [-0.13333333,  0.        ,  0.        ,  0.66666667],\n",
+       "       [ 0.        ,  0.        ,  0.        ,  0.        ],\n",
+       "       [ 0.        ,  0.        ,  0.        ,  0.        ]])"
+      ]
+     },
+     "execution_count": 220,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "T_fn = sp.lambdify(t, T)\n",
+    "dT_fn = sp.lambdify(t, T.diff(t))\n",
+    "dT_fn(0.)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "e4354a36-409a-40a6-8f4f-bb5a6c84f3b7",
+   "metadata": {},
+   "source": [
+    "Diese Methode funktioniert nicht, weil die Bedingung zur Bestimmung des Kontaktpunktes falsch ist.\n",
+    "Eine Bedingung für die generierung einer konstanten Übersetzung ist, dass die Normale auf die Kontaktfläche (Zahnstange) immer durch den Punkt p (Eingriffspunkt für Ersatzzahnrad (Zylinder) und Ersatzzahnstange (Quader) gehen muss.\n",
+    "Gesucht sind also Punkte auf der Fläche S welche verbunden mit P normal auf die Fläche stehen. Dies kann auch als minimaler Abstand von P zur Fläche gesehen werden.\n",
+    "\n",
+    "für jedes u: min(norm(S(u,v)-P)) -> d(norm(S(u,v)-P)/dv = 0\n",
+    "die Änderung des Abstands ist 0 -> die Gerade steht normal auf die Fläche\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "43f00b65-c05c-4734-8c5a-e7845a6f4dff",
+   "metadata": {},
+   "source": [
+    "## Vorgehensweise\n",
+    "\n",
+    "1. approximate the surface by a BSplineSurface\n",
+    "\n",
+    "- Erstellen eines \"Cross-Sektion\" objekts aus dem \"Werkzeug\"\n",
+    "\n",
+    "<img src=\"cross_section.png\">  \n",
+    "\n",
+    "- Draft downgrade um Kanten zu bekommen\n",
+    "das Cross-section Objekt beinhaltet nicht die anfangs und end Kanten. Diese müssen zusätzlich vom Werkzeug extrahiert werden\n",
+    "\n",
+    "- Part Loft zum erstellen einer schönen BSplinefläche\n",
+    "\n",
+    "<img src=\"loft_bspline_flaechen.png\"> \n",
+    "\n",
+    "3. Loft -> Surface\n",
+    "\n",
+    "```python\n",
+    "# select the cutting faces\n",
+    "face_1 = App.ActiveDocument.Loft.Shape.Faces[0].copy()\n",
+    "face_2 = App.ActiveDocument.Loft001.Shape.Faces[0].copy()\n",
+    "\n",
+    "# compute the contact curve:\n",
+    "bsp_1 = face_1.Surface\n",
+    "bsp_2 = face_2.Surface\n",
+    "\n",
+    "```\n",
+    "\n",
+    "4. Minimierung des Abstands zum \"Pitch-Punkt\"\n",
+    "\n",
+    "```python\n",
+    "import scipy as scp\n",
+    "point = App.Vector(5., 0., 1.15 - time) \n",
+    "xyz_1 = []\n",
+    "for v in np.linspace(0, 1, 5):\n",
+    "        def dist_1(u):\n",
+    "            distance = bsp_1.value(u, v) - point\n",
+    "            return distance.x ** 2 + distance.z ** 2\n",
+    "        u_1 = scp.optimize.minimize(dist_1, 0.5, tol=1e-6).x[0]\n",
+    "        xyz_1.append(bsp_1.value(u_1, v))\n",
+    "```\n",
+    "\n",
+    "5. erstellen einer B-Spline Kurve welche durch die Kinematik T transformiert wird\n",
+    "\n",
+    "```python\n",
+    "c_1 = Part.BSplineCurve()\n",
+    "c_1.interpolate(Points=xyz_1)\n",
+    "c_1 = c_1.toShape()\n",
+    "\n",
+    "Part.show(c_1.transformShape(T))\n",
+    "```\n",
+    "\n",
+    "6. Loft anwenden auf die erstellten BSpline Kurven\n",
+    "\n",
+    "<img src=\"loft_of_generated_bsplines.png\">\n",
+    "\n",
+    "7. Array für das Zahnrad\n",
+    "\n",
+    "<img src=\"gear_assembly.png\">"
+   ]
+  }
+ ],
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