{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Co-ordinate system and Angles\n", "\n", "In module 3, four distinct axes sets will be utilised; body, wind, stability, and earth axes. For this module, aircraft body axes is the only one required. Hopefully aircraft body axes have been covered in previous courses but a bit of revision never hurts.\n", "(Aircraft-Body-Axes)=\n", "## Aircraft Body Axes\n", "\n", "Aircraft body axes is a right-handed Cartesian axis set, centred at the aircraft centre of gravity. $x$ is defined positive along the aircraft longitudinal axis, positive forward. $y$ is positive over the starboard wing. $z$ is positive down, in accordance with the right hand rule.\n", "\n", "Along each of the $x, y, z$ axes the forces, moments, and velocities can be summarised:\n", "\n", "| Direction | Force | Linear Velocity | Description | Moment | Moment Coefficient | Angular Displacement | Angular Velocity | Nondimensional angular rate | Description |\n", "|-----------|-------|-----------------|-------------|--------|--------------------|----------------------|------------------|-----------------------------|-------------|\n", "| $x$ | $X$ | $U$ | Fore/aft | $L$ | $C_\\ell$ | $\\phi$ | $P$ | $\\bar{p}$ | Roll |\n", "| $y$ | $Y$ | $V$ | Sideward | $M$ | $C_m$ | $\\theta$ | $Q$ | $\\bar{q}$ | Pitch |\n", "| $z$ | $Z$ | $W$ | Heave | $N$ | $C_n$ | $\\psi$ | $R$ | $\\bar{r}$ | Yaw |\n", "\n", "The direction of positive rotations is defined in accordance with the right-hand screw rule - see the interactive figure below, which enables you to rotate an aircraft model, and click on the legend to show/hide different components.\n", "\n", "```{admonition} A note on lift vs rolling moment\n", ":class: dropdown\n", "You'll hear me lament about this occasionally, but it was a *terrible* choice to decide to start the moments alphabetically at \"L\", because $L$ is ambiguous - it's sometimes dimensional lift, and it's sometimes the rolling moment.\n", "\n", "It's almost always clear from context which is required, but I know this is often confusing to students who can just think \"I see an L, it must be lift\". The lesson is to *think* about the terms in front of you and what they're being used for.\n", "\n", "The nomenclature for the coefficients is inconsistent from text to text, also. Most texts use a lowercase $l$ in the rolling moment coefficient, but not all. Some texts use an 'extra curly' l, which is what I've used in this section. This is achieved in LaTeX via \\ell.\n", "\n", "So: $C_\\ell$ - RM coefficient, $C_L$ - Lift coefficient, $C_l$ - 2D lift coefficient\n", "\n", "But just be careful when comparing texts. I've seen $C_{LM}$ for 'rolling moment', and a fair few texts keep the pitching and yawing moment coefficients as uppercase subscripts (though many use the nomenclature I've gone with). For consistency, I've made all the moment coefficients lowercase, and the rolling moment is the 'extra curly L'.\n", "\n", "Just be careful, and make sure you're certain about which part you need.\n", "```\n", "\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "tags": [ "remove-input" ] }, "outputs": [ { "data": { "text/html": [ " \n", " " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "application/vnd.plotly.v1+json": { "config": { "plotlyServerURL": "https://plot.ly" }, "data": [ { "i": [ 3, 3, 5, 5, 7, 7, 9, 9, 11, 11, 13, 13, 15, 15, 17, 17, 19, 19, 21, 21, 23, 23, 25, 25, 27, 27, 29, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 39, 41, 41, 43, 43, 45, 45, 47, 47, 49, 49, 51, 51, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 63, 63, 65, 65, 67, 67, 69, 69, 71, 71, 73, 73, 75, 75, 77, 77, 79, 79, 2, 2, 83, 83, 82, 82, 85, 85, 87, 87, 89, 89, 91, 91, 93, 93, 95, 95, 97, 97, 99, 99, 101, 101, 103, 103, 105, 105, 107, 107, 109, 109, 111, 111, 113, 113, 115, 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2884, 2885, 2885, 1378, 1378, 2183, 2183, 2189, 2189, 2146, 2146, 1680, 1680, 2894, 2894, 2892, 2892, 1958, 1958, 2893, 2893, 2859, 2859, 2882, 2882, 2896, 2896, 2489, 2489, 2633, 2633, 2898, 2898, 2525, 2525, 2899, 2899, 2689, 2689, 1576, 1576, 2141, 2141, 1635, 1635, 2651, 2651, 2904, 2904, 2540, 2540, 2898, 2898, 2749, 2749, 2906, 2906, 2610, 2610, 2907, 2907, 2878, 2878, 1963, 1963, 2909, 2909, 1964, 1964, 1667, 1667, 1574, 1574, 2109, 2109, 763, 763, 1617, 1617, 1616, 1616, 2159, 2159, 2161, 2161, 2757, 2757, 2907, 2907, 2615, 2615, 821, 821, 831, 831, 823, 823, 834, 834, 1965, 1965, 1576, 1576, 1552, 1552, 1663, 1663, 2086, 2086, 835, 835, 826, 826, 825, 825, 836, 836, 2738, 2738, 832, 832, 2602, 2602, 2906, 2906, 2892, 2892, 2405, 2405, 844, 844, 1950, 1950, 1574, 1574, 1805, 1805, 1566, 1566, 1324, 1324, 1968, 1968, 1966, 1966, 2879, 2879, 1967, 1967, 2531, 2531, 2563, 2563, 2552, 2552, 828, 828, 757, 757, 2914, 2914, 1714, 1714, 1144, 1144, 2912, 2912, 830, 830, 1968, 1968, 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1434, 1433, 1433, 1361, 1361, 706, 706, 701, 701, 2102, 2102, 1887, 1887, 1973, 1973, 2391, 2391, 2555, 2555, 2357, 2357, 1877, 1877, 1850, 1850, 2091, 2091, 2090, 2090, 2273, 2273, 2272, 2272, 1878, 1878, 2272, 2272, 1840, 1840, 2247, 2247, 2243, 2243, 2248, 2248, 2787, 2787, 720, 727, 2788, 2788, 2247, 1708, 1708, 1248, 1248, 1248, 1041, 1041, 2266, 2266, 1039, 1039, 2256, 2256, 1027, 1027, 2253, 2253, 1025, 1025, 1276, 1276, 1024, 1024, 1523, 1523, 1021, 1021, 1535, 1535, 1020, 1020, 1747, 1747, 1016, 1016, 1424, 1424, 1009, 1009, 727, 727, 1278, 1278, 1392, 1392, 2800, 2800, 1421, 1421, 1040, 1040, 1409, 1409, 1026, 1026, 2769, 2769, 2782, 2782, 2781, 2781, 1974, 1974, 1405, 1405, 1390, 1390, 2790, 2790, 723, 2766, 2766, 2767, 2767, 1939, 1939, 1975, 1975, 1008, 1008, 1399, 1399, 1023, 1023, 1392, 1392, 1007, 1007, 1385, 1385, 2242, 2242, 2241, 2241, 2239, 2239, 2238, 2238, 2237, 2237, 2236, 2236, 1859, 1859, 2760, 2760, 1859, 1859, 1355, 1355, 2231, 2231, 2229, 2229, 2228, 2228, 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2425, 2425, 2416, 2416, 2873, 2873, 2418, 2418, 2417, 2417, 2403, 2403, 2409, 2409, 2575, 2575, 2408, 2408, 2396, 2396, 2408, 2408, 2578, 2578, 2401, 2401, 2393, 2393, 2870, 2870, 2416, 2416, 2394, 2394, 2385, 2385, 2387, 2387, 2616, 2616, 2386, 2386, 2381, 2381, 2386, 2386, 2383, 2383, 2382, 2382, 2377, 2377, 2382, 2382, 2379, 2379, 2378, 2378, 2371, 2371, 2378, 2378, 2375, 2375, 2374, 2374, 2368, 2368, 2374, 2374, 2599, 2599, 2369, 2369, 2366, 2366, 2369, 2369, 2621, 2621, 2433, 2433, 2431, 2431, 2364, 2364, 2363, 2363, 2362, 2362, 2424, 2424, 2360, 2360, 2352, 2352, 2364, 2364, 2349, 2349, 2864, 2864, 2393, 2393, 2350, 2350, 2403, 2403, 2347, 2347, 2343, 2343, 2342, 2342, 2396, 2396, 2340, 2340, 2339, 2339, 2347, 2347, 726, 726, 726, 726, 728, 728, 728, 728, 2385, 2385, 2332, 2332, 2331, 2331, 2330, 2330, 2381, 2381, 2328, 2328, 2327, 2327, 2332, 2332, 2377, 2377, 2325, 2325, 2324, 2324, 2328, 2328, 2371, 2371, 2322, 2322, 2321, 2321, 2325, 2325, 2368, 2368, 2319, 2319, 2312, 2312, 2322, 2322, 2366, 2366, 2362, 2362, 2310, 2310, 2319, 2319, 2307, 2307, 2308, 2308, 2769, 2769, 1998, 1998, 834, 834, 2305, 2305, 2307, 2307, 1999, 1999, 2781, 2781, 2298, 2298, 2912, 2912, 1970, 1970, 1613, 1613, 1614, 1614, 2295, 1603, 1603, 2857, 2857, 2293, 2293, 2292, 2292, 2291, 2291, 2337, 2337, 2289, 2289, 2258, 2258, 2861, 2861, 2790, 2790, 2287, 2287, 2234, 2234, 2289, 2289, 1788, 1788, 1676, 2754, 2754, 1688, 1688, 1362, 1362, 1363, 1363, 2176, 2176, 2175, 2175, 2835, 2835, 2291, 2291, 2279, 2279, 2278, 2278, 2826, 2826, 2278, 2278, 2276, 2276, 2275, 2275, 1859, 1859, 2162, 2162, 2163, 2163, 2164, 2164, 2820, 2820, 2275, 2275, 2270, 2270, 2269, 2269, 2264, 2264, 2266, 2266, 2265, 2265, 2363, 2363, 2255, 2255, 2256, 2256, 2264, 2264, 2352, 2352, 2251, 2251, 2253, 2253, 2252, 2252, 2343, 2343, 630, 630, 629, 629, 2669, 2669, 2687, 2687, 2003, 2003, 641, 641, 630, 630, 642, 642, 2004, 2004, 642, 642, 2686, 2686, 644, 644, 2006, 2006, 2005, 2005, 649, 649, 2471, 2471, 2007, 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" ] }, "metadata": {}, "output_type": "display_data" }, { "data": {}, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "AircraftBodyAxes3d" } }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import meshio\n", "import plotly.graph_objects as go\n", "from myst_nb import glue\n", "with open(\"AircraftModels/hawk_t1.obj\", \"rb\") as file:\n", " obj_data = file.read().decode('utf-8')\n", "\n", "def obj_data_to_mesh3d(odata):\n", " # odata is the string read from an obj file\n", " vertices = []\n", " faces = []\n", " lines = odata.splitlines() \n", " \n", " for line in lines:\n", " slist = line.split()\n", " if slist:\n", " if slist[0] == 'v':\n", " vertex = np.array(slist[1:], dtype=float)\n", " vertices.append(vertex)\n", " elif slist[0] == 'f':\n", " face = []\n", " for k in range(1, len(slist)):\n", " face.append([int(s) for s in slist[k].replace('//','/').split('/')])\n", " if len(face) > 3: # triangulate the n-polyonal face, n>3\n", " faces.extend([[face[0][0]-1, face[k][0]-1, face[k+1][0]-1] for k in range(1, len(face)-1)])\n", " else: \n", " faces.append([face[j][0]-1 for j in range(len(face))])\n", " else: pass\n", " \n", " \n", " return np.array(vertices), np.array(faces) \n", "\n", "vertices, faces = obj_data_to_mesh3d(obj_data)\n", "\n", "x, y, z = vertices[:,:3].T\n", "I, J, K = faces.T\n", "\n", "mesh = go.Mesh3d(\n", " x=z+1.5,\n", " y=x,\n", " z=-y+1.64,\n", " i=I,\n", " j=J,\n", " k=K,\n", " name='',\n", " showscale=False)\n", "\n", "mesh.update(lighting=dict(ambient= 0.5,\n", " diffuse= 1,\n", " fresnel= 5,\n", " specular= 1,\n", " roughness= 1),\n", " \n", " lightposition=dict(x=-1000,\n", " y=1000,\n", " z=1000));\n", "tiprange = 8\n", "\n", "layout = go.Layout(title='Aircraft Body Axes', title_x=0.5,\n", " font=dict(size=14, color='black'),\n", " width=900,\n", " height=900,\n", " scene=dict(xaxis=dict(visible=False, title='X', range=[-tiprange*1.2, tiprange*1.2]),\n", " yaxis=dict(visible=False, title='Y', range=[tiprange*1.2, -tiprange*1.2]), \n", " zaxis=dict(visible=False, title='Z', range=[tiprange*1.2, -tiprange*1.2]), \n", " aspectratio=dict(x=1,\n", " y=1,\n", " z=1\n", " ),\n", " camera=dict(eye=dict(x=1., y=1., z=0.5)),\n", " ), \n", " margin=dict(t=175),\n", " paper_bgcolor='rgba(0,0,0,0)',\n", " plot_bgcolor='rgba(0,0,0,0)') \n", "\n", "\n", "\n", "\n", "fig = go.Figure(data=[mesh], layout=layout)\n", "\n", "\n", "\n", "## Add some axes\n", "fig.add_trace(go.Scatter3d(x=[0, tiprange], y=[0, 0], z=[0, 0],\n", " mode='lines',\n", " showlegend=False, line=dict(color='royalblue', width=4)))\n", "\n", "fig.add_trace(go.Cone(x=[tiprange], y=[0], z=[0], u=[1], v=[0], w=[0], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(65,105,225)'], [1, 'rgb(65,105,225)']],\\\n", " showscale=False))\n", "\n", "fig.add_trace(go.Scatter3d(x=[0, 0], y=[0, tiprange], z=[0, 0],\n", " mode='lines',\n", " showlegend=False, line=dict(color='royalblue', width=4)))\n", "\n", "fig.add_trace(go.Cone(x=[0], y=[tiprange], z=[0], u=[0], v=[1], w=[0], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(65,105,225)'], [1, 'rgb(65,105,225)']],\\\n", " showscale=False))\n", "\n", "fig.add_trace(go.Scatter3d(x=[0, 0], y=[0, 0], z=[0, tiprange],\n", " mode='lines',\n", " showlegend=False, line=dict(color='royalblue', width=4)))\n", "\n", "fig.add_trace(go.Cone(x=[0], y=[0], z=[tiprange], u=[0], v=[0], w=[1], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(65,105,225)'], [1, 'rgb(65,105,225)']],\\\n", " showscale=False))\n", "\n", "fig.add_trace(go.Scatter3d(x=[tiprange, 0, 0], y=[0, tiprange, .2], z=[0, 0, tiprange], mode='text', text=['x', 'y', 'z'], name='Directions', showlegend=True))\n", "\n", "# Draw coloured arrows to show rotation - first around X\n", "dist = 7\n", "rad = 1\n", "dTH = np.radians(np.linspace(45, 325, 100))\n", "xX = np.ones(dTH.shape) * dist\n", "yX = rad * np.sin(dTH)\n", "zX = -rad * np.cos(dTH)\n", "\n", "fig.add_trace(go.Scatter3d(x=xX, y=yX, z=zX,\n", " mode='lines',\n", " showlegend=False, line=dict(color='royalblue', width=4)))\n", "\n", "fig.add_trace(go.Cone(x=[xX[-1]], y=[yX[-1]], z=[zX[-1]], u=[0], v=[1], w=[-1], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(65,105,225)'], [1, 'rgb(65,105,225)']],\\\n", " showscale=False))\n", "\n", "# Then around y\n", "yY = np.ones(dTH.shape) * dist\n", "xY = -rad * np.sin(dTH)\n", "zY = -rad * np.cos(dTH)\n", "\n", "fig.add_trace(go.Scatter3d(x=xY, y=yY, z=zY,\n", " mode='lines',\n", " showlegend=False, line=dict(color='royalblue', width=4)))\n", "\n", "fig.add_trace(go.Cone(x=[xY[-1]], y=[yY[-1]], z=[zY[-1]], u=[-1], v=[0], w=[-1], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(65,105,225)'], [1, 'rgb(65,105,225)']],\\\n", " showscale=False))\n", "\n", "# Then around z\n", "zZ = np.ones(dTH.shape) * dist\n", "xZ = -rad * np.sin(dTH)\n", "yZ = rad * np.cos(dTH)\n", "\n", "fig.add_trace(go.Scatter3d(x=xZ, y=yZ, z=zZ,\n", " mode='lines',\n", " showlegend=False, line=dict(color='royalblue', width=4)))\n", "\n", "fig.add_trace(go.Cone(x=[xZ[-1]], y=[yZ[-1]], z=[zZ[-1]], u=[-1], v=[1], w=[0], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(65,105,225)'], [1, 'rgb(65,105,225)']],\\\n", " showscale=False))\n", "\n", "fig.add_trace(go.Scatter3d(x=[tiprange+1, 0, 0], y=[0, tiprange+1, 0], z=[0, 0, tiprange+1], mode='text', text=['X', 'Y', 'Z'],\\\n", " textposition='middle center', name='Forces', showlegend=True, textfont=dict(\n", " color=\"red\",\n", " size=12)))\n", "\n", "fig.add_trace(go.Scatter3d(x=[11.5, 0, 0], y=[0, 11.5, 0], z=[0, 0, 11.5], mode='text', text=['U', 'V', 'W'],\\\n", " textposition='middle center', name='Translational Velocity', showlegend=True, textfont=dict(\n", " color=\"darkviolet\",\n", " size=12)))\n", "\n", "fig.add_trace(go.Scatter3d(x=[dist, xY[50], xZ[50]], y=[yX[50], dist, yZ[50]], z=[zX[50], zY[50], dist*.95], mode='text', text=['L', 'M', 'N'],\\\n", " textposition='bottom center', name='Moments', showlegend=True, textfont=dict(\n", " color=\"green\",\n", " size=12)))\n", "\n", "fig.add_trace(go.Scatter3d(x=[dist, xY[75], xZ[75]], y=[yX[75], dist, yZ[75]], z=[zX[75], zY[75], dist*.95], mode='text', text=['P', 'Q', 'R'],\\\n", " textposition='bottom center', name='Angular Velocity', showlegend=True, textfont=dict(\n", " color=\"blue\",\n", " size=12)))\n", "\n", "\n", "fig.add_trace(go.Scatter3d(x=[dist, xY[25], xZ[25]], y=[yX[25], dist, yZ[25]], z=[zX[25], zY[25], dist*.95], mode='text', text=['ϕ', 'θ', 'Ψ'],\\\n", " textposition='bottom center', name='Angular Displacement', showlegend=True, textfont=dict(\n", " color=\"darkslategray\",\n", " size=12)))\n", "\n", "fig.add_trace(go.Scatter3d(x=[dist, xY[0], xZ[0]], y=[yX[0], dist, yZ[0]], z=[zX[0], zY[0], dist*.95], mode='text', text=['Roll', 'Pitch', 'Yaw'],\\\n", " textposition='bottom left', name='Angular Motions', showlegend=True, textfont=dict(\n", " color=\"maroon\",\n", " size=12)))\n", "\n", "fig.layout.scene.camera.projection.type = \"orthographic\"\n", "\n", "\n", "# fig.show()\n", "\n", "glue(\"AircraftBodyAxes3d\", fig, display=False)\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(aerodynamic-angles)=\n", "## Aerodynamic Angles\n", "\n", "\n", "The aerodynamic angles, angle of attack - 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You might not be so comfortable however, with their actual definitions." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "tags": [ "remove-input" ] }, "outputs": [ { "data": { "application/vnd.plotly.v1+json": { "config": { "plotlyServerURL": "https://plot.ly" }, "data": [ { "i": [ 702, 702, 1284, 1284, 702, 702, 1537, 1537, 700, 700, 1285, 1285, 734, 734, 1133, 1133, 4063, 4063, 4064, 4064, 1735, 1735, 736, 736, 1272, 1272, 700, 700, 1565, 1565, 696, 696, 1262, 1262, 1328, 821, 821, 760, 760, 1502, 1502, 1500, 1500, 1501, 1501, 1114, 1114, 1507, 1507, 1504, 1504, 1505, 1505, 1506, 1506, 1510, 1510, 1506, 1506, 1509, 1509, 1500, 1500, 1115, 1115, 1512, 1512, 1513, 1513, 1514, 1514, 1517, 1517, 1116, 1116, 1516, 1516, 1504, 1504, 1521, 1521, 1519, 1519, 1520, 1520, 1116, 1116, 1524, 1524, 1117, 1117, 1523, 1523, 1519, 1519, 1529, 1529, 1526, 1526, 1527, 1527, 1528, 1528, 1530, 1530, 1532, 1532, 1532, 1532, 0, 0, 1538, 1538, 1535, 1535, 1536, 1536, 1537, 1537, 1513, 1513, 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1552, 868, 868, 1238, 1238, 876, 876, 934, 934, 870, 870, 1250, 1250, 866, 866, 933, 933, 864, 864, 1239, 1239, 868, 868, 930, 930, 866, 866, 1242, 1242, 862, 862, 929, 929, 924, 924, 1240, 1240, 864, 864, 927, 927, 862, 862, 1243, 1243, 1597, 1597, 926, 926, 1577, 1577, 32, 32, 783, 783, 780, 780, 1484, 1484, 856, 856, 1168, 1168, 1396, 1396, 677, 677, 689, 689, 642, 642, 2620, 2620, 613, 613, 620, 620, 602, 602, 848, 848, 850, 850, 849, 849, 1428, 1428, 845, 845, 846, 846, 848, 848, 1425, 1425, 842, 842, 1431, 1431, 849, 849, 843, 843, 845, 845, 1422, 1422, 1419, 1419, 840, 840, 836, 836, 838, 838, 837, 837, 839, 839, 833, 833, 834, 834, 1975, 1975, 835, 835, 837, 837, 831, 831, 833, 833, 832, 832, 747, 747, 1870, 1870, 829, 829, 1254, 836, 836, 828, 828, 1254, 1254, 827, 827, 1115, 1115, 772, 772, 1252, 1252, 1732, 1732, 778, 778, 1828, 1828, 104, 104, 105, 1782, 1779, 1779, 95, 1783, 1783, 1777, 1777, 1778, 1778, 1781, 1780, 1780, 819, 819, 1820, 1820, 816, 816, 1678, 1678, 816, 816, 1818, 1818, 814, 814, 1681, 1681, 814, 814, 1814, 1814, 812, 812, 1684, 1684, 1257, 1257, 810, 810, 812, 812, 1808, 1808, 1258, 1258, 1845, 1845, 808, 808, 1234, 1234, 891, 891, 908, 941, 941, 939, 939, 1237, 1237, 877, 877, 871, 871, 937, 937, 2021, 2021, 2020, 2020, 1201, 1201, 577, 577, 2023, 2023, 2022, 2022, 2027, 2027, 2028, 2028, 2029, 2029, 2030, 2030, 2019, 2019, 2024, 2024, 1970, 1970, 1108, 1108, 1107, 1107, 1106, 1106, 2026, 2026, 2031, 2031, 1100, 1100, 1099, 1099, 1098, 1098, 1097, 1097, 1983, 1983, 1105, 1105, 1102, 1102, 1103, 1103, 1190, 1190, 1344, 1344, 1253, 1253, 1182, 1182, 1096, 1096, 1190, 1190, 1344, 1344, 1253, 1253, 1096, 1096, 577, 577, 2011, 2011, 1097, 1097, 1106, 1106, 2030, 2030, 2023, 2023, 2018, 2018, 1234, 1234, 1201, 1201, 1236, 1236, 364, 364, 1235, 1235, 365, 365, 1246, 1246, 2018, 2018, 2017, 2017, 2013, 256, 256, 1175, 1175, 1971, 1971, 492, 492, 484, 484, 1988, 1988, 1986, 1986, 1042, 1042, 1041, 1041, 1039, 1039, 1038, 1038, 1036, 1036, 1035, 1035, 1033, 1033, 1032, 1032, 1030, 1030, 1029, 1029, 1027, 1027, 1026, 1026, 1024, 1024, 1023, 1023, 1021, 1021, 1020, 1020, 1018, 1018, 1017, 1017, 1015, 1015, 1014, 1014, 1013, 1013, 1012, 1012, 1010, 1010, 2068, 2068, 2064, 2064, 2063, 2063, 2062, 2062, 2061, 2065, 2065, 2066, 2066, 2067, 2067, 2036, 2069, 2069, 2070, 2070, 2071, 2071, 795, 795, 890, 890, 2061, 2061, 2062, 2062, 2063, 2063, 2064, 2064, 2065, 2065, 2066, 2066, 2067, 2067, 2068, 2068, 2069, 2069, 2070, 2070, 2071, 2071, 795, 795, 2050, 372, 43, 2072, 2072, 1606, 1606, 3275, 3275, 2709, 2709, 3517, 3517, 2705, 2705, 3276, 3276, 2707, 2707, 3126, 3126, 3196, 3196, 614, 614, 2243, 2243, 2740, 2740, 3265, 3265, 3707, 3707, 3544, 3544, 2701, 2701, 3278, 3278, 2773, 2773, 2801, 2763, 2763, 3676, 3676, 3481, 3481, 3482, 3482, 3107, 3107, 3483, 3483, 3485, 3485, 3486, 3486, 3487, 3487, 3488, 3488, 3487, 3487, 3490, 3490, 3481, 3481, 3491, 3491, 3493, 3493, 3494, 3494, 1514, 1514, 3108, 3108, 3109, 3109, 3496, 3496, 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3135, 3643, 3643, 3650, 3542, 3542, 2937, 3662, 3662, 3648, 3648, 3649, 3649, 3138, 3138, 3650, 3650, 3652, 3652, 3653, 3653, 3654, 3654, 3139, 3139, 3654, 3654, 3656, 3656, 3657, 3657, 3140, 3140, 3657, 3657, 3659, 3659, 3660, 3660, 3661, 3661, 603, 603, 612, 612, 3901, 2263, 2263, 3665, 3665, 3666, 3666, 3142, 3142, 3649, 3649, 3668, 3668, 1695, 1695, 3143, 3143, 3666, 3666, 3670, 3670, 3671, 3671, 3672, 3672, 3144, 3144, 3671, 3671, 3674, 3674, 2774, 2774, 2772, 2772, 3255, 3255, 2770, 2770, 2771, 2771, 2773, 2773, 3256, 3256, 2768, 2768, 2770, 2770, 3257, 3257, 3686, 3686, 3143, 3143, 1714, 1714, 3687, 3687, 3689, 3689, 3142, 3142, 3686, 3686, 3690, 3690, 3692, 3692, 3693, 3693, 3145, 3145, 3694, 3694, 3696, 3696, 3146, 3146, 3559, 2083, 2083, 3700, 3700, 3701, 3701, 3702, 3702, 3147, 3147, 2649, 2649, 3705, 3705, 3706, 3706, 3706, 3708, 3708, 3710, 3710, 1739, 1739, 341, 341, 3713, 3713, 3714, 3714, 3709, 3709, 3715, 3715, 3717, 3717, 3718, 3718, 3713, 3713, 3719, 3719, 3721, 3721, 3591, 3591, 3717, 3717, 3722, 3722, 3724, 3724, 3725, 3725, 3726, 3726, 3727, 3727, 3729, 3729, 3730, 3730, 3731, 3731, 3732, 3732, 3734, 3734, 3735, 3735, 3736, 3736, 3725, 3725, 3738, 3738, 3739, 3739, 3740, 3740, 3735, 3735, 3742, 3742, 3743, 3743, 3744, 3744, 3739, 3739, 3746, 3746, 3732, 3732, 3747, 3747, 3743, 3743, 2781, 2781, 2780, 2780, 778, 778, 2778, 2778, 2892, 2892, 2872, 2872, 2779, 2779, 2776, 2776, 3756, 3756, 3757, 3757, 3758, 3758, 3759, 3759, 3761, 3761, 3759, 3759, 3762, 3762, 3730, 3730, 3764, 3764, 3727, 3727, 3765, 3765, 3766, 3766, 3768, 3768, 3766, 3766, 3769, 3769, 3770, 3770, 3771, 3771, 3772, 3772, 3773, 3773, 3774, 3774, 3775, 3775, 1805, 1805, 3776, 3776, 3772, 3772, 3778, 3778, 3779, 3779, 3780, 3780, 3152, 3152, 3782, 3782, 3783, 3783, 3779, 3779, 3784, 3784, 3786, 3786, 3787, 3787, 3782, 3782, 3788, 3788, 3790, 3790, 3791, 3791, 3792, 3792, 3786, 3786, 3794, 3794, 3795, 3795, 3796, 3796, 3791, 3791, 3797, 3797, 3774, 3774, 3799, 3799, 3795, 3795, 3801, 3801, 3802, 3802, 1833, 1833, 3803, 3803, 3805, 3805, 3806, 3806, 3801, 3801, 3807, 3807, 3809, 3809, 3810, 3810, 3805, 3805, 3811, 3811, 3153, 3153, 3813, 3813, 3809, 3809, 3814, 3814, 3773, 3773, 3816, 3816, 3817, 3817, 3818, 3818, 3799, 3799, 3818, 3818, 3820, 3820, 3821, 3821, 3823, 3823, 3824, 3824, 3825, 3825, 3780, 3780, 3827, 3827, 3828, 3828, 3823, 3823, 3783, 3783, 3830, 3830, 3831, 3831, 3827, 3827, 3787, 3787, 3833, 3833, 3834, 3834, 3830, 3830, 3792, 3792, 3821, 3821, 3836, 3836, 3833, 3833, 3796, 3796, 3776, 3776, 1869, 1869, 3838, 3838, 3816, 3816, 3840, 3840, 3841, 3841, 1874, 1874, 3802, 3802, 3843, 3843, 3844, 3844, 3840, 3840, 3806, 3806, 3846, 3846, 3847, 3847, 3843, 3843, 3810, 3810, 3825, 3825, 3849, 3849, 3846, 3846, 3813, 3813, 3851, 3851, 3852, 3852, 3853, 3853, 3854, 3854, 3853, 3853, 3856, 3856, 3857, 3857, 3858, 3858, 3857, 3857, 3860, 3860, 3861, 3861, 3862, 3862, 3861, 3861, 3864, 3864, 3865, 3865, 3866, 3866, 3865, 3865, 3868, 3868, 3869, 3869, 3870, 3870, 3869, 3869, 3872, 3872, 3873, 3873, 3874, 3874, 3876, 3876, 3877, 3877, 3878, 3878, 3879, 3879, 3878, 3878, 3881, 3881, 3851, 3851, 3882, 3882, 3884, 3884, 3885, 3885, 3852, 3852, 3886, 3886, 3888, 3888, 3889, 3889, 3856, 3856, 3885, 3885, 3633, 3633, 3891, 3891, 3860, 3860, 3889, 3889, 3893, 3893, 3894, 3894, 3864, 3864, 3891, 3891, 3769, 3769, 3896, 3896, 3868, 3868, 3894, 3894, 3765, 3765, 3898, 3898, 3872, 3872, 3896, 3896, 3762, 3762, 3900, 3900, 3877, 3877, 3901, 3901, 3758, 3758, 3886, 3886, 3881, 3881, 3900, 3900, 2467, 2467, 4013, 4013, 4013, 3936, 3936, 3905, 3905, 162, 162, 2470, 2470, 3951, 3951, 2469, 2469, 4012, 4012, 4011, 4011, 3936, 3936, 1987, 1987, 4007, 4007, 3997, 3997, 2468, 2468, 3205, 3205, 4024, 4024, 4023, 4023, 3351, 3351, 3220, 4022, 4022, 604, 604, 3201, 4015, 605, 605, 2612, 2612, 3999, 3999, 3998, 3998, 622, 622, 1967, 1967, 3997, 3997, 2565, 2565, 1974, 1974, 1951, 1951, 606, 606, 2713, 2713, 3274, 3274, 1143, 1143, 596, 596, 3257, 4021, 4021, 3701, 3701, 596, 596, 2709, 2709, 597, 597, 3221, 3221, 598, 598, 2754, 2754, 599, 599, 3197, 3197, 600, 600, 2752, 2752, 601, 601, 3264, 3264, 602, 602, 2754, 2754, 4020, 4020, 2766, 2766, 4019, 4019, 3661, 3661, 4018, 4018, 3263, 3263, 4017, 4017, 2705, 2705, 3537, 3537, 3270, 3270, 3279, 3279, 603, 603, 1981, 1981, 2566, 2566, 2512, 2512, 2743, 2743, 2510, 2510, 3134, 3134, 2506, 2506, 2712, 2712, 2473, 2473, 3271, 3271, 2370, 2370, 2713, 2713, 2365, 2365, 2714, 2714, 2303, 2303, 2716, 2716, 2302, 2302, 2717, 2717, 2301, 2301, 2756, 2756, 604, 604, 2772, 2772, 2300, 2300, 2701, 2701, 3280, 3280, 2725, 2725, 2721, 2721, 2698, 2698, 605, 605, 570, 570, 606, 606, 2695, 2695, 2694, 2694, 3268, 3268, 3273, 3273, 1286, 1286, 1959, 1959, 2720, 2720, 3605, 3605, 2074, 3513, 3513, 2534, 2718, 2718, 2690, 2690, 2728, 2728, 2714, 2714, 3258, 3258, 2545, 2545, 1275, 1275, 3736, 3736, 2717, 2717, 2687, 2687, 2716, 2716, 2712, 2712, 3283, 3283, 2698, 2698, 2690, 2690, 2687, 2687, 3284, 3284, 1275, 1275, 1951, 1951, 3383, 3383, 2692, 2692, 3707, 3707, 3703, 3703, 3198, 3198, 3115, 3115, 3511, 3511, 2415, 2415, 3907, 2299, 2299, 2889, 2889, 2264, 2264, 673, 673, 3935, 3935, 2297, 2297, 2313, 2313, 3953, 3953, 3262, 3262, 2969, 2969, 3261, 3261, 2265, 2265, 2513, 2513, 2486, 3293, 3293, 3948, 3948, 2728, 2728, 594, 594, 3132, 3132, 1170, 1170, 3608, 3608, 2232, 2232, 2731, 2731, 2275, 2275, 3562, 3562, 2580, 2580, 3267, 3267, 2581, 2581, 3593, 3593, 2608, 2608, 3613, 3613, 2509, 2509, 3565, 3565, 2588, 2588, 2731, 2731, 2610, 2610, 3611, 3611, 2965, 2965, 2736, 2736, 2266, 2266, 3714, 3714, 199, 199, 2734, 2734, 188, 188, 3614, 3614, 4080, 4080, 3590, 3590, 4081, 4081, 3718, 3718, 4082, 4082, 2736, 2736, 4083, 4083, 197, 197, 4084, 4084, 187, 187, 3941, 3941, 2115, 202, 2720, 2720, 2507, 2507, 3460, 3460, 3946, 3946, 3955, 3955, 3182, 3182, 2677, 2677, 3616, 3181, 3181, 2675, 2675, 3824, 3824, 2677, 2677, 3180, 3180, 2673, 2673, 3849, 3849, 2675, 2675, 3179, 3179, 2671, 2671, 3847, 3847, 2673, 2673, 3177, 3177, 2669, 2669, 3844, 3844, 2671, 2671, 3176, 3176, 661, 661, 3841, 3841, 2669, 2669, 3834, 3834, 2323, 2323, 3538, 3511, 2399, 2366, 2721, 2721, 2262, 2262, 3596, 2498, 2498, 2721, 2721, 2260, 2260, 3267, 3267, 3967, 3967, 2725, 2725, 2259, 2259, 3119, 3119, 213, 213, 3270, 3270, 3963, 3963, 3269, 3269, 2258, 2258, 3223, 3223, 2665, 2665, 3224, 3224, 3695, 3695, 2661, 2661, 3259, 3259, 2659, 2659, 2658, 2658, 2659, 2659, 2656, 2661, 2661, 2662, 2662, 3698, 3698, 2658, 2658, 2656, 2656, 2798, 2798, 2738, 2738, 2611, 2611, 3128, 3128, 2267, 3603, 3603, 2734, 2734, 2232, 2232, 2967, 2967, 3206, 3206, 2664, 2664, 3117, 3117, 3114, 3114, 563, 607, 607, 548, 548, 609, 2786, 2786, 2652, 2652, 3113, 3113, 2652, 2652, 3710, 3710, 728, 728, 3461, 3154, 3268, 3268, 3398, 3398, 2261, 2261, 3962, 3962, 2740, 2740, 622, 622, 180, 180, 2974, 3600, 3600, 2718, 2718, 186, 186, 1191, 1191, 3097, 3097, 2741, 2741, 3230, 3230, 3094, 3094, 548, 548, 3194, 3194, 2236, 2236, 2225, 2225, 2744, 2744, 3127, 3127, 2270, 2270, 592, 592, 2554, 2554, 563, 563, 2765, 2765, 2620, 2620, 2147, 2147, 4085, 4085, 3660, 3660, 2809, 2809, 3140, 3140, 2547, 2547, 2619, 2619, 2618, 2618, 2613, 2613, 621, 621, 2274, 2274, 2274, 2621, 2621, 2270, 2270, 2205, 569, 569, 1112, 1112, 2240, 2240, 3105, 4153, 4153, 4122, 4122, 4086, 4086, 4087, 4087, 4088, 4088, 4089, 4089, 2241, 2241, 3259, 3259, 3607, 3607, 3721, 3721, 2759, 2759, 3130, 3130, 2759, 3169, 3169, 2291, 2291, 3709, 3709, 2554, 3482, 3482, 2315, 2315, 2295, 3170, 3170, 2317, 2317, 3873, 3873, 2842, 2842, 3169, 3169, 3876, 3876, 2844, 2844, 3164, 3164, 2845, 2845, 202, 202, 4058, 4058, 4128, 4128, 4129, 4129, 2723, 2723, 3261, 3378, 3378, 3670, 3670, 4130, 4130, 4131, 4131, 4059, 4059, 4132, 4132, 1174, 1174, 4060, 4060, 4133, 4133, 2845, 2845, 2844, 2844, 3464, 3464, 3465, 3465, 3106, 3106, 4134, 4134, 4135, 4135, 4061, 4061, 4136, 4136, 2278, 2278, 2277, 2277, 3945, 3945, 3962, 3962, 2261, 2261, 2258, 2258, 3963, 3963, 213, 213, 2259, 2259, 3967, 3967, 2260, 2260, 2498, 2498, 2262, 2262, 185, 185, 4062, 4062, 4137, 4137, 4138, 4138, 4127, 4127, 3989, 3989, 4139, 4139, 2322, 2322, 2322, 2322, 4140, 4140, 607, 607, 2285, 2285, 2971, 2267, 2267, 3171, 3171, 2869, 2869, 3524, 3524, 3493, 3493, 3172, 3172, 3173, 3173, 2877, 2877, 2869, 2869, 3173, 3173, 2872, 2872, 2873, 2873, 2874, 2874, 2875, 2875, 2874, 2874, 2883, 2883, 3110, 3110, 2884, 2884, 2319, 2319, 2892, 2892, 2892, 2892, 3879, 3879, 3882, 3882, 2873, 2873, 2873, 2873, 3882, 3882, 3879, 3879, 2087, 2087, 2883, 2883, 2084, 2084, 2891, 2891, 2086, 2086, 2087, 2087, 3858, 3858, 2230, 2230, 3500, 3500, 3500, 3500, 2314, 2314, 3866, 3866, 3496, 3496, 3496, 3496, 3866, 3866, 2079, 2079, 3617, 3617, 3486, 3486, 2082, 2082, 3874, 3874, 3464, 3464, 3486, 3486, 2292, 2292, 2842, 2842, 3898, 3898, 3726, 3726, 3171, 3171, 3518, 2931, 2931, 3332, 3332, 2304, 2304, 3836, 3836, 3644, 3644, 3540, 3540, 2659, 190, 190, 3176, 3176, 2237, 2237, 2242, 2242, 2237, 2237, 3177, 3177, 3159, 3159, 2251, 2251, 2283, 2283, 1980, 1980, 3174, 3174, 2330, 2330, 3159, 3159, 3179, 3179, 2963, 2963, 2973, 2973, 2963, 2963, 3180, 3180, 2972, 2972, 2970, 2970, 2972, 2972, 3181, 3181, 3959, 3959, 2964, 2964, 3959, 3959, 3182, 3182, 3166, 3166, 3943, 3943, 3946, 3946, 2505, 2505, 2508, 2508, 3943, 3943, 3828, 3828, 3968, 3968, 2376, 3616, 3616, 3817, 3817, 2751, 2751, 1480, 1480, 4141, 4141, 3820, 3820, 2750, 2750, 2415, 2415, 2707, 2707, 3960, 3960, 3174, 3174, 3174, 3831, 3831, 2747, 2747, 2609, 2609, 2312, 2312, 1437, 1437, 2242, 2242, 2647, 2647, 2646, 2646, 2609, 2609, 2251, 2251, 2640, 2640, 2639, 2639, 2647, 2647, 2973, 2973, 2637, 2637, 2636, 2636, 2640, 2640, 2970, 2970, 2508, 2508, 2643, 2643, 2637, 2637, 2964, 2964, 2645, 2645, 2643, 2643, 2644, 2644, 2642, 2642, 2644, 2644, 2505, 2505, 2862, 2862, 2849, 2849, 3958, 3958, 3186, 3186, 1994, 1994, 3957, 3957, 3474, 3474, 3187, 3187, 3958, 3958, 3473, 3473, 3471, 3471, 2639, 2639, 3470, 3470, 3469, 3469, 3470, 3470, 2636, 2636, 2645, 2645, 3467, 3467, 2862, 2862, 3460, 3460, 1477, 1477, 3459, 3459, 3457, 3457, 3188, 3188, 3474, 3474, 3469, 3469, 3455, 3455, 3189, 3189, 3457, 3457, 3467, 3467, 3453, 3453, 3190, 3190, 3455, 3455, 2642, 2642, 3451, 3451, 3191, 3191, 3453, 3453, 2849, 2849, 1466, 1466, 3192, 3192, 3451, 3451, 3459, 3459, 3448, 3448, 2646, 2646, 3471, 3471, 3473, 3473, 1462, 1462, 2312, 2312, 3448, 3448, 3957, 3957, 3445, 3445, 3444, 3444, 1458, 1458, 3443, 3443, 2891, 2891, 2359, 2359, 2089, 2089, 2344, 2344, 3870, 3870, 2358, 2358, 3874, 3874, 2345, 2345, 2894, 2894, 2360, 2360, 2885, 2341, 2341, 2079, 2361, 2361, 2320, 2320, 2340, 2340, 3870, 3870, 2362, 2362, 3403, 3403, 2342, 2342, 2894, 2894, 2363, 2363, 2088, 2088, 304, 304, 3422, 3422, 3421, 3421, 3445, 3445, 3439, 3439, 3419, 3419, 3418, 3418, 3422, 3422, 3436, 3436, 3416, 3416, 3415, 3415, 3419, 3419, 3433, 3433, 3413, 3413, 3412, 3412, 3416, 3416, 3430, 3430, 3410, 3410, 3409, 3409, 3413, 3413, 3427, 3427, 1420, 1420, 3407, 3407, 3410, 3410, 3424, 3424, 2927, 2927, 328, 328, 2085, 2085, 2343, 2343, 2807, 2807, 3807, 3807, 2803, 2803, 3665, 3665, 3399, 3399, 3398, 3398, 3700, 3700, 3193, 3193, 3404, 3403, 3403, 2927, 2927, 3617, 3617, 3393, 3393, 3770, 3770, 3893, 3893, 3632, 3632, 3391, 3391, 3628, 3628, 3390, 3390, 3690, 3690, 1400, 1400, 3624, 3624, 3391, 3391, 3687, 3687, 4142, 4142, 4143, 4143, 3129, 3129, 3387, 3387, 3385, 3385, 2561, 2561, 2148, 2148, 3383, 3383, 3319, 3319, 2558, 3707, 3707, 3379, 3379, 3252, 3252, 3378, 3378, 3380, 3380, 3376, 3376, 3702, 3702, 2766, 2766, 3377, 3377, 3108, 3108, 2765, 2765, 2763, 2763, 3375, 3375, 3372, 3372, 3199, 3199, 3371, 3371, 3370, 3370, 3640, 3640, 3135, 3135, 3399, 3399, 3200, 3200, 3145, 3145, 3145, 3367, 3367, 3364, 3364, 3581, 3581, 3692, 3692, 3629, 3629, 1373, 1373, 3578, 3578, 3364, 3364, 3625, 3625, 3123, 3123, 3361, 3361, 3360, 3360, 3202, 3202, 3358, 3358, 3634, 3634, 3888, 3888, 3357, 3357, 2403, 2403, 3580, 3580, 3507, 3507, 3356, 3356, 3203, 3203, 3204, 3204, 3352, 3352, 3587, 3587, 3205, 3205, 2409, 3349, 3349, 3347, 3347, 3650, 3650, 3206, 3401, 3401, 3503, 3503, 2958, 2958, 3854, 3854, 3242, 3242, 2489, 2489, 3241, 3241, 2501, 2501, 3371, 3371, 3209, 3209, 3696, 3696, 3340, 3340, 3338, 3338, 3210, 3210, 3372, 3372, 3337, 3337, 3153, 3153, 3335, 3335, 3152, 3152, 3166, 3166, 583, 583, 2600, 2600, 2846, 2846, 2778, 2778, 3172, 3172, 3558, 3558, 3211, 3211, 3586, 3586, 3330, 3330, 3689, 3689, 3328, 3328, 3212, 3212, 3138, 3138, 3328, 3328, 3390, 3390, 3326, 3326, 3213, 3213, 3326, 3326, 3627, 3627, 3694, 3694, 3214, 3214, 2784, 2784, 3527, 3527, 2564, 2564, 3384, 3384, 3583, 3583, 2409, 3321, 3321, 3367, 3367, 2817, 2817, 3150, 3150, 2155, 2155, 3652, 3652, 3317, 3317, 3147, 3147, 3316, 3316, 3240, 3240, 3314, 3314, 3193, 3193, 3317, 3317, 3216, 3216, 3312, 3312, 3200, 3200, 3314, 3314, 3217, 3217, 2404, 3133, 3133, 3312, 3312, 3219, 3219, 3309, 3309, 3357, 3357, 3884, 3884, 3757, 3757, 3221, 3220, 3220, 3307, 3307, 2426, 2426, 3218, 3360, 3360, 3220, 3220, 3361, 3361, 3201, 3201, 3222, 3303, 3303, 3214, 3214, 3321, 3321, 3124, 3124, 3301, 3301, 3213, 3213, 3303, 3303, 3211, 3211, 2487, 3225, 3301, 3301, 3118, 3118, 2937, 2937, 3698, 3318, 3318, 3316, 3316, 2797, 2797, 3559, 2487, 2487, 3295, 3295, 3294, 3294, 3293, 3293, 3225, 3146, 3146, 3291, 3291, 3215, 3215, 2796, 2796, 3292, 3292, 3289, 3289, 3216, 3216, 3291, 3291, 3209, 3209, 3287, 3287, 3217, 3217, 3289, 3289, 3199, 3199, 3219, 3219, 3287, 3287, 3210, 3210, 3286, 3286, 3637, 3637, 3204, 3204, 3338, 3338, 3281, 3281, 3202, 3202, 2404, 2404, 3203, 3203, 3584, 3584, 3350, 3350, 3226, 3226, 3636, 3636, 3103, 3103, 3131, 3131, 3226, 3226, 3122, 3122, 3352, 3352, 2414, 2414, 2414, 2414, 3208, 3208, 2932, 2932, 2296, 2296, 3224, 3397, 2311, 2311, 3299, 3299, 3298, 3298, 3297, 3297, 2490, 2490, 2413, 2413, 2935, 2935, 3087, 3087, 3086, 3086, 3085, 3085, 3084, 3084, 3085, 3085, 3082, 3082, 3081, 3081, 3080, 3080, 3081, 3081, 3078, 3078, 3077, 3077, 3076, 3076, 3077, 3077, 3074, 3074, 3073, 3073, 3072, 3072, 3070, 3070, 3073, 3073, 3069, 3069, 3068, 3068, 3066, 3066, 3065, 3065, 3337, 3337, 3064, 3064, 3062, 3062, 3227, 3227, 3061, 3061, 3060, 3060, 3061, 3061, 3228, 3228, 3058, 3058, 3057, 3057, 3058, 3058, 3229, 3229, 3055, 3055, 3054, 3054, 3055, 3055, 3230, 3230, 3052, 3052, 3051, 3051, 3052, 3052, 3231, 3231, 3049, 3049, 3048, 3048, 3049, 3049, 3232, 3232, 3046, 3046, 3045, 3045, 3046, 3046, 3233, 3233, 3043, 3043, 3042, 3042, 3043, 3043, 3234, 3234, 3087, 3087, 3040, 3040, 3038, 3038, 3037, 3037, 3040, 3040, 3036, 3036, 3950, 3950, 2597, 3174, 3174, 3033, 3033, 3034, 3034, 3968, 3749, 3749, 3030, 3030, 3031, 3031, 2434, 2434, 2163, 2163, 3027, 3027, 3028, 3028, 3703, 3703, 3024, 3024, 3025, 3025, 2371, 2371, 2371, 2371, 3021, 3021, 3022, 3022, 3722, 3722, 2372, 2372, 3018, 3018, 3019, 3019, 3719, 3719, 2373, 2373, 3015, 3015, 3016, 3016, 3715, 3715, 2758, 2758, 3012, 3012, 3013, 3013, 3491, 3201, 3009, 3009, 3010, 3010, 3158, 2427, 2427, 3038, 3038, 3007, 3007, 3641, 3641, 3311, 3311, 3004, 3004, 3005, 3005, 2419, 2419, 3002, 3002, 3004, 3004, 3000, 3000, 3042, 3042, 3037, 3037, 3084, 3084, 2998, 2998, 3009, 3009, 3036, 3036, 3080, 3080, 2996, 2996, 3012, 3012, 2998, 2998, 3076, 3076, 2994, 2994, 3015, 3015, 2996, 2996, 3072, 3072, 2992, 2992, 3018, 3018, 2994, 2994, 3070, 3070, 2990, 2990, 3021, 3021, 2992, 2992, 2988, 2988, 2987, 2987, 3024, 3024, 2990, 2990, 3060, 3060, 2985, 2985, 3027, 3027, 2987, 2987, 3057, 3057, 2983, 2983, 3030, 3030, 2985, 2985, 3054, 3054, 2981, 2981, 3033, 3033, 2983, 2983, 3051, 3051, 2979, 2979, 3948, 3948, 2981, 2981, 3048, 3048, 2977, 2977, 3953, 3953, 2979, 2979, 3045, 3045, 3000, 3000, 3002, 3002, 2977, 2977, 2752, 2752, 3945, 3945, 3643, 3643, 4144, 4144, 4145, 4145, 4146, 4146, 1481, 4147, 4147, 3139, 3139, 4148, 4148, 2546, 2546, 4149, 4149, 4150, 4150, 4151, 4151, 4152, 4152, 1169, 1169, 3144, 3144, 1202, 1202, 2556, 2556, 2500, 2500, 1161, 1161, 233, 2222, 2222, 4123, 4123, 3261, 3261, 4124, 4124, 2555, 2555, 4125, 4125, 4126, 4126, 976, 976, 183, 183, 2224, 2224, 3235, 3235, 3854, 3854, 2958, 2958, 3235, 3235, 2958, 2958, 3330, 3330, 2956, 2956, 3236, 3236, 2956, 2956, 3588, 3588, 2954, 2954, 3121, 3121, 2954, 2954, 3349, 3349, 2952, 2952, 2951, 2951, 2952, 2952, 3103, 3103, 2949, 2949, 2948, 2948, 2949, 2949, 3638, 3638, 2946, 2946, 2945, 2945, 2946, 2946, 3281, 3281, 3065, 3065, 2943, 2943, 3062, 3062, 2988, 2988, 3068, 3068, 3237, 3237, 3064, 3064, 3370, 3370, 2940, 2940, 3238, 3238, 2940, 2940, 3340, 3340, 2938, 2938, 3239, 3239, 2321, 2321, 3698, 3698, 3618, 3618, 2366, 2366, 2366, 3223, 3223, 2399, 2399, 3212, 3212, 3300, 3300, 3118, 3118, 3556, 3556, 2934, 2934, 3243, 3243, 2486, 2486, 2475, 2475, 2491, 2491, 2929, 2929, 2928, 2928, 2805, 2805, 3803, 3803, 2924, 2924, 2923, 2923, 2929, 2929, 2922, 2922, 2920, 2920, 2919, 2919, 2924, 2924, 2918, 2918, 3571, 3571, 2916, 2916, 2920, 2920, 2915, 2915, 3234, 3234, 2913, 2913, 3570, 3570, 2911, 2911, 3086, 3086, 3120, 3120, 2951, 2951, 2909, 2909, 3082, 3082, 2911, 2911, 2948, 2948, 2907, 2907, 3078, 3078, 2909, 2909, 2945, 2945, 2905, 2905, 3074, 3074, 2907, 2907, 2943, 2943, 3069, 3069, 2905, 2905, 3066, 3066, 3244, 3244, 2902, 2902, 2901, 2901, 3244, 3244, 2900, 2900, 802, 802, 3668, 3668, 2902, 2902, 2896, 2896, 2803, 2803, 3258, 3258, 2156, 2156, 2819, 2819, 3672, 3672, 3335, 3335, 3251, 3251, 2798, 2798, 2887, 2887, 3237, 3237, 2901, 2901, 3250, 3250, 2124, 2124, 3215, 3215, 3252, 3252, 3238, 3238, 2881, 2881, 2900, 2900, 2796, 2796, 2887, 2887, 2897, 2897, 2403, 3239, 3239, 3999, 2098, 2881, 2881, 3512, 3512, 2534, 2793, 2793, 3158, 3158, 3253, 3253, 3540, 3540, 2793, 2793, 2884, 2884, 2790, 2790, 3515, 3515, 3253, 3253, 3504, 3504, 3254, 3254, 3545, 3545, 2790, 2790, 3501, 3501, 3254, 3254, 3497, 3497, 2787, 2787, 2786, 2786, 3230, 3230, 2787, 2787, 3488, 3488, 2865, 2865, 3535, 3535, 2784, 2784, 3531, 3531, 3530, 3530, 3111, 2860, 2860, 2926, 2926, 3111, 3111, 3231, 3231, 2865, 2865, 2923, 2923, 2858, 2858, 3243, 3243, 2856, 2856, 2922, 2922, 2860, 2860, 3232, 3232, 2858, 2858, 2919, 2919, 2854, 2854, 3235, 3235, 2852, 2852, 2918, 2918, 2856, 2856, 3233, 3233, 2854, 2854, 2916, 2916, 2913, 2913, 3236, 3236, 3575, 3575, 2915, 2915, 2852, 2852, 2781, 2781, 2783, 2783, 2094, 2094, 3112, 3112, 2846, 2846, 3161, 3161, 3465, 3465, 2649, 2649, 2695, 2695, 2649, 2649, 2684, 2684, 2267, 2267, 3163, 3163, 219, 219, 2886, 2886, 850, 850, 2840, 2840, 3415, 3415, 2839, 2839, 846, 846, 2839, 2839, 3412, 3412, 2837, 2837, 3418, 3418, 2840, 2840, 843, 843, 2835, 2835, 3409, 3409, 3407, 3407, 840, 840, 2837, 2837, 838, 838, 2832, 2832, 2749, 2749, 2750, 2750, 834, 834, 3939, 3939, 2667, 2667, 2323, 2323, 831, 831, 2829, 2829, 2325, 2325, 3644, 3644, 3838, 3838, 829, 829, 3247, 3247, 2827, 3615, 3615, 2751, 2751, 827, 827, 2831, 2831, 772, 772, 3245, 3245, 2776, 2776, 3148, 3148, 3551, 3551, 2162, 2162, 3151, 3151, 3794, 2800, 2157, 2157, 2820, 2800, 2800, 2435, 2435, 2818, 2818, 2817, 2817, 2817, 3790, 3790, 2815, 2815, 3653, 3653, 2818, 2818, 3788, 3788, 2813, 2813, 3656, 3656, 2815, 2815, 3784, 3784, 2811, 2811, 3659, 3659, 2813, 2813, 2809, 2809, 2811, 2811, 3778, 3778, 3250, 3250, 3814, 3814, 2807, 2807, 3347, 3347, 2878, 2878, 2896, 2896, 2896, 3970, 3970, 3969, 3969, 3973, 3973, 3971, 3971, 3975, 3975, 3974, 3974, 3976, 3976, 3978, 3978, 3973, 3973, 3971, 3971, 3981, 3981, 3977, 3977, 3983, 3983, 3984, 3984, 3985, 3985, 3986, 3986, 3980, 3980, 3979, 3979, 3990, 3990, 3991, 3991, 3992, 3992, 3993, 3993, 3988, 3988, 3989, 3989, 3947, 3947, 3934, 3934, 3101, 3101, 4093, 4093, 4092, 4092, 4090, 4090, 4091, 4091, 4121, 4121, 4120, 4120, 4119, 4119, 4118, 4118, 4117, 4117, 4116, 4116, 4115, 4115, 4114, 4114, 4113, 4113, 4112, 4112, 4111, 4111, 4110, 4110, 4109, 4109, 4108, 4108, 4107, 4107, 4106, 4106, 4105, 4105, 4104, 4104, 4103, 4103, 4102, 4102, 4101, 4101, 4100, 4100, 4099, 4099, 4098, 4098, 4097, 4097, 4096, 4096, 4095, 4095, 4094, 4094, 4058, 4058, 4059, 4059, 4060, 4060, 4061, 4061, 4062, 4062, 4063, 4063, 4064, 4064, 4065, 4065, 4066, 4066, 4067, 4067, 4068, 4068, 4069, 4069, 4070, 4070, 4071, 4071, 4072, 4072, 4073, 4073, 4074, 4074, 4075, 4075, 4076, 4076, 4077, 4077, 4078, 4078, 4079, 4079, 4080, 4080, 4081, 4081, 4082, 4082, 4083, 4083, 4084, 4084, 4085, 4085, 4086, 4086, 4087, 4087, 4088, 4088, 4089, 4089, 4091, 4091, 4090, 4090, 4092, 4092, 4093, 4093, 4094, 4094, 4095, 4095, 4096, 4096, 4097, 4097, 4098, 4098, 4099, 4099, 4100, 4100, 4101, 4101, 4102, 4102, 4103, 4103, 4104, 4104, 4105, 4105, 4106, 4106, 4107, 4107, 4108, 4108, 4109, 4109, 4110, 4110, 4111, 4111, 4112, 4112, 4113, 4113, 4114, 4114, 4115, 4115, 4116, 4116, 4117, 4117, 4118, 4118, 4119, 4119, 4120, 4120, 4121, 4121, 4122, 4122, 4123, 4123, 4124, 4124, 4125, 4125, 4126, 4126, 4127, 4127, 4128, 4128, 4129, 4129, 4130, 4130, 4131, 4131, 4132, 4132, 4133, 4133, 4134, 4134, 4135, 4135, 4136, 4136, 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\n", " \n", "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "with open(\"AircraftModels/harrier.obj\", \"rb\") as file:\n", " obj_data = file.read().decode('utf-8')\n", "\n", "def obj_data_to_mesh3d(odata):\n", " # odata is the string read from an obj file\n", " vertices = []\n", " faces = []\n", " lines = odata.splitlines() \n", " \n", " for line in lines:\n", " slist = line.split()\n", " if slist:\n", " if slist[0] == 'v':\n", " vertex = np.array(slist[1:], dtype=float)\n", " vertices.append(vertex)\n", " elif slist[0] == 'f':\n", " face = []\n", " for k in range(1, len(slist)):\n", " face.append([int(s) for s in slist[k].replace('//','/').split('/')])\n", " if len(face) > 3: # triangulate the n-polyonal face, n>3\n", " faces.extend([[face[0][0]-1, face[k][0]-1, face[k+1][0]-1] for k in range(1, len(face)-1)])\n", " else: \n", " faces.append([face[j][0]-1 for j in range(len(face))])\n", " else: pass\n", " \n", " \n", " return np.array(vertices), np.array(faces) \n", "\n", "vertices, faces = obj_data_to_mesh3d(obj_data)\n", "\n", "x, y, z = vertices[:,:3].T\n", "I, J, K = faces.T\n", "\n", "# Normalise the aircraft\n", "# x = z, y = x, z = -y\n", "degangle = 5\n", "rot = np.radians(degangle)\n", "cos = np.cos(rot)\n", "sin = np.sin(rot)\n", "Tmatrix = np.array([[cos, 0, -sin],[0, 1, 0],[sin, 0, cos]])\n", "\n", "X = z + 1.5\n", "Y = x\n", "Z= -y + 1.5\n", "\n", "for i in range(len(X)):\n", " X[i] = cos * X[i] - sin * Z[i]\n", " Z[i] = sin * X[i] + cos * Z[i]\n", "\n", "mesh = go.Mesh3d(\n", " x=X,\n", " y=Y,\n", " z=Z,\n", " i=I,\n", " j=J,\n", " k=K,\n", " name='',\n", " showscale=False)\n", "\n", "mesh.update(lighting=dict(ambient= 0.5,\n", " diffuse= 1,\n", " fresnel= 5,\n", " specular= 1,\n", " roughness= 1),\n", " \n", " lightposition=dict(x=-1000,\n", " y=1000,\n", " z=1000));\n", "tiprange = 12\n", "\n", "layout = go.Layout(title='Aerodynamic Angles', title_x=0.5,\n", " font=dict(size=14, color='black'),\n", " width=900,\n", " height=900,\n", " scene=dict(xaxis=dict(visible=False, title='X', range=[-tiprange*1.2, tiprange*1.2]),\n", " yaxis=dict(visible=False, title='Y', range=[tiprange*1.2, -tiprange*1.2]), \n", " zaxis=dict(visible=False, title='Z', range=[tiprange*1.2, -tiprange*1.2]), \n", " aspectratio=dict(x=1,\n", " y=1,\n", " z=1\n", " ),\n", " camera=dict(eye=dict(x=1., y=1., z=0.5)),\n", " ), \n", " margin=dict(t=175),\n", " paper_bgcolor='rgba(0,0,0,0)',\n", " plot_bgcolor='rgba(0,0,0,0)') \n", "\n", "\n", "\n", "\n", "fig = go.Figure(data=[mesh], layout=layout)\n", "\n", "\n", "\n", "## Add some axes\n", "fig.add_trace(go.Scatter3d(x=[0, tiprange], y=[0, 0], z=[0, 0],\n", " mode='lines',\n", " showlegend=False, line=dict(color='royalblue', width=4)))\n", "\n", "fig.add_trace(go.Cone(x=[tiprange], y=[0], z=[0], u=[1], v=[0], w=[0], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(65,105,225)'], [1, 'rgb(65,105,225)']],\\\n", " showscale=False))\n", "\n", "fig.add_trace(go.Scatter3d(x=[0, 0], y=[0, tiprange], z=[0, 0],\n", " mode='lines',\n", " showlegend=False, line=dict(color='royalblue', width=4)))\n", "\n", "fig.add_trace(go.Cone(x=[0], y=[tiprange], z=[0], u=[0], v=[1], w=[0], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(65,105,225)'], [1, 'rgb(65,105,225)']],\\\n", " showscale=False))\n", "\n", "fig.add_trace(go.Scatter3d(x=[0, 0], y=[0, 0], z=[0, tiprange],\n", " mode='lines',\n", " showlegend=False, line=dict(color='royalblue', width=4)))\n", "\n", "fig.add_trace(go.Cone(x=[0], y=[0], z=[tiprange], u=[0], v=[0], w=[1], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(65,105,225)'], [1, 'rgb(65,105,225)']],\\\n", " showscale=False))\n", "\n", "fig.add_trace(go.Scatter3d(x=[tiprange, 0, 0], y=[0, tiprange, .2], z=[0, 0, tiprange], mode='text', text=['x', 'y', 'z'], name='Directions', showlegend=False))\n", "\n", "\n", "## Add in teh flightspeed\n", "Vf = tiprange*.8\n", "alfa = np.radians(20)\n", "beta = np.radians(30)\n", "U = Vf * np.cos(alfa) * np.cos(beta)\n", "V = Vf * np.sin(beta)\n", "W = Vf * np.cos(beta) * np.sin(alfa)\n", "\n", "## Add on the flightspeed vector\n", "fig.add_trace(go.Scatter3d(x=[0, U], y=[0, V], z=[0, W],\n", " mode='lines', name='$V_f\\\\text{ - Flightspeed}$',\n", " showlegend=True, line=dict(color='red', width=4)))\n", "\n", "fig.add_trace(go.Cone(x=[U], y=[V], z=[W], u=[U/Vf], v=[V/Vf], w=[W/Vf], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(255,0,0)'], [1, 'rgb(255,0,0)']],\\\n", " showscale=False))\n", "\n", "# Now do the same for the velocity components - first U\n", "fig.add_trace(go.Scatter3d(x=[0, U], y=[0, 0], z=[0, 0],\n", " mode='lines', name='$U$', legendgroup=\"Velocity Components\",\n", " showlegend=True, line=dict(color='red', width=10, dash='solid')))\n", "\n", "fig.add_trace(go.Cone(x=[U], y=[0], z=[0], u=[U/Vf], v=[0/Vf], w=[0/Vf], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(255,0,0)'], [1, 'rgb(255,0,0)']],\\\n", " showscale=False, legendgroup=\"Velocity Components\"))\n", "\n", "# Now do the same for the velocity components - aecond V\n", "fig.add_trace(go.Scatter3d(x=[0, 0], y=[0, V], z=[0, 0],\n", " mode='lines', name='$V$', legendgroup=\"Velocity Components\",\n", " showlegend=True, line=dict(color='rgb(0,255,0)', width=10, dash='solid')))\n", "\n", "fig.add_trace(go.Cone(x=[0], y=[V], z=[0], u=[0/Vf], v=[V/Vf], w=[0/Vf], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(0,255,0)'], [1, 'rgb(0,255,0)']],\\\n", " showscale=False, legendgroup=\"Velocity Components\"))\n", "\n", "# Now do the same for the velocity components - last W\n", "fig.add_trace(go.Scatter3d(x=[0, 0], y=[0, 0], z=[0, W],\n", " mode='lines', name='$W$', legendgroup=\"Velocity Components\",\n", " showlegend=True, line=dict(color='rgb(0,0,255)', width=10, dash='solid')))\n", "\n", "fig.add_trace(go.Cone(x=[0], y=[0], z=[W], u=[0/Vf], v=[0/Vf], w=[W/Vf], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(0,0,255)'], [1, 'rgb(0,0,255)']],\\\n", " showscale=False, legendgroup=\"Velocity Components\"))\n", "\n", "# Add in cosine beta\n", "## Add on the flightspeed vector\n", "fig.add_trace(go.Scatter3d(x=[0, U], y=[0, 0], z=[0, W],\n", " mode='lines', name='$V_f\\cos\\\\beta$',\n", " showlegend=True, line=dict(color='red', width=4, dash='dash')))\n", "\n", "fig.add_trace(go.Cone(x=[U], y=[0], z=[W], u=[U/Vf], v=[0/Vf], w=[W/Vf], cmin=1, cmax=1,\\\n", " colorscale=[[0, 'rgb(255,0,0)'], [1, 'rgb(255,0,0)']],\\\n", " showscale=False))\n", "\n", "## Add on sideslip angle\n", "Ub = np.zeros(100)\n", "Vb = np.zeros(100)\n", "Wb = np.zeros(100)\n", "for i, b in enumerate(np.linspace(0, beta, 100)):\n", " Ub[i] = .8*Vf * np.cos(alfa) * np.cos(b)\n", " Vb[i] = .8*Vf * np.sin(b)\n", " Wb[i] = .8*Vf * np.cos(b) * np.sin(alfa)\n", " \n", "# Put SS plot in\n", "fig.add_trace(go.Scatter3d(x=Ub, y=Vb, z=Wb,\n", " mode='lines', name='$\\\\beta$', legendgroup=\"Sideslip\",\n", " showlegend=True, line=dict(color='blue', width=10, dash='solid'))) \n", "\n", "# And label it\n", "fig.add_trace(go.Scatter3d(x=[Ub[50]], y=[Vb[50]], z=[Wb[50]], mode='text', text=['β'],\\\n", " textposition='bottom center', name='β - Sideslip', showlegend=False, textfont=dict(\n", " color=\"blue\",\n", " size=12), legendgroup=\"Sideslip\"))\n", "\n", "## Add on AoA\n", "Ua = np.zeros(100)\n", "Va = np.zeros(100)\n", "Wa = np.zeros(100)\n", "for i, a in enumerate(np.linspace(0, alfa, 100)):\n", " Ua[i] = .9*Vf * np.cos(a) * np.cos(beta)\n", " Va[i] = 0\n", " Wa[i] = .9*Vf * np.cos(beta) * np.sin(a)\n", " \n", "# Put AoA plot in\n", "fig.add_trace(go.Scatter3d(x=Ua, y=Va, z=Wa,\n", " mode='lines', name='$\\\\alpha$', legendgroup=\"AoA\",\n", " showlegend=True, line=dict(color='rgb(0,0,0)', width=10, dash='solid'))) \n", "\n", "fig.add_trace(go.Scatter3d(x=[Ua[50]], y=[Va[50]], z=[Wa[50]], mode='text', text=['α'],\\\n", " textposition='middle right', name='AoA', showlegend=False, textfont=dict(\n", " color=\"blue\",\n", " size=12), legendgroup=\"AoA\"))\n", "\n", "\n", "fig.layout.scene.camera.projection.type = \"orthographic\"\n", "\n", "fig.show()\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The aircraft flightspeed is a vector, $\\vec{V}_f$, when defined in body axes is:\n", "\n", "$$\\vec{V}_f=\\begin{bmatrix}U\\\\V\\\\W\\end{bmatrix}=\\begin{bmatrix}\\left|V_f\\right|\\cdot\\cos\\alpha\\cos\\beta\\\\\\left|V_f\\right|\\cdot\\sin\\beta\\\\\\left|V_f\\right|\\cdot\\sin\\alpha\\cos\\beta\\end{bmatrix}$$\n", "\n", "From the diagram above, you can see the sign convention can be remembered by $V_f$ being drawn between the $x-y$ axes, and the projection onto the aircraft $x-z$ plane, $V_f\\cos\\beta$, as drawn between the $x-z$ axes.\n", "\n", "This can be recalled as **positive angle of attack is nose up** and **positive sideslip is nose port**, as in the aircraft movement with respect to the flightspeed vector.\n", "\n", "```{warning} Be careful with the definition of $V_f$ - \n", "$V_f$ is the aircraft velocity vector, and hence defined as moving *away* from the aircraft. It is not the vector defining the inclement wind - which is actually colinear to $V_f$ but opposite in sign.\n", "```\n", "\n", "```{admonition} Why I don't use 'plane' for 'aircraft'\n", ":class: dropdown\n", "'plane' is acceptable parlance for aircraft in North America, but it really should not be part of a good aeronautical engineer's speech. It can be confusing because *plane* has a strict definition in geometry, and we use that meaning **all the time**.\n", "\n", "Furthermore, and this might just be my British sensibilities - it's informal and lazy to use in speech when talking in aeronautical engineer terms, but much worse to use in technical written English. Would you trust a cardiac surgeon who referred to your heart as your ticker? Or an automotive engineer referring to a car, in professional speed, as 'wheels'? Try and use aeroplane/airplane/aircraft.\n", "```\n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3", "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": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }