How can I create a 3D cubic-bezier curved triangle from 3D points in Three.js?

Question

Following this topic, I am trying to generate a 3D curved triangle as a NURBS surface, but I don\'t understand how to set up my 3D points to do that.

Here is the cur

Answer 1

I modified Kamil Kiełczewski's code and separated it into 2 classes:

1. BarycentricBufferGeometry based on ParametricBufferGeometry
2. BezierTriangle based on NURBSSurface

Now it functions similar to NURBSSurface.js and is more efficient.

BarycentricBufferGeometry.js

import { BufferGeometry, Float32BufferAttribute, Vector3 } from './three.module.js';

class BarycentricBufferGeometry extends BufferGeometry {

    constructor(func, slices) {

        super();

        this.type = 'BezierTriangleGeometry';

        this.parameters = {
            func: func,
            slices: slices
        };

        // buffers
        const indices = [];
        const vertices = [];
        const normals = [];
        const uvs = [];

        const EPS = 0.00001;

        const normal = new Vector3();

        const p0 = new Vector3(), p1 = new Vector3();
        const pu = new Vector3(), pv = new Vector3();

        if (func.length < 3) {

            console.error('THREE.ParametricGeometry: Function must now modify a Vector3 as third parameter.');

        }

        // generate vertices, normals and uvs
        for (let i = 0; i <= slices; i++) {

            for (let j = 0; j <= slices - i; j++) {

                const u = j / slices;
                const v = i / slices;

                // vertex
                func(u, v, p0);
                vertices.push(p0.x, p0.y, p0.z);

                // normal
                // approximate tangent vectors via finite differences
                if (u - EPS >= 0) {

                    func(u - EPS, v, p1);
                    pu.subVectors(p0, p1);

                } else {

                    func(u + EPS, v, p1);
                    pu.subVectors(p1, p0);

                }

                if (v - EPS >= 0) {

                    func(u, v - EPS, p1);
                    pv.subVectors(p0, p1);

                } else {

                    func(u, v + EPS, p1);
                    pv.subVectors(p1, p0);

                }

                // cross product of tangent vectors returns surface normal
                normal.crossVectors(pu, pv).normalize();
                normals.push(normal.x, normal.y, normal.z);

                // uv
                uvs.push(u, v);

            }

        }

        // generate indices
        let st = 0;
        let m = slices;

        for (let j = slices; j > 0; j--) {

            for (let i = 0; i < m; i++) {

                const a = st + i;
                const b = st + i + 1;
                const c = st + i + 1 + m;

                indices.push(a, b, c);

                if (i < m - 1)
                    indices.push(st + i + 1, st + m + i + 2, st + m + i + 1);
            }

            m = m - 1;
            st += j + 1;
        }

        // build geometry
        this.setIndex(indices);
        this.setAttribute('position', new Float32BufferAttribute(vertices, 3));
        this.setAttribute('normal', new Float32BufferAttribute(normals, 3));
        this.setAttribute('uv', new Float32BufferAttribute(uvs, 2));

    }
}

// BarycentricBufferGeometry.prototype = Object.create( BufferGeometry.prototype );
;


export { BarycentricBufferGeometry };

BezierTriangle.js

class BezierTriangle {

    constructor(controlPoints) {

        this.controlPoints = controlPoints;
    }

    static bp(i, j, k, r, s, t, n = 3) {
        const f = x => x ? f(x - 1) * x : 1;
        return r ** i * s ** j * t ** k * f(n) / (f(i) * f(j) * f(k));
    }

    static calcSurfacePoint(p, u, v, target) {

        const t = 1 - u - v;
        let b = [];

        b[0] = BezierTriangle.bp(0, 0, 3, u, v, t);
        b[1] = BezierTriangle.bp(1, 0, 2, u, v, t);
        b[2] = BezierTriangle.bp(2, 0, 1, u, v, t);
        b[3] = BezierTriangle.bp(3, 0, 0, u, v, t);
        b[4] = BezierTriangle.bp(2, 1, 0, u, v, t);
        b[5] = BezierTriangle.bp(1, 2, 0, u, v, t);
        b[6] = BezierTriangle.bp(0, 3, 0, u, v, t);
        b[7] = BezierTriangle.bp(0, 2, 1, u, v, t);
        b[8] = BezierTriangle.bp(0, 1, 2, u, v, t);
        b[9] = BezierTriangle.bp(1, 1, 1, u, v, t);

        let x = 0,
            y = 0,
            z = 0;

        for (let i = 0; i < 10; i++) {
            x += p[i].x * b[i];
            y += p[i].y * b[i];
            z += p[i].z * b[i];
        }

        target.set(x, y, z);
    }

    getPoint(u, v, target) {

        BezierTriangle.calcSurfacePoint(this.controlPoints, u, v, target);
    }
}


export { BezierTriangle };

Example:

import * as THREE from './three.module.js';

import { BarycentricBufferGeometry } from './BarycentricBufferGeometry.js';
import { BezierTriangle } from './BezierTriangle.js';


//setup
const scene = new THREE.Scene();
const camera = new THREE.PerspectiveCamera(45, window.innerWidth / window.innerHeight, .01, 10000);
camera.position.set(2, 2, 6)

const renderer = new THREE.WebGLRenderer();
renderer.setSize(window.innerWidth, window.innerHeight);
document.body.appendChild(renderer.domElement);


// bezier triangle points
const points = [
    { x: 0, y: 0, z: 0, c: 'red' },

    { x: 0, y: 1, z: 0, c: 'grey' },
    { x: 0, y: 2, z: 0, c: 'grey' },
    { x: 0, y: 3, z: 1, c: 'green' },

    { x: 1, y: 3, z: 1, c: 'grey' },
    { x: 2, y: 3, z: 1, c: 'grey' },
    { x: 3, y: 3, z: 2, c: 'blue' },

    { x: 2, y: 2, z: 0, c: 'grey' },
    { x: 1, y: 1, z: 0, c: 'grey' },

    { x: 1, y: 2, z: 0, c: 'yellow' },
];

// add some colored spheres to help identify points
points.forEach(p => {
    const sphere = new THREE.Mesh(
        new THREE.SphereBufferGeometry(.1, 32, 32),
        new THREE.MeshBasicMaterial({ color: p.c ? p.c : 'white' })
    );
    sphere.position.set(p.x, p.y, p.z);
    scene.add(sphere);
});

// draw bezier triangle
const triangle = new BezierTriangle(points);

function getSurfacePoint(u, v, target) {
    return triangle.getPoint(u, v, target);
}

const geometry = new BarycentricBufferGeometry(getSurfacePoint, 3);
const material = new THREE.MeshBasicMaterial({ color: 'gold', wireframe: true });
const mesh = new THREE.Mesh(geometry, material);

scene.add(mesh);

renderer.render(scene, camera);