three.js/examples/jsm/geometries/ParametricFunctions.js

/** @module ParametricFunctions */

/**
 * A parametric function representing the Klein bottle.
 *
 * @param {number} v - The `v` coordinate on the surface in the range `[0,1]`.
 * @param {number} u - The `u` coordinate on the surface in the range `[0,1]`.
 * @param {Vector3} target - The target vector that is used to store the method's result.
 */
function klein( v, u, target ) {

	u *= Math.PI;
	v *= 2 * Math.PI;

	u = u * 2;
	let x, z;
	if ( u < Math.PI ) {

		x = 3 * Math.cos( u ) * ( 1 + Math.sin( u ) ) + ( 2 * ( 1 - Math.cos( u ) / 2 ) ) * Math.cos( u ) * Math.cos( v );
		z = - 8 * Math.sin( u ) - 2 * ( 1 - Math.cos( u ) / 2 ) * Math.sin( u ) * Math.cos( v );

	} else {

		x = 3 * Math.cos( u ) * ( 1 + Math.sin( u ) ) + ( 2 * ( 1 - Math.cos( u ) / 2 ) ) * Math.cos( v + Math.PI );
		z = - 8 * Math.sin( u );

	}

	const y = - 2 * ( 1 - Math.cos( u ) / 2 ) * Math.sin( v );

	target.set( x, y, z );

}

/**
 * A parametric function representing a flat plane.
 *
 * @param {number} u - The `u` coordinate on the surface in the range `[0,1]`.
 * @param {number} v - The `v` coordinate on the surface in the range `[0,1]`.
 * @param {Vector3} target - The target vector that is used to store the method's result.
 */
function plane( u, v, target ) {

	target.set( u, 0, v );

}

/**
 * A parametric function representing a flat mobius strip.
 *
 * @param {number} u - The `u` coordinate on the surface in the range `[0,1]`.
 * @param {number} t - The `v` coordinate on the surface in the range `[0,1]`.
 * @param {Vector3} target - The target vector that is used to store the method's result.
 */
function mobius( u, t, target ) {

	// http://www.wolframalpha.com/input/?i=M%C3%B6bius+strip+parametric+equations&lk=1&a=ClashPrefs_*Surface.MoebiusStrip.SurfaceProperty.ParametricEquations-
	u = u - 0.5;
	const v = 2 * Math.PI * t;

	const a = 2;

	const x = Math.cos( v ) * ( a + u * Math.cos( v / 2 ) );
	const y = Math.sin( v ) * ( a + u * Math.cos( v / 2 ) );
	const z = u * Math.sin( v / 2 );

	target.set( x, y, z );

}

/**
 * A parametric function representing a volumetric mobius strip.
 *
 * @param {number} u - The `u` coordinate on the surface in the range `[0,1]`.
 * @param {number} t - The `v` coordinate on the surface in the range `[0,1]`.
 * @param {Vector3} target - The target vector that is used to store the method's result.
 */
function mobius3d( u, t, target ) {

	u *= Math.PI;
	t *= 2 * Math.PI;

	u = u * 2;
	const phi = u / 2;
	const major = 2.25, a = 0.125, b = 0.65;

	let x = a * Math.cos( t ) * Math.cos( phi ) - b * Math.sin( t ) * Math.sin( phi );
	const z = a * Math.cos( t ) * Math.sin( phi ) + b * Math.sin( t ) * Math.cos( phi );
	const y = ( major + x ) * Math.sin( u );
	x = ( major + x ) * Math.cos( u );

	target.set( x, y, z );

}

export { klein, plane, mobius, mobius3d };