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    "# Parametric Surfaces - All About\n",
    "\n",
    "This document describes how parametric surfaces are implemented in OSCARS.\n",
    "\n",
    "A parametric surface in OSCARS is represented by a python class.  This class must contain the variables *u* and *v*, start and stop points *ustart*, *ustop*, *vstart*, *vstop*, and the number of points in each dimension *nu*, *nv*.  Later in this notebook we develop a custom surface.  First we start with surfaces in OSCARS."
   ]
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   "cell_type": "code",
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   "source": [
    "# matplotlib plots inline\n",
    "%matplotlib inline\n",
    "\n",
    "# Import the OSCARS SR module\n",
    "import oscars.sr\n",
    "\n",
    "# Import OSCARS 3D tools (matplotlib)\n",
    "from oscars.plots3d_mpl import *\n",
    "\n",
    "# Import OSCARS parametric surfaces\n",
    "from oscars.parametric_surfaces import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Plot the rectangle\n",
    "rectangle = PSRectangle(L=0.01, W=0.01, nu=51, nv=51)\n",
    "plot_surface(rectangle)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Plot the sphere\n",
    "sphere = PSSphere(R=0.010, nu=51, nv=51)\n",
    "plot_surface(sphere)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Plot the cylinder\n",
    "cylinder = PSCylinder(R=0.010, L=0.1, nu=51, nv=51)\n",
    "plot_surface(cylinder)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Plot the torus\n",
    "torus = PSTorus(R=0.010, r=0.005, nu=51, nv=51)\n",
    "plot_surface(torus)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Eample - Custom Parametric Surface in OSCARS \n",
    "\n",
    "The example that follows shows how to create a custom parametric surface.  The parametric surface is created in a simple python class.  OSCARS requires the class to have the following defined:\n",
    "\n",
    "Functions: position(u, v), normal(u, v) which each return a 3-element list [x, y, z]\n",
    "\n",
    "Integers: nu, nv\n",
    "\n",
    "Floats: ustart, ustop, vstart, vstop\n",
    "\n",
    "At the moment you must provide the normal.  You can use the following recipe for the components of the normal:\n",
    "\n",
    "In the following, let x = x(u, v) = position(u, v)[0], and so on\n",
    "\n",
    "$$\n",
    "N_x = \\frac{\\partial y}{\\partial u}\\frac{\\partial z}{\\partial v} - \\frac{\\partial z}{\\partial u}\\frac{\\partial y}{\\partial v}\\\\\n",
    "N_y = \\frac{\\partial z}{\\partial u}\\frac{\\partial x}{\\partial v} - \\frac{\\partial x}{\\partial u}\\frac{\\partial z}{\\partial v}\\\\\n",
    "N_y = \\frac{\\partial x}{\\partial u}\\frac{\\partial y}{\\partial v} - \\frac{\\partial y}{\\partial u}\\frac{\\partial x}{\\partial v}\n",
    "$$\n",
    "\n",
    "*You* need to decide if you want to invert this or not since it's not clear which is the front and back side.  It is also a good idea to normalize this.\n",
    "\n",
    "The position in this example is given by the following:\n",
    "$$\n",
    "x = u \\sin(v)\\\\\n",
    "y = u \\cos(v)\\\\\n",
    "z = A \\sin( 2 \\pi u / \\lambda )\n",
    "$$\n",
    "\n",
    "which gives a normal of:\n",
    "$$\n",
    "N_x = 2 A \\pi u \\cos(2 \\pi u / \\lambda) \\sin(v) / \\lambda\\\\\n",
    "N_y = 2 A \\pi u \\cos(2 \\pi u / \\lambda) \\cos(v) / \\lambda\\\\\n",
    "N_z = -u\n",
    "$$\n",
    "and in this example I did have to invert the normal vector\n"
   ]
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   "source": [
    "from math import sin, cos, pi, sqrt, exp\n",
    "class PSWavyDisk:\n",
    "    \"\"\"A Parametric surface - wavy disk with a hole in the center\"\"\"\n",
    "\n",
    "    # This shape specific parameters\n",
    "    R1 = 0.001\n",
    "    R2 = 0.005\n",
    "    A = 0.001\n",
    "    Lambda = 0.01\n",
    "\n",
    "    # Required for all PS shapes\n",
    "    # Start, stop, and number of points for the u and v parameters\n",
    "    # All PSShapes must have these defined\n",
    "    ustart = R1\n",
    "    ustop  = R2\n",
    "    vstart = 0\n",
    "    vstop  = 2 * pi\n",
    "    nu = 21\n",
    "    nv = 21\n",
    "\n",
    "    def __init__ (self, R1=0.001, R2=0.005, nu=21, nv=21):\n",
    "        self.R1 = R1\n",
    "        self.R2 = R2\n",
    "        self.ustart = R1\n",
    "        self.ustop  = R2\n",
    "        self.nu = nu\n",
    "        self.vstart = 0\n",
    "        self.vstop = 2.*pi\n",
    "        self.nv = nv\n",
    "\n",
    "\n",
    "    def position (self, u, v):\n",
    "        \"\"\"Return the position in 3D at this u and v\"\"\"\n",
    "\n",
    "        x = u * sin(v)\n",
    "        y = u * cos(v)\n",
    "        z = self.A * sin( 2. * pi * u / self.Lambda )\n",
    "\n",
    "        return [x, y, z]\n",
    "\n",
    "\n",
    "    def normal (self, u, v):\n",
    "        \"\"\"Return a unit normal in 3D at this u and v position\"\"\"\n",
    "\n",
    "        xn = -2. * self.A * pi * u * cos(2. * pi * u / self.Lambda) * sin(v) / self.Lambda\n",
    "        yn = -2. * self.A * pi * u * cos(2. * pi * u / self.Lambda) * cos(v) / self.Lambda\n",
    "        zn = u\n",
    "\n",
    "        mag = sqrt(xn*xn + yn*yn + zn*zn)\n",
    "\n",
    "        return [xn / mag, yn / mag, zn / mag]"
   ]
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   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "# Create a parametric shape and plot it\n",
    "wavydisk = PSWavyDisk(R1=0.01, R2=0.04, nu=51, nv=101)\n",
    "plot_surface(wavydisk, zlim=[-0.01, 0.01])"
   ]
  },
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   "execution_count": null,
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