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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "76430a19-e222-4c2c-9e4e-743641a5e12e",
+   "metadata": {},
+   "source": [
+    "# Equivalence of various equiangular samplings\n",
+    "\n",
+    "In this demo we show the equivalence between Fejer/Clenshaw-Curtis sampling and other equiangular sampling schemes, including MW, MWSS and DH."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "be9189dc-7827-416a-9340-fb66fbc390e1",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "JAX is not using 64-bit precision. This will dramatically affect numerical precision at even moderate L.\n"
+     ]
+    }
+   ],
+   "source": [
+    "import s2fft \n",
+    "import numpy as np"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "81b92ac3-36e8-40fc-93f6-45394eb673dd",
+   "metadata": {},
+   "source": [
+    "Set band-limit for tests."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "463a7044-99c3-45e6-bce2-3b3c7dd0e259",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "L = 4"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "f59ae250-b286-4a9a-8a45-32098fa134a1",
+   "metadata": {},
+   "source": [
+    "## Equivalence of MW and Fejer type 1 sampling"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "8559d609-67d5-4561-bb43-2de124f49e25",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "thetas_mw = s2fft.sampling.s2_samples.thetas(L, sampling=\"mw\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "bed1f19f-d015-4284-ade6-eeb8954d34b9",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([0.44879895, 1.34639685, 2.24399475, 3.14159265])"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "thetas_mw"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "ad22a641-3940-489e-98f5-315f0ca4c5e6",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n = L - 1/2"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "de96933f-6b89-4796-bdcd-1eb4959a1a55",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "thetas_mw_fejer1 = (np.arange(0, n) + 0.5) * np.pi / n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "974e4e4d-0753-45da-8eab-ecf0cf21f501",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([0.44879895, 1.34639685, 2.24399475, 3.14159265])"
+      ]
+     },
+     "execution_count": 7,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "thetas_mw_fejer1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "01fd0a9d-180d-4275-b4bb-4e269aa1d466",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "assert np.allclose(thetas_mw, thetas_mw_fejer1)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "0f13b08a-3df9-4999-a1a0-f022b36c7aa8",
+   "metadata": {},
+   "source": [
+    "## Equivalence of MWSS and Clenshaw-Curtis (Fejer type 2 with end-points) sampling "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "3df7cacf-90c1-44c5-9d84-4c0164b5df4e",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([0.        , 0.78539816, 1.57079633, 2.35619449, 3.14159265])"
+      ]
+     },
+     "execution_count": 9,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "thetas_mwss = s2fft.sampling.s2_samples.thetas(L, sampling=\"mwss\")\n",
+    "thetas_mwss"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "id": "c7e7af1f-d1dd-4d32-bbaf-3bf61f6026b1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n = L"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "5334dba6-f247-45ea-8340-9e13d1321ab9",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "thetas_mwss_fejer2 = np.arange(1, n) * np.pi / n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "id": "c8dbc543-71a7-4b4d-b8d0-3ca1af2fb360",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([0.78539816, 1.57079633, 2.35619449])"
+      ]
+     },
+     "execution_count": 12,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "thetas_mwss_fejer2"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "ae69b19f-c6cb-4f94-a591-946ab9c235fa",
+   "metadata": {},
+   "source": [
+    "Note that Fejer tyle 2 sampling does not include end-points."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "id": "6484fd92-7c6c-45dd-87c6-170c2cd94699",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "assert np.allclose(thetas_mwss[1:-1], thetas_mwss_fejer2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "c80ef1dd-1b41-401c-a506-95da1e842bc1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "thetas_mwss_cc = np.arange(0, n + 1) * np.pi / n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "id": "2c4b38b9-8e40-4050-8942-77cc9dbc2b52",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([0.        , 0.78539816, 1.57079633, 2.35619449, 3.14159265])"
+      ]
+     },
+     "execution_count": 15,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "thetas_mwss_cc"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "id": "c5818c01-bcb5-4df6-9a94-097af75c504f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "assert np.allclose(thetas_mwss, thetas_mwss_cc)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "303b49fa-ceff-4f36-99fd-adc714d93140",
+   "metadata": {},
+   "source": [
+    "## Equivalence of DH and Fejer type 1 sampling"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "d412cec5-47e1-4146-955b-3aed62318647",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([0.19634954, 0.58904862, 0.9817477 , 1.37444679, 1.76714587,\n",
+       "       2.15984495, 2.55254403, 2.94524311])"
+      ]
+     },
+     "execution_count": 17,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "thetas_dh = s2fft.sampling.s2_samples.thetas(L, sampling=\"dh\")\n",
+    "thetas_dh"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "id": "19336652-2ccb-48d5-9bbe-1de0b352d5d7",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n = 2 * L"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "id": "c58e6bbe-4892-46d3-8478-747d282630a0",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "thetas_dh_fejer1 = (np.arange(0, n) + 0.5) * np.pi / n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "id": "383a53ad-0fbe-40e2-8c4f-fdb3d630596c",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "array([0.19634954, 0.58904862, 0.9817477 , 1.37444679, 1.76714587,\n",
+       "       2.15984495, 2.55254403, 2.94524311])"
+      ]
+     },
+     "execution_count": 20,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "thetas_dh_fejer1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "id": "ed2f9425-bf04-45b6-bb94-1c5546dd1417",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "assert np.allclose(thetas_dh, thetas_dh_fejer1)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "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": "3.11.11"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}