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diff --git a/examples/worm_cutting_tool/cutting_tool_worm_assembly.ipynb b/examples/worm_cutting_tool/cutting_tool_worm_assembly.ipynb
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+{
+ "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",
+ "![](\"../../docs/computing)
"
+ ]
+ },
+ {
+ "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",
+ "![](\"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",
+ "![](\"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",
+ "![](\"loft_of_generated_bsplines.png\")
\n",
+ "\n",
+ "7. Array für das Zahnrad\n",
+ "\n",
+ "![](\"gear_assembly.png\")
"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3 (ipykernel)",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.11.7"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
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