{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Web Mercator Tiles\n", "\n", "Tyler Erickson \n", "2024-09-09\n", "\n", "The internet is full of interactive tiled *slippy* web maps. These maps\n", "load tiles as needed in response to the user zooming and panning around,\n", "which allows users to explore extremely large datasets with minimal\n", "lagtime. This style of web map was first popularized by [Google Maps in\n", "2005](https://en.wikipedia.org/wiki/Google_Maps#2005%E2%80%932010), and\n", "since then the approach has been adopted by other mapping libraries,\n", "such as [Leaflet.js](https://leafletjs.com/),\n", "[OpenLayers](https://openlayers.org/),\n", "[Mapbox](https://docs.mapbox.com/api/maps/static-tiles/) and many\n", "others. These interactive maps utilize an [XYZ tiling\n", "scheme](https://en.wikipedia.org/wiki/Tiled_web_map#Defining_a_tiled_web_map),\n", "where Z represents the zoom level and X and Y are denote the tile\n", "indices.\n", "\n", "
\n", "\n", "
Figure 1: Overview of web mercator tiles zoom levels 0\n", "through 5 with an example tile for each zoom level. Source: File:XYZ\n", "Tiles.png. License CC0.\n", "If you prefer an interactive 2-D map, check out MapTiler’s Tiles\n", "à la Google Maps.
\n", "
\n", "\n", "These tiles use a [Web (pseudo)\n", "Mercator](https://en.wikipedia.org/wiki/Web_Mercator_projection)\n", "projection ([EPSG:3857](https://spatialreference.org/ref/epsg/3857/)),\n", "which is defined using a strange mix of ellipsoidal and spherical\n", "properties. If you care about positional accuracy, [use this projection\n", "at your own\n", "risk](https://web.archive.org/web/20170329065451/https://earth-info.nga.mil/GandG/wgs84/web_mercator/index.html).\n", "Also the projection strongly distorts areas. For example, compare how\n", "Greenland and Africa appear to be similar in size (2.2 million\n", "$\\text{km}^2$ and 30.4 million $\\text{km}^2$, respectively).\n", "\n", "But if you are creating an interactive map for the web, the projection\n", "can be very useful.\n", "\n", "# Mathematical Description\n", "\n", "The Web Mercator projection is\n", "[defined](https://en.wikipedia.org/wiki/Web_Mercator_projection#Formulas)\n", "by the following formulas. The tile’s $x$-index is a function of\n", "longitude ($\\lambda$) and zoom level ($z$):\n", "$$\n", "x = {\\frac {1}{2\\pi }}\\cdot 2^z\\left(\\pi +\\lambda \\right)\n", " \\qquad(1)$$ and the tile’s $y$-index is a function of latitude\n", "($\\varphi$): $$\n", "y = \\frac{1}{2\\pi } \\cdot 2^z\\left(\\pi -\\ln \\left[\\tan \\left({\\frac {\\pi }{4}} + \\frac{\\varphi }{2}\\right)\\right]\\right)\n", " \\qquad(2)$$\n", "\n", "Rearranging\n", "Equation 1 to solve\n", "for longitude ($\\lambda$) yields: $$\n", "\\lambda = \\frac{2\\pi \\cdot x}{2^z} - \\pi\n", " \\qquad(3)$$\n", "\n", "and rearranging\n", "Equation 2 to solve\n", "for latitude ($\\varphi$) yields: $$\n", "\\varphi = 2 \\cdot \\arctan(e^{\\pi - \\frac{2\\pi \\cdot y}{2^z}}) - {\\frac {\\pi }{2}}\n", " \\qquad(4)$$\n", "\n", "# Python Implementation\n", "\n", "In Python those equations can be written as:" ], "id": "3c681263-fed0-45a2-a804-930af64faeac" }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import numpy.typing as npt\n", "\n", "def xyz_to_lonlat(\n", " x: npt.NDArray[np.floating], # horizontal index\n", " y: npt.NDArray[np.floating], # vertical index\n", " z: int, # zoom level\n", "):\n", " lon = 2 * np.pi * x / 2**z - np.pi\n", " lat = 2 * np.arctan(np.exp(np.pi - (2 * np.pi * y / 2**z))) - np.pi / 2\n", " return (lon * 180 / np.pi, lat * 180 / np.pi)" ], "id": "cell-6" }, { "cell_type": "markdown", "metadata": {}, "source": [ "To understand how tiles differ between levels and across the Earth, we\n", "create a GeoDataFrame object containing the tile information." ], "id": "4011b7cb-0618-401d-979b-92aa35100d3b" }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "import geopandas as gpd\n", "import shapely\n", "\n", "zoom_levels = [0, 1, 2, 3, 4]\n", "\n", "# for layer in layer_config_list:\n", "tile_data_list = []\n", "for zoom in zoom_levels:\n", " tilerange = range(2**zoom + 1)\n", "\n", " # Create arrays of cell vertices.\n", " x_vertices = np.array(tilerange)\n", " y_vertices = np.array(tilerange)\n", "\n", " # Convert vertices to lon/lat coordinates.\n", " lons, lats = xyz_to_lonlat(x_vertices, y_vertices, zoom)\n", "\n", " for ix in tilerange[:-1]:\n", " for iy in tilerange[:-1]:\n", " poly_coords = [\n", " (lons[ix], lats[iy]),\n", " (lons[ix + 1], lats[iy]),\n", " (lons[ix + 1], lats[iy + 1]),\n", " (lons[ix], lats[iy + 1])\n", " ]\n", " # print('poly_coords', poly_coords)\n", "\n", " tile_data = {\n", " 'zoom': zoom,\n", " 'x_index': ix,\n", " 'y_index': iy,\n", " 'geometry': shapely.Polygon(poly_coords),\n", " }\n", " tile_data_list.append(tile_data)\n", "\n", "gdf = gpd.GeoDataFrame(\n", " data=tile_data_list,\n", " crs=\"EPSG:4326\"\n", ")\n" ], "id": "cell-8" }, { "cell_type": "markdown", "metadata": {}, "source": [ "Based on code described in [this earlier\n", "post](../on-areas-on-earth/index.ipynb), we can calculate the area\n", "covered by each tile." ], "id": "2bb34e99-db99-4650-91e4-b4ea096c290a" }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "import math\n", "\n", "def patch_area_single_rectangle(\n", " R: float, # Authalic radius, in meters\n", " lon1: float, # Minimum longitude, in degrees\n", " lat1: float, # Minimum latitude, in degrees\n", " lon2: float, # Maximum longitude, in degrees\n", " lat2: float, # Maximum latitude, in degrees\n", ") -> float:\n", " # Convert from degrees to radians\n", " lon1_rad = lon1 * (np.pi / 180)\n", " lon2_rad = lon2 * (np.pi / 180)\n", " lat1_rad = lat1 * (np.pi / 180)\n", " lat2_rad = lat2 * (np.pi / 180)\n", " \n", " h = R * (math.sin(lat2_rad) - math.sin(lat1_rad))\n", " area_patch = R * h * (lon2_rad - lon1_rad)\n", " return area_patch\n", "\n", "gdf['area_km2'] = gdf.apply(\n", " lambda row:patch_area_single_rectangle(\n", " 6371007.180918, # Authalic radius, in meters\n", " *row['geometry'].bounds, # tuple of (minx, miny, maxx, maxy)\n", " ) * 1e-6,\n", " axis=1\n", ")" ], "id": "cell-10" }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let’s see a summary of this tile dataset." ], "id": "2b0afd69-f248-4d6e-8f79-43a775978bad" }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "output_type": "display_data", "metadata": {}, "data": { "text/html": [ "\n", "

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\n", "" ] } } ], "source": [ "gdf" ], "id": "cell-12" }, { "cell_type": "markdown", "metadata": {}, "source": [ "Among other things, the data table shows that tile areas vary by y_index\n", "and zoom level. But it is difficult to grasp how these values change\n", "around the Earth from a table.\n", "\n", "# Tile Visualization\n", "\n", "This section creates visualizations of Web Mercator tiles for various\n", "zoom levels. We start by formatting the tile data so that can be used by\n", "[PyDeck](https://deckgl.readthedocs.io/)." ], "id": "5b4aaa86-55bf-4dfd-93ae-6bdb3ed0d305" }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "tile_colors = [\n", " [0, 0, 0],\n", " [255, 255, 255],\n", "]\n", "\n", "def add_pydeck_colums(gdf):\n", " \"\"\"Add columns for PyDeck visualization\"\"\"\n", " \n", " # Convert geodataframe geometries into a PyDeck compatible format\n", " gdf['pydeck_polygon'] = [\n", " [\n", " vertex + (200000, ) # Add a third dimension\n", " for vertex in list(geom.exterior.coords)\n", " ]\n", " for geom in gdf['geometry'].geometry\n", " ]\n", " gdf['pydeck_centroid'] = [\n", " list(geom.centroid.coords[0]) + [800000] # Add a third dimension\n", " for geom in gdf['geometry'].geometry\n", " ]\n", " gdf['centroid_label'] = '(' + gdf['x_index'].astype(str) + ',' + gdf['y_index'].astype(str) + ')'\n", " gdf['bounds'] = [\n", " [f'{x:.2f}' for x in geom.bounds]\n", " for geom in gdf['geometry'].geometry\n", " ]\n", "\n", " gdf['fill_color'] = gdf.apply(\n", " lambda row: tile_colors[(row['x_index'] + row['y_index']) % 2],\n", " axis=1\n", " )\n", "\n", " gdf['area_desc'] = gdf.apply(\n", " lambda row: f\"{row['area_km2']:.2e}\", \n", " axis=1\n", " )\n", "\n", " return gdf\n", "\n", "gdf = add_pydeck_colums(gdf)" ], "id": "cell-15" }, { "cell_type": "markdown", "metadata": {}, "source": [ "We then create a function that can generate PyDeck objects for a\n", "specified zoom level." ], "id": "a3aab3e3-1141-4a5c-ad72-31915428cd5d" }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "import pydeck as pdk\n", "from pydeck.types import String\n", "import pandas as pd\n", "\n", "def get_globe_view():\n", " \"\"\"Return a pyDeck view for the globe.\"\"\"\n", " view = pdk.View(\n", " type=\"_GlobeView\",\n", " controller=True,\n", " width='100%',\n", " )\n", " return view\n", "\n", "def get_initial_view_state():\n", " \"\"\"Specify the initial pyDeck view state.\"\"\"\n", " initial_view_state = pdk.ViewState(\n", " latitude=45,\n", " longitude=-45,\n", " zoom=0,\n", " min_zoom=0,\n", " )\n", " return initial_view_state\n", "\n", "def get_basemap_layer():\n", " \"\"\"Get a pyDeck layer based on Natural Earth country boundaries.\"\"\"\n", " basemap_layer = pdk.Layer(\n", " \"GeoJsonLayer\",\n", " id=\"base-map\",\n", " data=\"https://d2ad6b4ur7yvpq.cloudfront.net/naturalearth-3.3.0/ne_50m_admin_0_scale_rank.geojson\",\n", " stroked=False,\n", " filled=True,\n", " extruded=False,\n", " get_fill_color=[200, 200, 200],\n", " get_line_color=[255, 255, 255],\n", " opacity=0.5,\n", " )\n", " return basemap_layer\n", "\n", "def get_tiles(zoom):\n", " \"\"\"Create a pyDeck polygon and text layers for tiles of a specified zoom level.\"\"\"\n", " poly_layer = pdk.Layer(\n", " \"SolidPolygonLayer\",\n", " gdf[gdf['zoom']==zoom],\n", " opacity=0.25,\n", " visible=True,\n", " get_polygon=\"pydeck_polygon\",\n", " get_fill_color=\"fill_color\",\n", " extruded=True,\n", " )\n", "\n", " text_layer = pdk.Layer(\n", " \"TextLayer\",\n", " gdf[gdf['zoom']==zoom],\n", " get_position=\"pydeck_centroid\",\n", " get_text=\"centroid_label\",\n", " get_size=12,\n", " # get_color=[127, 255, 255],\n", " get_color=[255, 255, 0],\n", " get_angle=0,\n", " get_text_anchor=String(\"middle\"),\n", " get_alignment_baseline=String(\"center\"),\n", " pickable=True,\n", " extruded=True,\n", " )\n", " return [poly_layer, text_layer]\n", "\n", "\n", "def create_globe_with_xyz_tiles(zoom):\n", " \n", " # gdf = layer_config['gdf']\n", " deck = pdk.Deck(\n", " views=[get_globe_view()],\n", " initial_view_state=get_initial_view_state(),\n", " layers=[\n", " get_basemap_layer(),\n", " get_tiles(zoom),\n", " ],\n", " map_provider=None,\n", " tooltip={\n", " \"html\": (\n", " \"\"\n", " \"\"\n", " \"\"\n", " \"\"\n", " \"\"\n", " \"\"\n", " \"
Zoom:{zoom}
X, Y:{x_index}, {y_index}
Area, km2:{area_desc}
Longitude:{bounds.0}° to {bounds.2}°
Latitude:{bounds.1}° to {bounds.3}°
\"\n", " ),\n", " \"style\": {\n", " \"text-align\": \"right\",\n", " }\n", " },\n", " # Note: set to True for the globe to be opaque\n", " parameters={\"cull\": True},\n", " )\n", " return deck" ], "id": "cell-17" }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following figures show how the Web Mercator tiles are distributed on\n", "the globe for zoom levels 1 through 4. Viewing Web Mercator tiles this\n", "way clearly shows that the tiles do not reach the poles, regardless of\n", "the zoom level. Rather the tiles end at latitudes $85.051\\degree$ N and\n", "$85.051\\degree$ S[1].\n", "\n", "## Tiles - Zoom Level 1\n", "\n", "For zoom level 1, the `x` and `y` tile indices range from 0 to 1[2].\n", "There are a total of 4 tiles[3].\n", "\n", "[1] $\\varphi_{max} = 2 \\cdot \\arctan(e^{\\pi}) - {\\frac {\\pi }{2}} = 85.051\\degree$\n", "(which is\n", "Equation 4\n", "simplified with $y=0$)\n", "\n", "[2] $z=1; \\ \\ 2^z - 1 = 1$\n", "\n", "[3] $z=1; \\ \\ 4^z = 4$ tiles" ], "id": "e0aaca93-c611-4b11-80c3-4831c845436e" }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "output_type": "display_data", "metadata": {}, "data": { "text/html": [ "\n", " \n", " " ] } } ], "source": [ "create_globe_with_xyz_tiles(1)" ], "id": "cell-20" }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Tiles - Zoom Level 2\n", "\n", "For zoom level 2, the `x` and `y` tile indices range from 0 to 3[1].\n", "There are a total of 16 tiles[2].\n", "\n", "[1] $z=2; \\ \\ 2^z - 1 = 3$\n", "\n", "[2] $z=2; \\ \\ 4^z = 16$ tiles" ], "id": "1ad9f935-62eb-4325-a027-5146c6c5ada8" }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "output_type": "display_data", "metadata": {}, "data": { "text/html": [ "\n", " \n", " " ] } } ], "source": [ "create_globe_with_xyz_tiles(2)" ], "id": "cell-22" }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Tiles - Zoom Level 3\n", "\n", "For zoom level 3, the `x` and `y` tile indices range from 0 to 7[1].\n", "There are a total of 64 tiles[2].\n", "\n", "[1] $z=3; \\ \\ 2^z - 1 = 7$\n", "\n", "[2] $z=3; \\ \\ 4^z = 64$ tiles" ], "id": "aafca418-2141-475f-83d6-03bdd8d3df94" }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "output_type": "display_data", "metadata": {}, "data": { "text/html": [ "\n", " \n", " " ] } } ], "source": [ "create_globe_with_xyz_tiles(3)" ], "id": "cell-24" }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Tiles - Zoom Level 4\n", "\n", "For zoom level 4, the `x` and `y` tile indices range from 0 to 15[1].\n", "There are a total of 256 tiles[2].\n", "\n", "[1] $z=4; \\ \\ 2^z - 1 = 15$\n", "\n", "[2] $z=4; \\ \\ 4^z = 256$ tiles" ], "id": "bd093aee-05cc-4913-874b-c526ef46939f" }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "output_type": "display_data", "metadata": {}, "data": { "text/html": [ "\n", " \n", " " ] } } ], "source": [ "create_globe_with_xyz_tiles(4)" ], "id": "cell-26" }, { "cell_type": "markdown", "metadata": {}, "source": [ "The preceding globes for level 2 and higher show how web mercator tiles\n", "can distort the appearent size of regions at different latitudes. Roll\n", "your cursor over the tile numbers for a quantitative comparison." ], "id": "19abcdc8-edea-4cdb-a3b6-1cd20a584244" } ], "nbformat": 4, "nbformat_minor": 5, "metadata": { "kernelspec": { "name": "python3", "display_name": "default", "language": "python" }, "language_info": { "name": "python", "codemirror_mode": { "name": "ipython", "version": "3" }, "file_extension": ".py", "mimetype": "text/x-python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.9" }, "widgets": { "application/vnd.jupyter.widget-state+json": { "state": { "06702ea9a3fd4399a0e956248d55197f": { "model_module": "@jupyter-widgets/base", "model_module_version": "2.0.0", "model_name": "LayoutModel", "state": { "_model_module": "@jupyter-widgets/base", "_model_module_version": "2.0.0", "_model_name": "LayoutModel", "_view_count": null, "_view_module": "@jupyter-widgets/base", "_view_module_version": "2.0.0", "_view_name": "LayoutView", "align_content": null, "align_items": null, 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