distorts an icosahedron sphere into a superellipsoid

superellipsoid

// https://en.wikipedia.org/wiki/Superellipsoid

function c(w, m) {
  const cv = Math.cos(w);
  return Math.sign(cv) * Math.abs(cv) ** m;
}

function s(w, m) {
  const sv = Math.sin(w);
  return Math.sign(sv) * Math.abs(sv) ** m;
}

function getSuperEllipsoidPoint(theta, phi) {
  // radii
  const A = 1;
  const B = 1;
  const C = 1;
  // https://en.wikipedia.org/wiki/Superellipsoid#/media/File:Superellipsoid_collection.png
  // r = 2/e, t = 2/n
  const r = 4;
  const t = 4;
  const x = A * c(theta, 2 / t) * c(phi, 2 / r);
  const y = B * c(theta, 2 / t) * s(phi, 2 / r);
  const z = C * s(theta, 2 / t);
  return { x, y, z };
}

const v0 = new Vector3();
const v1 = new Vector3();
const v2 = new Vector3();

function getSuperEllipsoidNormal(theta, phi) {
  const e = 0.00001;
  const { x: x0, y: y0, z: z0 } = getSuperEllipsoidPoint(theta, phi);
  const { x: x1, y: y1, z: z1 } = getSuperEllipsoidPoint(theta + e, phi);
  const { x: x2, y: y2, z: z2 } = getSuperEllipsoidPoint(theta, phi + e);
  v0.set(x0, y0, z0);
  v1.set(x1, y1, z1).sub(v0);
  v2.set(x2, y2, z2).sub(v0);
  v0.crossVectors(v1, v2);
  return { x: v0.x, y: v0.y, z: v0.z };
}

function createSuperEllipsoid() {
  const geometry = new IcosahedronBufferGeometry(1, 10);
  const vertices = geometry.attributes.position.array;
  const normals = geometry.attributes.normal.array;
  const v = new Vector3();

  for (let i = 0; i < vertices.length; i += 3) {
    v.set(vertices[i], vertices[i + 1], vertices[i + 2]);
    const theta = Math.atan2(Math.sqrt(v.x ** 2 + v.y ** 2), v.z) + Math.PI / 2;
    const phi = Math.atan2(v.y, v.x);

    const { x, y, z } = getSuperEllipsoidPoint(theta, phi);
    vertices[i] = x;
    vertices[i + 1] = y;
    vertices[i + 2] = z;

    const { x: nx, y: ny, z: nz } = getSuperEllipsoidNormal(theta, phi);
    normals[i] = nx;
    normals[i + 1] = ny;
    normals[i + 2] = nz;
  }
  return geometry;
}

const geometry2 = createSuperEllipsoid();