Jupyters/Quaternion/Quaternion_Cube.ipynb

Quaternion_Cube.ipynb

{
  "cells": [
    {
      "metadata": {},
      "cell_type": "markdown",
      "source": "# Quaternion: Rotation of Cube"
    },
    {
      "metadata": {
        "ExecuteTime": {
          "start_time": "2019-12-27T05:16:29.904730Z",
          "end_time": "2019-12-27T05:16:30.587317Z"
        },
        "trusted": true
      },
      "cell_type": "code",
      "source": "import numpy as np\nimport matplotlib.pyplot as plt\nfrom mpl_toolkits.mplot3d import Axes3D\n%matplotlib inline",
      "execution_count": 1,
      "outputs": []
    },
    {
      "metadata": {
        "ExecuteTime": {
          "start_time": "2019-12-27T05:16:30.789196Z",
          "end_time": "2019-12-27T05:16:30.929659Z"
        },
        "trusted": true
      },
      "cell_type": "code",
      "source": "# quaternion multiplication function\ndef QbyQ(q1, q2): # q = [w, x, y, z]\n    w = q1[0]*q2[0] - q1[1]*q2[1] - q1[2]*q2[2] - q1[3]*q2[3]\n    i = q1[0]*q2[1] + q1[1]*q2[0] + q1[2]*q2[3] - q1[3]*q2[2]\n    j = q1[0]*q2[2] - q1[1]*q2[3] + q1[2]*q2[0] + q1[3]*q2[1]\n    k = q1[0]*q2[3] + q1[1]*q2[2] - q1[2]*q2[1] + q1[3]*q2[0]\n    return np.array([w, i, j, k])\n\n# quaternion rotation function\ndef QPQc(P, UV, t): # P: quaternion, UV: unit vector, t: rotation angle theta\n    q =  [np.cos(t/2),  UV[0] * np.sin(t/2),  UV[1] * np.sin(t/2),  UV[2] * np.sin(t/2)]\n    qc = [np.cos(t/2), -UV[0] * np.sin(t/2), -UV[1] * np.sin(t/2), -UV[2] * np.sin(t/2)]\n    qP = QbyQ(q, P)\n    result = QbyQ(qP, qc)\n    return result\n\n# plot unit sphere\ndef plot_base(elev=25, azim=-70):\n    fig = plt.figure(figsize=(10, 10))\n    ax = fig.add_subplot(111, projection='3d')\n    ax.view_init(elev=elev, azim=azim)\n    ax.set(xlim=(-1, 1), ylim=(-1 ,1), zlim=(-1, 1))\n    ax.set(xlabel='X', ylabel='Y', zlabel='Z')\n    ax.xaxis.pane.fill = ax.yaxis.pane.fill = ax.zaxis.pane.fill = False\n\n    t = np.linspace(0, 2*np.pi, 128+1)\n    alpha = 0.7\n    ax.plot(np.cos(t), np.sin(t), [0]*len(t), linestyle=':', c='red', alpha=alpha)\n    ax.plot(np.cos(t), [0]*len(t), np.sin(t), linestyle=':', c='red', alpha=alpha)\n    ax.plot([0]*len(t), np.cos(t), np.sin(t), linestyle=':', c='red', alpha=alpha)\n    ax.plot([-1, 1], [0, 0], [0, 0], linestyle=':', c='red', alpha=alpha)\n    ax.plot([0, 0], [-1, 1], [0, 0], linestyle=':', c='red', alpha=alpha)\n    ax.plot([0, 0], [0, 0], [-1, 1], linestyle=':', c='red', alpha=alpha)\n    return ax",
      "execution_count": 2,
      "outputs": []
    },
    {
      "metadata": {
        "ExecuteTime": {
          "end_time": "2019-12-24T02:45:54.704700Z",
          "start_time": "2019-12-24T02:45:54.699747Z"
        }
      },
      "cell_type": "markdown",
      "source": "## Rotating Cube with ipywidgets"
    },
    {
      "metadata": {
        "ExecuteTime": {
          "start_time": "2019-12-27T05:16:44.200242Z",
          "end_time": "2019-12-27T05:16:44.726548Z"
        },
        "trusted": true
      },
      "cell_type": "code",
      "source": "from ipywidgets import interact\nfrom mpl_toolkits.mplot3d.art3d import Poly3DCollection\n\n@interact(theta=(-np.pi, np.pi, np.pi/100), w=(0.1, 1, 0.1), l=(0.1, 1, 0.1), h=(0.1, 1, 0.1)) # theta: rotation angle on unit vector\ndef cube(theta=0, w=0.5, l=0.5, h=0.5):\n    scale = np.array([w, l, h])  # scale of the cube\n    pos = np.array([-w/2, -l/2, 1])  # position of the cube\n    UV = [1, 0, 0]                     # unit vector: rotation axis\n\n    CUBE = np.array([[0,0,0],[1,0,0],[1,1,0],[0,1,0], \n                     [0,0,1],[1,0,1],[1,1,1],[0,1,1]]) * scale + pos            # cube vertices with scale and position\n    CUBEs = np.vstack([np.linalg.norm(c) for c in CUBE])                        # scalar\n    CUBEn = np.array([v / s if s != 0 else v * 0 for v, s in zip(CUBE, CUBEs)]) # normalization\n    QCUBE = np.insert(CUBEn, 0, 0, axis=1)                                      # quaternion\n\n    UV = np.array(UV) / np.linalg.norm(UV)                    # normalization of unit vector\n\n    QCUBE_R = np.array([QPQc(qc, UV, theta) for qc in QCUBE]) # quaternion rotation\n    CX, CY, CZ = (QCUBE_R[:, 1:] * CUBEs).T                   # vectors of cube\n\n    ax = plot_base()\n    ax.plot([0, UV[0]], [0, UV[1]], [0, UV[2]], c='r')\n    ax.scatter(UV[0], UV[1], UV[2], c='r')\n    ax.plot(np.r_[CX[:4], CX[0]], np.r_[CY[:4], CY[0]], np.r_[CZ[:4], CZ[0]], c='b')         # 4 bottom edges\n    ax.plot(np.r_[CX[4:], CX[4]], np.r_[CY[4:], CY[4]], np.r_[CZ[4:], CZ[4]], c='g')         # 4 top edges\n    [ax.plot([CX[i], CX[i+4]], [CY[i], CY[i+4]], [CZ[i], CZ[i+4]], c='m') for i in range(4)] # 4 colomns edges\n    \n    bottom = list(zip(CX[:4], CY[:4], CZ[:4]))\n    ax.add_collection3d(Poly3DCollection([bottom], color='#ddddff'))  # fill in color : bottom side\n    top = list(zip(CX[4:], CY[4:], CZ[4:]))\n    ax.add_collection3d(Poly3DCollection([top], color='#ddffdd'))     # fill in color : top side",
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": "interactive(children=(FloatSlider(value=0.0, description='theta', max=3.141592653589793, min=-3.14159265358979…",
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "805616efe6394cd2b648d4d961e8d128"
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          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "ExecuteTime": {
          "end_time": "2019-12-26T10:16:10.686158Z",
          "start_time": "2019-12-26T10:16:10.175231Z"
        },
        "trusted": true
      },
      "cell_type": "code",
      "source": "from ipywidgets import interact\nfrom mpl_toolkits.mplot3d.art3d import Poly3DCollection\n\n@interact(theta=(-np.pi, np.pi, np.pi/100), ux=(-1, 1, 0.1), uy=(-1, 1, 0.1), uz=(-1, 1, 0.1)) # theta: rotation angle on unit vector\ndef cube(theta=0, ux=1, uy=0, uz=0):\n    scale = np.array([1/2, 1/2, 1/2])  # scale of the cube\n    pos = np.array([-1/4, -1/4, 1])  # position of the cube\n    UV = [ux, uy, uz]                     # unit vector: rotation axis\n\n    CUBE = np.array([[0,0,0],[1,0,0],[1,1,0],[0,1,0], \n                     [0,0,1],[1,0,1],[1,1,1],[0,1,1]]) * scale + pos            # cube vertices with scale and position\n    CUBEs = np.vstack([np.linalg.norm(c) for c in CUBE])                        # scalar\n    CUBEn = np.array([v / s if s != 0 else v * 0 for v, s in zip(CUBE, CUBEs)]) # normalization\n    QCUBE = np.insert(CUBEn, 0, 0, axis=1)                                      # quaternion\n\n    UV = np.array(UV) / np.linalg.norm(UV)                    # normalization of unit vector\n\n    QCUBE_R = np.array([QPQc(qc, UV, theta) for qc in QCUBE]) # quaternion rotation\n    CX, CY, CZ = (QCUBE_R[:, 1:] * CUBEs).T                   # vectors of cube\n\n    ax = plot_base()\n    ax.plot([0, UV[0]], [0, UV[1]], [0, UV[2]], c='r')\n    ax.scatter(UV[0], UV[1], UV[2], c='r')\n    ax.plot(np.r_[CX[:4], CX[0]], np.r_[CY[:4], CY[0]], np.r_[CZ[:4], CZ[0]], c='b')         # 4 bottom edges\n    ax.plot(np.r_[CX[4:], CX[4]], np.r_[CY[4:], CY[4]], np.r_[CZ[4:], CZ[4]], c='g')         # 4 top edges\n    [ax.plot([CX[i], CX[i+4]], [CY[i], CY[i+4]], [CZ[i], CZ[i+4]], c='m') for i in range(4)] # 4 colomns edges\n    ax.text(ux, uy, uz, s='  unit vector')\n    \n    bottom = list(zip(CX[:4], CY[:4], CZ[:4]))\n    ax.add_collection3d(Poly3DCollection([bottom], color='#ddddff'))  # fill in color : bottom side\n    top = list(zip(CX[4:], CY[4:], CZ[4:]))\n    ax.add_collection3d(Poly3DCollection([top], color='#ddffdd'))     # fill in color : top side",
      "execution_count": 3,
      "outputs": [
        {
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            "text/plain": "interactive(children=(FloatSlider(value=0.0, description='theta', max=3.141592653589793, min=-3.14159265358979…"
          },
          "metadata": {},
          "output_type": "display_data"
        }
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requirements.txt

numpy
matplotlib
ipywidgets