{
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   "source": [
    "# Symmetry Rendering\n",
    "\n",
    "When solving on a reduced domain with symmetry boundary conditions, the `Symmetry` class renders the full solution by applying mirror transformations on the GPU."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b2c3d4e5",
   "metadata": {},
   "source": [
    "## Mirror Symmetry\n",
    "\n",
    "Render a quarter-domain solution as a full domain:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c3d4e5f6",
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   "outputs": [],
   "source": [
    "from ngsolve import *\n",
    "from ngsolve_webgpu import MeshData, FunctionData, CFRenderer, Symmetry\n",
    "from webgpu.jupyter import Draw\n",
    "\n",
    "# Solve on a quarter of the domain\n",
    "mesh = Mesh(unit_square.GenerateMesh(maxh=0.1))\n",
    "cf = sin(3.14159 * x) * sin(3.14159 * y)\n",
    "\n",
    "md = MeshData(mesh)\n",
    "fd = FunctionData(md, cf, order=3)\n",
    "\n",
    "sym = Symmetry()\n",
    "sym.mirror_x()  # reflect across x=0 -> 2 copies\n",
    "sym.mirror_y()  # reflect across y=0 -> 4 copies (group closure)\n",
    "\n",
    "cfr = CFRenderer(fd, symmetry=sym)\n",
    "Draw(cfr)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d4e5f6a7",
   "metadata": {},
   "source": [
    "Each call to `mirror_x/y/z` doubles the number of rendered copies by computing the group closure of all generator combinations. The transforms are applied in the vertex shader — no mesh duplication needed."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e5f6a7b8",
   "metadata": {},
   "source": [
    "## Antisymmetric Fields (Parity)\n",
    "\n",
    "For fields that change sign under a mirror operation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f6a7b8c9",
   "metadata": {},
   "outputs": [],
   "source": [
    "from ngsolve import *\n",
    "from ngsolve_webgpu import MeshData, FunctionData, CFRenderer, Symmetry, Colorbar\n",
    "from webgpu.jupyter import Draw\n",
    "\n",
    "mesh = Mesh(unit_square.GenerateMesh(maxh=0.1))\n",
    "cf = x * sin(3.14159 * y)  # odd in x\n",
    "\n",
    "md = MeshData(mesh)\n",
    "fd = FunctionData(md, cf, order=3)\n",
    "\n",
    "sym = Symmetry()\n",
    "sym.mirror_x()\n",
    "sym_odd = sym.with_parity([-1])  # field is antisymmetric under mirror_x\n",
    "cfr = CFRenderer(fd, symmetry=sym_odd)\n",
    "cfr.colormap.set_min(-1)\n",
    "colorbar = Colorbar(cfr.colormap)\n",
    "Draw([cfr, colorbar])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a7b8c9d0",
   "metadata": {},
   "source": [
    "`with_parity([-1])` tells the renderer to negate function values for copies involving that mirror generator. The parity list has one entry per generator in the order they were added."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "vec1",
   "metadata": {},
   "source": [
    "## Vector Symmetry\n",
    "\n",
    "Arrow glyphs (`SurfaceVectors`, `ClippingVectors`) support the same symmetry system. Positions and directions are expanded on the GPU via a compute shader — no CPU overhead."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "vec2",
   "metadata": {},
   "outputs": [],
   "source": [
    "from ngsolve import *\n",
    "from ngsolve_webgpu import MeshData, FunctionData, SurfaceVectors, Symmetry\n",
    "from webgpu.jupyter import Draw\n",
    "\n",
    "mesh = Mesh(unit_cube.GenerateMesh(maxh=0.3))\n",
    "cf = CF((x, y, 0))\n",
    "\n",
    "md = MeshData(mesh)\n",
    "fd = FunctionData(md, cf, order=1)\n",
    "\n",
    "sym = Symmetry()\n",
    "sym.mirror_x()\n",
    "sym.mirror_y()\n",
    "\n",
    "vecs = SurfaceVectors(fd, symmetry=sym)\n",
    "Draw(vecs)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "vec3",
   "metadata": {},
   "source": [
    "## Electromagnetic Symmetry (Polar vs Axial Vectors)\n",
    "\n",
    "For electromagnetic simulations, vectors transform differently depending on whether they are **polar** (E-field, displacement) or **axial/pseudovectors** (B-field, H-field):\n",
    "\n",
    "| `vector_symmetry` | Transform | Use case |\n",
    "|---|---|---|\n",
    "| `\"polar\"` (default) | d' = M · d | E-field with PMC wall, B-field with PEC wall |\n",
    "| `\"axial\"` | d' = det(M) · M · d | E-field with PEC wall, B-field with PMC wall |\n",
    "\n",
    "```python\n",
    "# B-field mirrored at a PMC boundary:\n",
    "vecs_B = SurfaceVectors(fd_B, symmetry=sym, vector_symmetry=\"axial\")\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "vec4",
   "metadata": {},
   "source": [
    "## API Reference\n",
    "\n",
    "**Symmetry()**\n",
    "- `.mirror_x()`, `.mirror_y()`, `.mirror_z()` — add mirror generators (chainable)\n",
    "- `.with_parity(list)` — returns new Symmetry with per-generator sign flips for scalar fields\n",
    "- `.n_copies` — total number of symmetry copies (read-only)\n",
    "\n",
    "**Scalar renderers** (`CFRenderer`, `ClippingCF`, `MeshElements2d`, `MeshElements3d`, `IsoSurfaceRenderer`, `NegativeSurfaceRenderer`, `NegativeClippingRenderer`, `GeometryRenderer`):\n",
    "- Pass `symmetry=sym` to constructor\n",
    "\n",
    "**Vector renderers** (`SurfaceVectors`, `ClippingVectors`):\n",
    "- Pass `symmetry=sym` to constructor\n",
    "- Pass `vector_symmetry=\"axial\"` for pseudovectors (B-field at PMC, E-field at PEC)"
   ]
  }
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