Handling Pole Distortion in Spherical Mapping

The distortion you're experiencing at the poles is a common issue when mapping 2D tiles onto a sphere. This is due to the fundamental challenge of projecting a flat surface onto a spherical one, especially near the poles where the meridians converge.

Understanding the Problem

When using standard spherical mapping, the rectangular tiles get increasingly distorted as they approach the poles because a single tile needs to cover a smaller and smaller area of longitude. This is visible in your image where the tiles appear stretched horizontally near the poles.

Potential Solutions

1. Custom UV Mapping

Instead of using the built-in SphereGeometry with partial segments, consider creating a complete sphere and then applying custom UV coordinates for each tile:

JAVASCRIPTCopied
// Create a full sphere
const fullSphere = new THREE.SphereGeometry(2, 64, 64);
const geometry = fullSphere.clone();

// Calculate UV coordinates for the specific tile
const [zoom, x, y] = tileKey.split('/').map(Number);
const tilesPerRow = Math.pow(2, zoom);

// Modify the UV coordinates of the geometry
const uvs = geometry.attributes.uv;
for (let i = 0; i < uvs.count; i++) {
  const u = uvs.getX(i);
  const v = uvs.getY(i);
  
  // Map to the specific tile
  uvs.setXY(i, 
    uOffset + u * uvScale,
    vOffset + v * uvScale
  );
}

2. Use a Different Projection Method

For planetary mapping, consider using a different projection method that's better suited for spherical representation:

JAVASCRIPTCopied
// Create a custom geometry with adjusted vertices
const geometry = new THREE.BufferGeometry();
const vertices = [];
const uvs = [];

// Generate vertices with a projection that reduces pole distortion
// This would involve mathematical transformations based on latitude/longitude
// ...

geometry.setAttribute('position', new THREE.Float32BufferAttribute(vertices, 3));
geometry.setAttribute('uv', new THREE.Float32BufferAttribute(uvs, 2));

3. Adaptive Mesh Resolution

Increase the mesh resolution near the poles to better handle the distortion:

JAVASCRIPTCopied
// Create a sphere with more segments near the poles
const geometry = new THREE.SphereGeometry(
  2,
  64,  // horizontal segments
  Math.max(64, 128 * Math.abs(Math.cos(vOffset * Math.PI + uvScale * Math.PI/2))), // more segments near poles
  uOffset * Math.PI * 2,
  uvScale * Math.PI * 2,
  vOffset * Math.PI,
  uvScale * Math.PI
);

4. Use Specialized Libraries

Consider using specialized libraries for planetary mapping that handle these issues, such as:

• Cesium.js (though it's more focused on Earth)
• Custom implementations of map projections like the Quadrilateralized Spherical Cube

Recommended Approach

The most effective approach would be to use a combination of:

1. A complete sphere geometry
2. Custom UV mapping that accounts for the spherical distortion
3. Potentially pre-processing your tile images to counter-distort them before mapping

This would involve more complex mathematics to properly map the tiles, but would result in a more accurate representation, especially at the poles.

For the mathematical approach, you would need to implement a projection that accounts for the convergence of meridians at the poles, such as an equal-area projection or a modified Mercator projection that's adjusted for your specific use case.