threejs/examples/jsm/geometries/ParametricGeometries.js

import {
	Curve,
	Vector3
} from 'three';

import { ParametricGeometry } from './ParametricGeometry.js';

/**
 * Experimenting of primitive geometry creation using Surface Parametric equations
 */

const ParametricGeometries = {

	klein: function ( 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 );

	},

	plane: function ( width, height ) {

		return function ( u, v, target ) {

			const x = u * width;
			const y = 0;
			const z = v * height;

			target.set( x, y, z );

		};

	},

	mobius: function ( u, t, target ) {

		// flat mobius strip
		// 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 );

	},

	mobius3d: function ( u, t, target ) {

		// volumetric mobius strip

		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 );

	}

};


/*********************************************
 *
 * Parametric Replacement for TubeGeometry
 *
 *********************************************/

ParametricGeometries.TubeGeometry = class TubeGeometry extends ParametricGeometry {

	constructor( path, segments = 64, radius = 1, segmentsRadius = 8, closed = false ) {

		const numpoints = segments + 1;

		const frames = path.computeFrenetFrames( segments, closed ),
			tangents = frames.tangents,
			normals = frames.normals,
			binormals = frames.binormals;

		const position = new Vector3();

		function ParametricTube( u, v, target ) {

			v *= 2 * Math.PI;

			const i = Math.floor( u * ( numpoints - 1 ) );

			path.getPointAt( u, position );

			const normal = normals[ i ];
			const binormal = binormals[ i ];

			const cx = - radius * Math.cos( v ); // TODO: Hack: Negating it so it faces outside.
			const cy = radius * Math.sin( v );

			position.x += cx * normal.x + cy * binormal.x;
			position.y += cx * normal.y + cy * binormal.y;
			position.z += cx * normal.z + cy * binormal.z;

			target.copy( position );

		}

		super( ParametricTube, segments, segmentsRadius );

		// proxy internals

		this.tangents = tangents;
		this.normals = normals;
		this.binormals = binormals;

		this.path = path;
		this.segments = segments;
		this.radius = radius;
		this.segmentsRadius = segmentsRadius;
		this.closed = closed;

	}

};


/*********************************************
  *
  * Parametric Replacement for TorusKnotGeometry
  *
  *********************************************/
ParametricGeometries.TorusKnotGeometry = class TorusKnotGeometry extends ParametricGeometries.TubeGeometry {

	constructor( radius = 200, tube = 40, segmentsT = 64, segmentsR = 8, p = 2, q = 3 ) {

		class TorusKnotCurve extends Curve {

			getPoint( t, optionalTarget = new Vector3() ) {

				const point = optionalTarget;

				t *= Math.PI * 2;

				const r = 0.5;

				const x = ( 1 + r * Math.cos( q * t ) ) * Math.cos( p * t );
				const y = ( 1 + r * Math.cos( q * t ) ) * Math.sin( p * t );
				const z = r * Math.sin( q * t );

				return point.set( x, y, z ).multiplyScalar( radius );

			}

		}

		const segments = segmentsT;
		const radiusSegments = segmentsR;
		const extrudePath = new TorusKnotCurve();

		super( extrudePath, segments, tube, radiusSegments, true, false );

		this.radius = radius;
		this.tube = tube;
		this.segmentsT = segmentsT;
		this.segmentsR = segmentsR;
		this.p = p;
		this.q = q;

	}

};

/*********************************************
  *
  * Parametric Replacement for SphereGeometry
  *
  *********************************************/
ParametricGeometries.SphereGeometry = class SphereGeometry extends ParametricGeometry {

	constructor( size, u, v ) {

		function sphere( u, v, target ) {

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

			const x = size * Math.sin( u ) * Math.cos( v );
			const y = size * Math.sin( u ) * Math.sin( v );
			const z = size * Math.cos( u );

			target.set( x, y, z );

		}

		super( sphere, u, v );

	}

};


/*********************************************
  *
  * Parametric Replacement for PlaneGeometry
  *
  *********************************************/

ParametricGeometries.PlaneGeometry = class PlaneGeometry extends ParametricGeometry {

	constructor( width, depth, segmentsWidth, segmentsDepth ) {

		function plane( u, v, target ) {

			const x = u * width;
			const y = 0;
			const z = v * depth;

			target.set( x, y, z );

		}

		super( plane, segmentsWidth, segmentsDepth );

	}

};

export { ParametricGeometries };