Changing the way three.js projects a texture onto a halfsphere

Question

I am currently trying to project an image onto the inside of a halfsphere in a three.js project. The half sphere is created via

const geometry = new THREE.SphereGeometry(Component.radius, this.resolution, this.resolution,
        Math.PI, Math.PI, 0, Math.PI);
this.material = new THREE.MeshStandardMaterial({color: 0xffffff});
this.material.side = THREE.DoubleSide;

this.sphere = new THREE.Mesh(geometry, this.material);

// The image of the texture is set later dynamically via 
// this.material.map = textureLoader.load(...); 

With radius and resolution being constants. This works fine, except for one issue: The image becomes distorted around the "top" and "bottom" of the sphere, like this: Simple example of distorted texture:

The texture originally had the value "0,0" in the bottom left and "0,1" in the bottom right, and with the camera facing down from the center of the demisphere the bottom left and bottom right corner are squished onto the "bottom" point of the sphere.

I want to change this behavior so the texture corners are instead on where they would be if you place the square texture into a sphere, with the corners touching the circle, then stretching the lines between the corners to match the circle. Simple mock of what i mean:

I have tried playing with the mapping Attribute of my texture, but that doesn't change the behaviour from my understanding.

Answer 1

After changing the UV coordinates, my half sphere texture is't stretching on border as well:

this.sphereGeometry = new THREE.SphereGeometry(
  10,
  32,
  24,
  0,
  Math.PI,
  0,
  Math.PI
);

const {
  uv: uvAttribute,
  normal
} = this.sphereGeometry.attributes;

for (let i = 0; i < normal.count; i += 1) {
  let u = normal.getX(i);
  let v = normal.getY(i);

  u = u / 2 + 0.5;
  v = v / 2 + 0.5;

  uvAttribute.setXY(i, u, v);
}

const texture = new THREE.TextureLoader().load(
  'https://i.imgur.com/CslEXIS.jpg'
);

texture.flipY = false;
texture.mapping = THREE.CubeUVRefractionMapping;
texture.needsUpdate = true;

const basicMaterial = new THREE.MeshBasicMaterial({
  map: texture,
  side: THREE.DoubleSide,
});

this.sphere = new THREE.Mesh(this.sphereGeometry, basicMaterial);
this.scene.add(this.sphere);