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Showing content from https://developers.google.com/maps/documentation/javascript/examples/map-projection-simple below:

Custom Map Projections | Maps JavaScript API

TypeScript
// This example defines an image map type using the Gall-Peters
// projection.
// https://en.wikipedia.org/wiki/Gall%E2%80%93Peters_projection

function initMap(): void {
  // Create a map. Use the Gall-Peters map type.
  const map = new google.maps.Map(
    document.getElementById("map") as HTMLElement,
    {
      zoom: 0,
      center: { lat: 0, lng: 0 },
      mapTypeControl: false,
    }
  );

  initGallPeters();
  map.mapTypes.set("gallPeters", gallPetersMapType);
  map.setMapTypeId("gallPeters");

  // Show the lat and lng under the mouse cursor.
  const coordsDiv = document.getElementById("coords") as HTMLElement;

  map.controls[google.maps.ControlPosition.TOP_CENTER].push(coordsDiv);
  map.addListener("mousemove", (event: google.maps.MapMouseEvent) => {
    coordsDiv.textContent =
      "lat: " +
      Math.round(event.latLng!.lat()) +
      ", " +
      "lng: " +
      Math.round(event.latLng!.lng());
  });

  // Add some markers to the map.
  map.data.setStyle((feature) => {
    return {
      title: feature.getProperty("name") as string,
      optimized: false,
    };
  });
  map.data.addGeoJson(cities);
}

let gallPetersMapType;

function initGallPeters() {
  const GALL_PETERS_RANGE_X = 800;
  const GALL_PETERS_RANGE_Y = 512;

  // Fetch Gall-Peters tiles stored locally on our server.
  gallPetersMapType = new google.maps.ImageMapType({
    getTileUrl: function (coord, zoom) {
      const scale = 1 << zoom;

      // Wrap tiles horizontally.
      const x = ((coord.x % scale) + scale) % scale;

      // Don't wrap tiles vertically.
      const y = coord.y;

      if (y < 0 || y >= scale) return "";

      return (
        "https://developers.google.com/maps/documentation/" +
        "javascript/examples/full/images/gall-peters_" +
        zoom +
        "_" +
        x +
        "_" +
        y +
        ".png"
      );
    },
    tileSize: new google.maps.Size(GALL_PETERS_RANGE_X, GALL_PETERS_RANGE_Y),
    minZoom: 0,
    maxZoom: 1,
    name: "Gall-Peters",
  });

  // Describe the Gall-Peters projection used by these tiles.
  gallPetersMapType.projection = {
    fromLatLngToPoint: function (latLng) {
      const latRadians = (latLng.lat() * Math.PI) / 180;
      return new google.maps.Point(
        GALL_PETERS_RANGE_X * (0.5 + latLng.lng() / 360),
        GALL_PETERS_RANGE_Y * (0.5 - 0.5 * Math.sin(latRadians))
      );
    },
    fromPointToLatLng: function (point, noWrap) {
      const x = point.x / GALL_PETERS_RANGE_X;
      const y = Math.max(0, Math.min(1, point.y / GALL_PETERS_RANGE_Y));

      return new google.maps.LatLng(
        (Math.asin(1 - 2 * y) * 180) / Math.PI,
        -180 + 360 * x,
        noWrap
      );
    },
  };
}

// GeoJSON, describing the locations and names of some cities.
const cities = {
  type: "FeatureCollection",
  features: [
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [-87.65, 41.85] },
      properties: { name: "Chicago" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [-149.9, 61.218] },
      properties: { name: "Anchorage" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [-99.127, 19.427] },
      properties: { name: "Mexico City" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [-0.126, 51.5] },
      properties: { name: "London" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [28.045, -26.201] },
      properties: { name: "Johannesburg" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [15.322, -4.325] },
      properties: { name: "Kinshasa" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [151.207, -33.867] },
      properties: { name: "Sydney" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [0, 0] },
      properties: { name: "0°N 0°E" },
    },
  ],
};

declare global {
  interface Window {
    initMap: () => void;
  }
}
window.initMap = initMap;
Note: Read the guide on using TypeScript and Google Maps. JavaScript
// This example defines an image map type using the Gall-Peters
// projection.
// https://en.wikipedia.org/wiki/Gall%E2%80%93Peters_projection
function initMap() {
  // Create a map. Use the Gall-Peters map type.
  const map = new google.maps.Map(document.getElementById("map"), {
    zoom: 0,
    center: { lat: 0, lng: 0 },
    mapTypeControl: false,
  });

  initGallPeters();
  map.mapTypes.set("gallPeters", gallPetersMapType);
  map.setMapTypeId("gallPeters");

  // Show the lat and lng under the mouse cursor.
  const coordsDiv = document.getElementById("coords");

  map.controls[google.maps.ControlPosition.TOP_CENTER].push(coordsDiv);
  map.addListener("mousemove", (event) => {
    coordsDiv.textContent =
      "lat: " +
      Math.round(event.latLng.lat()) +
      ", " +
      "lng: " +
      Math.round(event.latLng.lng());
  });
  // Add some markers to the map.
  map.data.setStyle((feature) => {
    return {
      title: feature.getProperty("name"),
      optimized: false,
    };
  });
  map.data.addGeoJson(cities);
}

let gallPetersMapType;

function initGallPeters() {
  const GALL_PETERS_RANGE_X = 800;
  const GALL_PETERS_RANGE_Y = 512;

  // Fetch Gall-Peters tiles stored locally on our server.
  gallPetersMapType = new google.maps.ImageMapType({
    getTileUrl: function (coord, zoom) {
      const scale = 1 << zoom;
      // Wrap tiles horizontally.
      const x = ((coord.x % scale) + scale) % scale;
      // Don't wrap tiles vertically.
      const y = coord.y;

      if (y < 0 || y >= scale) return "";
      return (
        "https://developers.google.com/maps/documentation/" +
        "javascript/examples/full/images/gall-peters_" +
        zoom +
        "_" +
        x +
        "_" +
        y +
        ".png"
      );
    },
    tileSize: new google.maps.Size(GALL_PETERS_RANGE_X, GALL_PETERS_RANGE_Y),
    minZoom: 0,
    maxZoom: 1,
    name: "Gall-Peters",
  });
  // Describe the Gall-Peters projection used by these tiles.
  gallPetersMapType.projection = {
    fromLatLngToPoint: function (latLng) {
      const latRadians = (latLng.lat() * Math.PI) / 180;
      return new google.maps.Point(
        GALL_PETERS_RANGE_X * (0.5 + latLng.lng() / 360),
        GALL_PETERS_RANGE_Y * (0.5 - 0.5 * Math.sin(latRadians)),
      );
    },
    fromPointToLatLng: function (point, noWrap) {
      const x = point.x / GALL_PETERS_RANGE_X;
      const y = Math.max(0, Math.min(1, point.y / GALL_PETERS_RANGE_Y));
      return new google.maps.LatLng(
        (Math.asin(1 - 2 * y) * 180) / Math.PI,
        -180 + 360 * x,
        noWrap,
      );
    },
  };
}

// GeoJSON, describing the locations and names of some cities.
const cities = {
  type: "FeatureCollection",
  features: [
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [-87.65, 41.85] },
      properties: { name: "Chicago" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [-149.9, 61.218] },
      properties: { name: "Anchorage" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [-99.127, 19.427] },
      properties: { name: "Mexico City" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [-0.126, 51.5] },
      properties: { name: "London" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [28.045, -26.201] },
      properties: { name: "Johannesburg" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [15.322, -4.325] },
      properties: { name: "Kinshasa" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [151.207, -33.867] },
      properties: { name: "Sydney" },
    },
    {
      type: "Feature",
      geometry: { type: "Point", coordinates: [0, 0] },
      properties: { name: "0°N 0°E" },
    },
  ],
};

window.initMap = initMap;
Note: The JavaScript is compiled from the TypeScript snippet. CSS
/* 
 * Always set the map height explicitly to define the size of the div element
 * that contains the map. 
 */
#map {
  height: 100%;
}

/* 
 * Optional: Makes the sample page fill the window. 
 */
html,
body {
  height: 100%;
  margin: 0;
  padding: 0;
}

#coords {
  background-color: black;
  color: white;
  padding: 5px;
}
HTML
<html>
  <head>
    <title>Custom Map Projections</title>

    <link rel="stylesheet" type="text/css" href="./style.css" />
    <script type="module" src="./index.js"></script>
  </head>
  <body>
    <div id="map"></div>
    <div id="coords"></div>

    <!-- 
      The `defer` attribute causes the script to execute after the full HTML
      document has been parsed. For non-blocking uses, avoiding race conditions,
      and consistent behavior across browsers, consider loading using Promises. See
      https://developers.google.com/maps/documentation/javascript/load-maps-js-api
      for more information.
      -->
    <script
      src="https://maps.googleapis.com/maps/api/js?key=AIzaSyB41DRUbKWJHPxaFjMAwdrzWzbVKartNGg&callback=initMap&v=weekly"
      defer
    ></script>
  </body>
</html>

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