{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "ac87e6fb-030a-4c57-8cdc-20a9450c0a4f",
   "metadata": {
    "tags": []
   },
   "source": [
    "# Assignment 3.1: Differential Geometry of smooth surfaces"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "eb14a5c8-13ce-4d21-8167-ddbb674225e2",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "/opt/conda/bin/python\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import sys\n",
    "print(sys.executable)\n",
    "import meshplot as mp\n",
    "sys.path.append('../src/')\n",
    "from smooth_surfaces import *\n",
    "from utils import *"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4949242e-7532-4744-836c-3da903f42ea9",
   "metadata": {},
   "source": [
    "## Paraboloid surface"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5290233d-e1c7-464f-b116-63917aaa8df3",
   "metadata": {},
   "source": [
    "To evaluate the paraboloid surface, we first sample the domain $\\Omega$ with a regular grid. For that, we specify the $u$ and $v$ intervals, and the sampling in the two directions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "b06d705d-1164-496f-9f58-00267da59e81",
   "metadata": {},
   "outputs": [],
   "source": [
    "u_interval = [-1, 1]\n",
    "v_interval = [-1, 1]\n",
    "u_sampling = 21\n",
    "v_sampling = 21"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "23442be0-a0bc-42ea-83bd-23693de97a49",
   "metadata": {},
   "source": [
    "We use now the following provided function that creates the points $p\\in\\Omega\\subseteq\\mathbb R^2$, collected in the array `P`, and a triangulation `F` for visualization:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "ae541330-6a57-46d4-a8cd-7005fe8cd70e",
   "metadata": {},
   "outputs": [],
   "source": [
    "P, F = uv_grid(u_interval, v_interval, u_sampling, v_sampling)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "600fbd25-4295-4074-a979-9d2a7aac8b45",
   "metadata": {},
   "source": [
    "We can now choose the parameters of the paraboloid:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "id": "a0169d4c-ae67-4e23-92ec-798a7012e649",
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 0.2\n",
    "b = 0.2\n",
    "c = -0.2\n",
    "d = -0.2\n",
    "e = 0"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "35a40d7a-2ac8-4248-b497-39dafac57078",
   "metadata": {},
   "source": [
    "Try to change the parameters to get and then compute the corresponding paraboloid points $\\mathbf x(p)\\in\\mathbb R^3$ and the derivatives of the parametrization at that points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "id": "75d4933e-2a64-4c01-91d7-3c9c47ace5a1",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = compute_paraboloid_points(P, a, b, c, d, e)\n",
    "\n",
    "x_u, x_v = compute_paraboloid_first_derivatives(P, a, b, c, d, e)\n",
    "\n",
    "x_uu, x_uv, x_vv = compute_paraboloid_second_derivatives(P, a, b, c, d, e)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e41fefc0-dbd9-4fb7-bf4a-ae0e034bfb9b",
   "metadata": {},
   "source": [
    "### Principal curvatures"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc4a49ce-a517-4a5b-a6bb-18aad4be33fc",
   "metadata": {},
   "source": [
    "We define now a function that computes the principal curvatures of the surface at the points by using your functions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "id": "0577a574-95c2-4d0b-97a2-b5ca04235552",
   "metadata": {},
   "outputs": [],
   "source": [
    "def principal_curvatures(x_u, x_v, x_uu, x_uv, x_vv):\n",
    "\n",
    "    n = compute_surface_normal(x_u, x_v)\n",
    "\n",
    "    I = compute_first_fundamental_form(x_u, x_v)\n",
    "\n",
    "    II = compute_second_fundamental_form(x_uu, x_uv, x_vv, n)\n",
    "\n",
    "    S = compute_shape_operator(I, II)\n",
    "\n",
    "    return compute_principal_curvatures(S, x_u, x_v)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "064c8eba-7b20-4b30-8064-40db7a2e3e4d",
   "metadata": {},
   "source": [
    "and use it to evaluate the principal curvatures $\\kappa_1$ and $\\kappa_2$ with the corresponding principal directions $\\mathbf e_1$ and $\\mathbf e_2$ at each surface point:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "id": "26483943-2117-4b33-b13f-54351ea4ee1e",
   "metadata": {},
   "outputs": [],
   "source": [
    "k_1, k_2, e_1, e_2 = principal_curvatures(x_u, x_v, x_uu, x_uv, x_vv)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ae42813-cd7f-4b27-b9c4-9407a5a79e44",
   "metadata": {},
   "source": [
    "We can compute now the Gaussian curvature $K$ and the mean curvature $H$ as"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "id": "d365f3c0-c069-4645-a360-ba4f2219f3c3",
   "metadata": {},
   "outputs": [],
   "source": [
    "K = k_1 * k_2\n",
    "\n",
    "H = (k_1 + k_2)/2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "34710736-2847-42e3-9468-3d620bb82d4b",
   "metadata": {},
   "source": [
    "To display the surface and the curvature, we define a plot function that colors points according to the curvature values with a blue-green-red color scale, with blue for negative values, green for zero, and red for positive values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "id": "4ac8a92c-ded6-485b-b0d0-9d692fc96a73",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_curvature(x, F, values):\n",
    "    color = bgr_color(values)\n",
    "    shading_options = {\n",
    "        \"flat\":False,\n",
    "        \"wireframe\":False,\n",
    "        \"metalness\": 0,\n",
    "    }\n",
    "    mp.plot(x, F, c=color, shading=shading_options)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "97dfabad-89d6-4d16-a028-5809aa4d3d22",
   "metadata": {},
   "source": [
    "and use it to visualize the Gaussian curvature of the surface"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "id": "ccb2dde3-cf23-4c0f-81bc-1de8f0910eb3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "fd0f312ceca0422583d581083529866e",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(3.1415927…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_curvature(x, F, K)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1d166a4e-04dd-4d0d-a248-b5baf5e7dcf8",
   "metadata": {},
   "source": [
    "and the mean curvature"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "id": "400c14b0-e551-4363-ac36-a71d8bd74fb8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "b0f1ccdc2b3c43dcb18857be4c6d7083",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(3.1415927…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_curvature(x, F, H)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "afcb4e17-7677-4bb5-acc4-2548e77cd8f9",
   "metadata": {},
   "source": [
    "We want now to visualize directions on the surface. For that define the plot function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "id": "75c72db5-cd8a-4a73-9c4b-ed17f6a870a2",
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_directions(x, F, d_1, d_2, scale=0.1):\n",
    "    color = np.ones((len(x), 3))\n",
    "    shading_options = {\n",
    "        \"flat\": False,\n",
    "        \"wireframe\":False,\n",
    "        \"metalness\": 0.05,\n",
    "    }\n",
    "    p = mp.plot(x, F, c=color, shading=shading_options)\n",
    "    p.add_lines(x+d_1*scale, x-d_1*scale, shading={\"line_color\": \"red\"})\n",
    "    p.add_lines(x+d_2*scale, x-d_2*scale, shading={\"line_color\": \"blue\"})\n",
    "    p.update_object()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "94fc28d4-795d-440a-9c86-373186989d81",
   "metadata": {},
   "source": [
    "We can now display the principal curvature directions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "id": "8435ea44-6114-4d96-9483-9b4d6ea1c36f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "851bbfdf30f44a228f72b790a8fb6a88",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(3.1415927…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_directions(x, F, e_1, e_2, scale=0.08)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "efc58d26-2eae-490c-b8f0-a0226c63b978",
   "metadata": {},
   "source": [
    "### Asymptotic directions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "22d70cd0-c1c1-496d-8513-e6f35a715be5",
   "metadata": {},
   "source": [
    "We estimate now the asymptotic directions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "id": "9f872d92-2dea-41fc-a273-f3e65d4e5086",
   "metadata": {},
   "outputs": [],
   "source": [
    "a_1, a_2 = compute_asymptotic_directions(k_1, k_2, e_1, e_2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd7f02ac-825c-4ed4-bc88-8569cd7696e2",
   "metadata": {
    "tags": []
   },
   "source": [
    "and display them"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "id": "98b09cbb-b742-4457-99d2-8c59b9ec40c5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "8091f6dfb14e4c2398bb2d503c8c12a3",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(3.1415927…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_directions(x, F, a_1, a_2, scale=0.08)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cd635566-05e4-48fc-93ec-0cee724e7751",
   "metadata": {},
   "source": [
    "## Torus surface"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "238ae4db-5c65-41fc-b5f5-cbea30683c9c",
   "metadata": {},
   "source": [
    "We display now the same quantities for thr torus surface, and start defining the domain interval and sampling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "4e1aa125-23ca-42f8-b846-7c9d8f7444bb",
   "metadata": {},
   "outputs": [],
   "source": [
    "u_interval = [0, 2*np.pi]\n",
    "v_interval = [0, 2*np.pi]\n",
    "u_sampling = 33\n",
    "v_sampling = 33"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4fd11272-b041-4880-88f0-398987b360da",
   "metadata": {},
   "source": [
    "We now create an $u$-$v$ grid"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "d298045e-3294-4ec4-ab22-28a9670c2ce2",
   "metadata": {},
   "outputs": [],
   "source": [
    "P, F = uv_grid(u_interval, v_interval, u_sampling, v_sampling)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "da8fb757-e162-463b-9226-46ff6c6ab90c",
   "metadata": {},
   "source": [
    "and choose the torus radii"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "aa3a5ac0-06ea-446a-ab6c-917265487c29",
   "metadata": {},
   "outputs": [],
   "source": [
    "R = 2\n",
    "\n",
    "r = 1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "40dda590-2553-4a8a-a2f7-e0242eb5fec7",
   "metadata": {},
   "source": [
    "We can now proceed as for the paraboloid:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "870c5ee0-dd7e-4a87-9a64-2820513ff38e",
   "metadata": {},
   "outputs": [],
   "source": [
    "x = compute_torus_points(P, R, r)\n",
    "\n",
    "x_u, x_v = compute_torus_first_derivatives(P, R, r)\n",
    "\n",
    "x_uu, x_uv, x_vv = compute_torus_second_derivatives(P, R, r)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "63c523f3-7db0-4bd1-8b04-8d7ddc66cfa9",
   "metadata": {},
   "source": [
    "### Principal curvatures"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "f8da5db4-74c3-4c68-ac58-f123ce75c446",
   "metadata": {},
   "outputs": [],
   "source": [
    "k_1, k_2, e_1, e_2 = principal_curvatures(x_u, x_v, x_uu, x_uv, x_vv)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "d0a4bab6-d7e8-46ee-a0e3-fbf32a4c12c5",
   "metadata": {},
   "outputs": [],
   "source": [
    "K = k_1 * k_2\n",
    "\n",
    "H = (k_1 + k_2)/2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bc627815-58fd-40a7-afe8-a68adc8bac04",
   "metadata": {},
   "source": [
    "We plot now the Gaussian curvature $K$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "2cea4c46-a698-4a3e-91ad-a86cd2484b54",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "cccde70c2ddb4a7896fe694fdd3f4f9a",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(0.0, 0.0,…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_curvature(x, F, K)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e05c34b0-770f-42fb-abd9-c8e790c2033a",
   "metadata": {},
   "source": [
    "the mean curvature $H$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "727be3f1-02b1-4db1-842d-c2ca5f619694",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "5cfec29c275e4ad09fd9ed50ca3e1048",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(0.0, 0.0,…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_curvature(x, F, H)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4e2e4c3f-1a8c-4120-befc-1f7a3ea377b0",
   "metadata": {},
   "source": [
    "and the principal curvature directions $\\mathbf e_1$ and $\\mathbf e_2$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "6255e260-df73-4d4a-b8ec-2cd922d9942f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "ef10f31b1d7d41c4a45129198f1a5ba4",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(0.0, 0.0,…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_directions(x, F, e_1, e_2, scale=0.12)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5be26065-90f5-4fba-9dc4-80e283117dc9",
   "metadata": {},
   "source": [
    "### Asymptotic directions\n",
    "\n",
    "Finally, we can plot the asymptotic directions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "7c9161e6-7ef0-42ea-b1ec-fc9dd49533e0",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "../src/smooth_surfaces.py:418: RuntimeWarning: invalid value encountered in sqrt\n",
      "  sinT = np.sqrt(f)\n"
     ]
    }
   ],
   "source": [
    "a_1, a_2 = compute_asymptotic_directions(k_1, k_2, e_1, e_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "ac71ac02-d0c0-4dd1-94af-227b3636ed9f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "3f3da2e0f0474e4aaac9e29e11068f4f",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Renderer(camera=PerspectiveCamera(children=(DirectionalLight(color='white', intensity=0.6, position=(0.0, 0.0,…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_directions(x, F, a_1, a_2, scale=0.12)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a64403c3",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "87a49e59",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6f639996",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "483b637b",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "99069ff6",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
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