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   "source": [
    "# Gaussian and Mean Curvature of a Graph Surface\n",
    "\n",
    "Let a surface in Euclidean three-space be the graph\n",
    "\n",
    "$$\n",
    "X(x,y)=(x,y,f(x,y)).\n",
    "$$\n",
    "\n",
    "This notebook derives both fundamental forms and then computes the\n",
    "Gaussian curvature $K$ and mean curvature $H$.  Egison keeps the\n",
    "derivatives of the arbitrary function $f$ symbolic.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "20fedc81",
   "metadata": {},
   "source": [
    "## Tangent plane\n",
    "\n",
    "The coordinate tangent vectors are\n",
    "\n",
    "$$\n",
    "X_x=(1,0,f_x),\\qquad X_y=(0,1,f_y).\n",
    "$$\n",
    "\n",
    "Their cross product is $(-f_x,-f_y,1)$, so its norm is\n",
    "$W=\\sqrt{1+f_x^2+f_y^2}$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "646ca5b2",
   "metadata": {},
   "outputs": [],
   "source": [
    "declare symbol x, y: MathValue\n",
    "\n",
    "def f : MathValue := function (x, y)\n",
    "def X : Vector MathValue := [| x, y, f x y |]\n",
    "\n",
    "def vx : Vector MathValue := [| 1, 0, ∂/∂ (f x y) x |]\n",
    "def vy : Vector MathValue := [| 0, 1, ∂/∂ (f x y) y |]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "05ae2f83",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 1 \\\\ 0 \\\\ \\frac{\\partial f}{\\partial 1}\\\\ \\end{pmatrix}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "vx\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "0b25d70b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 0 \\\\ 1 \\\\ \\frac{\\partial f}{\\partial 2}\\\\ \\end{pmatrix}$"
      ]
     },
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    }
   ],
   "source": [
    "vy\n"
   ]
  },
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   "cell_type": "markdown",
   "id": "3a274d02",
   "metadata": {},
   "source": [
    "## Oriented unit normal\n",
    "\n",
    "We choose the upward-pointing normal\n",
    "\n",
    "$$\n",
    "n=\\frac{X_x\\times X_y}{\\lVert X_x\\times X_y\\rVert}.\n",
    "$$\n",
    "\n",
    "Reversing this orientation changes the sign of $H$ but not of $K$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e6d2b7e5",
   "metadata": {},
   "outputs": [],
   "source": [
    "def normalNumerator : Vector MathValue := crossProduct vx vy\n",
    "def W : MathValue := sqrt (V.* normalNumerator normalNumerator)\n",
    "def normal : Vector MathValue := normalNumerator / W\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "cfbc0b09",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} -\\frac{\\partial f}{\\partial 1} \\\\ -\\frac{\\partial f}{\\partial 2} \\\\ 1\\\\ \\end{pmatrix}$"
      ]
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   "source": [
    "normalNumerator\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "ab6a31ad",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} -\\sqrt{\\frac{\\partial f}{\\partial 2}^{2} + \\frac{\\partial f}{\\partial 1}^{2} + 1}^{-1} \\frac{\\partial f}{\\partial 1} \\\\ -\\sqrt{\\frac{\\partial f}{\\partial 2}^{2} + \\frac{\\partial f}{\\partial 1}^{2} + 1}^{-1} \\frac{\\partial f}{\\partial 2} \\\\ \\sqrt{\\frac{\\partial f}{\\partial 2}^{2} + \\frac{\\partial f}{\\partial 1}^{2} + 1}^{-1}\\\\ \\end{pmatrix}$"
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   "source": [
    "normal\n"
   ]
  },
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   "id": "e9ad7a25",
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   "source": [
    "## First and second fundamental forms\n",
    "\n",
    "With\n",
    "\n",
    "$$\n",
    "I=E\\,dx^2+2F\\,dx\\,dy+G\\,dy^2,\n",
    "\\qquad\n",
    "II=L\\,dx^2+2M\\,dx\\,dy+N\\,dy^2,\n",
    "$$\n",
    "\n",
    "the coefficients are dot products of tangent vectors and derivatives of\n",
    "tangent vectors with the chosen normal.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "b7d717ed",
   "metadata": {},
   "outputs": [],
   "source": [
    "def E : MathValue := V.* vx vx\n",
    "def F : MathValue := V.* vx vy\n",
    "def G : MathValue := V.* vy vy\n",
    "\n",
    "def L : MathValue := V.* (∂/∂ vx x) normal\n",
    "def M : MathValue := V.* (∂/∂ vx y) normal\n",
    "def N : MathValue := V.* (∂/∂ vy y) normal\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "c0c97a16",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$(\\frac{\\partial f}{\\partial 1}^{2} + 1, \\frac{\\partial f}{\\partial 1} \\frac{\\partial f}{\\partial 2}, \\frac{\\partial f}{\\partial 2}^{2} + 1)$"
      ]
     },
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   ],
   "source": [
    "(E, F, G)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "08c7dbf4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$(\\sqrt{\\frac{\\partial f}{\\partial 2}^{2} + \\frac{\\partial f}{\\partial 1}^{2} + 1}^{-1} \\frac{\\partial^2 f}{\\partial 1^2}, \\sqrt{\\frac{\\partial f}{\\partial 2}^{2} + \\frac{\\partial f}{\\partial 1}^{2} + 1}^{-1} \\frac{\\partial^2 f}{\\partial 1 \\partial 2}, \\sqrt{\\frac{\\partial f}{\\partial 2}^{2} + \\frac{\\partial f}{\\partial 1}^{2} + 1}^{-1} \\frac{\\partial^2 f}{\\partial 2^2})$"
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   ],
   "source": [
    "(L, M, N)\n"
   ]
  },
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   "cell_type": "markdown",
   "id": "81e8100a",
   "metadata": {},
   "source": [
    "## Curvature\n",
    "\n",
    "The determinant and trace of the shape operator give\n",
    "\n",
    "$$\n",
    "K=\\frac{LN-M^2}{EG-F^2},\\qquad\n",
    "H=\\frac{EN-2FM+GL}{2(EG-F^2)}.\n",
    "$$\n",
    "\n",
    "These formulas apply wherever the graph chart is regular.  For a graph,\n",
    "$EG-F^2=1+f_x^2+f_y^2$ is always positive.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "5ac914cc",
   "metadata": {},
   "outputs": [],
   "source": [
    "def K : MathValue := (L * N - M^2) / (E * G - F^2)\n",
    "def H : MathValue := (E * N - 2 * F * M + G * L) / (2 * (E * G - F^2))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "c0fe438c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\frac{-\\frac{\\partial^2 f}{\\partial 1 \\partial 2}^{2} + \\frac{\\partial^2 f}{\\partial 1^2} \\frac{\\partial^2 f}{\\partial 2^2}}{\\frac{\\partial f}{\\partial 2}^{4} + \\frac{\\partial f}{\\partial 1}^{4} + 2 \\frac{\\partial f}{\\partial 1}^{2} \\frac{\\partial f}{\\partial 2}^{2} + 2 \\frac{\\partial f}{\\partial 2}^{2} + 2 \\frac{\\partial f}{\\partial 1}^{2} + 1}$"
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   "source": [
    "K\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "d7f9d8da",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\frac{\\sqrt{\\frac{\\partial f}{\\partial 2}^{2} + \\frac{\\partial f}{\\partial 1}^{2} + 1}^{-1} \\frac{\\partial^2 f}{\\partial 1^2} \\frac{\\partial f}{\\partial 2}^{2} + \\sqrt{\\frac{\\partial f}{\\partial 2}^{2} + \\frac{\\partial f}{\\partial 1}^{2} + 1}^{-1} \\frac{\\partial f}{\\partial 1}^{2} \\frac{\\partial^2 f}{\\partial 2^2} - 2 \\sqrt{\\frac{\\partial f}{\\partial 2}^{2} + \\frac{\\partial f}{\\partial 1}^{2} + 1}^{-1} \\frac{\\partial f}{\\partial 1} \\frac{\\partial^2 f}{\\partial 1 \\partial 2} \\frac{\\partial f}{\\partial 2} + \\sqrt{\\frac{\\partial f}{\\partial 2}^{2} + \\frac{\\partial f}{\\partial 1}^{2} + 1}^{-1} \\frac{\\partial^2 f}{\\partial 2^2} + \\sqrt{\\frac{\\partial f}{\\partial 2}^{2} + \\frac{\\partial f}{\\partial 1}^{2} + 1}^{-1} \\frac{\\partial^2 f}{\\partial 1^2}}{2 \\frac{\\partial f}{\\partial 2}^{2} + 2 \\frac{\\partial f}{\\partial 1}^{2} + 2}$"
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   "source": [
    "H\n"
   ]
  },
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   "id": "488fe260",
   "metadata": {},
   "source": [
    "## Interpretation\n",
    "\n",
    "$K$ is intrinsic: it can be recovered entirely from distances measured on\n",
    "the surface.  Positive, zero, and negative values correspond locally to\n",
    "elliptic, parabolic, and hyperbolic behavior.  $H$ is extrinsic and depends\n",
    "on the embedding and orientation; the equation $H=0$ is the minimal-surface\n",
    "equation for a graph.\n"
   ]
  }
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