In general, trigonometric functions are used when calculating circles and spheres.
But it is also very easy without these functions.
The construction of the sphere is made of 8 parts similar to my magic sphere in the addon THREEg. Addon to create special / extended geometries
The basis is the logo of discourse. threejs. org
Here I have now used indexed BufferGeometry.
Correction April 15th
Working on an example, I looked at the sphere again. I noticed a mistake. Since I often only worked with THREE.DoubleSide and wireframe: true, I didn’t notice it.
Half of the octants have the front inside! 
I have to reverse the order of the indices there.
I have corrected the function indicesPartSphere( p ) and use
spin = ( p === 0 || p === 2 || p === 5 || p === 7 ) ? true : false;
Here you can find it in the collection.
http://discourse.threejs.hofk.de/2019/SphereWithoutTrigonometry/SphereWithoutTrigonometry.html
The complete code:
<head>
<title> SphereWithoutTrigonometry </title>
<meta charset="utf-8" />
</head>
<body> </body>
<script src="../js/three.min.103.js"></script>
<script src="../js/OrbitControls.js"></script>
<script>
// @author hofk
'use strict';
const sumNN = ( n ) => ( n * ( n + 1 ) / 2 ); // Sum natural numbers
const sumON = ( n ) => ( n * n ); // Sum odd numbers
const scene = new THREE.Scene();
const camera = new THREE.PerspectiveCamera( 55, window.innerWidth / window.innerHeight, 0.1, 100 );
camera.position.set( 0, 1, 8 );
const renderer = new THREE.WebGLRenderer( { antialias: true } );
renderer.setSize( window.innerWidth, window.innerHeight );
renderer.setClearColor( 0xdddddd, 1 );
const container = document.createElement( 'div' );
document.body.appendChild( container );
container.appendChild( renderer.domElement );
const controls = new THREE.OrbitControls( camera, renderer.domElement );
const tex1 = new THREE.TextureLoader().load( 'uvgrid01.png' );
//const material = new THREE.MeshBasicMaterial( { map: tex1, side: THREE.DoubleSide } );
const tex2 = new THREE.TextureLoader().load( 'dahlia.png' );
const material = [
new THREE.MeshBasicMaterial( { map: tex1, side: THREE.DoubleSide } ),
new THREE.MeshBasicMaterial( { color: 0xff0000, side: THREE.DoubleSide, wireframe: true } ),
new THREE.MeshBasicMaterial( { color: 0xff00ff, side: THREE.DoubleSide } ),
new THREE.MeshBasicMaterial( { color: 0x00ffff, side: THREE.DoubleSide, wireframe: true } ),
new THREE.MeshBasicMaterial( { color: 0xffff00, side: THREE.DoubleSide } ),
new THREE.MeshBasicMaterial( { map: tex2, side: THREE.DoubleSide, wireframe: false } ),
new THREE.MeshBasicMaterial( { color: 0x33ff55, side: THREE.DoubleSide } ),
new THREE.MeshBasicMaterial( { map: tex1, side: THREE.DoubleSide } )
];
const g = new THREE.BufferGeometry( );
//SphereWithoutTrigonometry( g, 2.5, 10, [ 1,1,0,0, 1,0,1,0 ] ); // ( BufferGeometry, radius, equator, parts )
//equator = half of height segments = quarter of equatorial segments
SphereWithoutTrigonometry( g, 3,18 );
const mesh = new THREE.Mesh( g, material );
scene.add( mesh );
animate();
function animate() {
requestAnimationFrame( animate );
renderer.render( scene, camera );
controls.update();
}
function SphereWithoutTrigonometry( g, r, eqt, parts ) { // ( BufferGeometry, optional: radius, equator, parts )
r = r !== undefined ? r : 1;
eqt = eqt !== undefined ? eqt : 8;
// parts array, value 1 for octant, otherwise arbitrary - upper counterclockwise, lower clockwise seen from below
parts = parts !== undefined ? parts : [ 1,1,1,1, 1,1,1,1 ];
let pCount = 0;
for ( let p = 0; p < 8; p ++ ) {
pCount += parts[ p ] === 1 ? 1 : 0;
}
let posIdx = 0; // position index
let uvIdx = 0; // uv index
let fIdx = 0; // face index
const vertexCount = sumNN( eqt + 1 ) * pCount;
const faceCount = sumON( eqt ) * pCount;
g.faceIndices = new Uint32Array( faceCount * 3 );
g.vertices = new Float32Array( vertexCount * 3 );
g.uvs = new Float32Array( vertexCount * 2 );
g.setIndex( new THREE.BufferAttribute( g.faceIndices, 1 ) );
g.addAttribute( 'position', new THREE.BufferAttribute( g.vertices, 3 ) );
g.addAttribute( 'uv', new THREE.BufferAttribute( g.uvs, 2 ) );
pCount = 0;
for ( let p = 0; p < 8; p ++ ) {
if ( parts[ p ] === 1 ) {
indicesPartSphere( p );
verticesUVsPartSphere( p );
pCount += 1;
}
}
function indicesPartSphere( p ) {
let a0, a1, b0, b1; // indices
let spin = ( p === 0 || p === 2 || p === 5 || p === 7 ) ? true : false;
a0 = sumNN( eqt + 1 ) * pCount; // start vertex index per part
g.addGroup( fIdx, sumON( eqt ) * 3, p ); // write groups for multi material
for ( let i = 0; i < eqt; i ++ ) {
a1 = a0 + 1;
b0 = a0 + i + 1; // below ( i )
b1 = b0 + 1;
// each two faces
for ( let j = 0; j < i; j ++ ) {
g.faceIndices[ fIdx ] = j + a0; // left face
g.faceIndices[ fIdx + 1 ] = j + ( spin ? b1 : b0 );
g.faceIndices[ fIdx + 2 ] = j + ( spin ? b0 : b1 );
g.faceIndices[ fIdx + 3 ] = j + a0; // right face
g.faceIndices[ fIdx + 4 ] = j + ( spin ? a1 : b1 );
g.faceIndices[ fIdx + 5 ] = j + ( spin ? b1 : a1 );
fIdx += 6;
}
g.faceIndices[ fIdx ] = i + a0; // last face in row ( like a left face )
g.faceIndices[ fIdx + 1 ] = i + ( spin ? b1 : b0 );
g.faceIndices[ fIdx + 2 ] = i + ( spin ? b0 : b1 );
fIdx += 3;
a0 += i + 1; // next start index
}
}
function verticesUVsPartSphere( p ) {
const signX = ( p === 0 || p === 3 || p === 4 || p === 7 ) ? 1 : -1;
const signY = p < 4 ? 1 : -1;
const signZ = ( p === 2 || p === 3 || p === 6 || p === 7 ) ? 1 : -1;
let xp, yp, zp, len;
let ux = 0.5;
let du = 1 / eqt;
let uy = 1 + du;
for ( let i = 0 ; i <= eqt; i ++ ) {
yp = 1 - i / eqt;
ux = 0.5 + signX * signY * signZ * ( i + 2 ) * du / 2;
uy -= du;
for ( let j = 0; j <= i ; j ++ ) {
xp = ( i - j ) / eqt;
zp = j / eqt;
len = Math.sqrt( xp * xp + yp * yp + zp * zp ); // to normalize
g.vertices[ posIdx ] = r * signX * xp / len;
g.vertices[ posIdx + 1 ] = r * signY * yp / len;
g.vertices[ posIdx + 2 ] = r * signZ * zp / len;
posIdx += 3;
ux += -signX * signY * signZ * du;
g.uvs[ uvIdx ] = ux;
g.uvs[ uvIdx + 1 ] = uy;
uvIdx += 2;
}
}
}
}
</script>
</html>