如何从 Three.js 中的 3D 点创建 3D 三次贝塞尔曲线三角形? [英] How can I create a 3D cubic-bezier curved triangle from 3D points in Three.js?

问题描述

遵循

我尝试了许多不同的设置，但没有找到任何可以正常工作的设置.

注意:白点是边缘；红点是贝塞尔曲线的中点.
注2:dots[0]指的是示例图片中的点0，依此类推.

这是工作片段(和小提琴版本

在这段代码中，我们不使用法线，b 点名称更改为 p(例如 b003 到 p003).我们使用以下公式(对于三次贝塞尔三角形n=3)

其中 p_ijk 是控制点(对于上面的 n=3，总和有 10 个元素，所以我们有 10 个控制点)，其中 B^n_ijk(r,s,t) 是定义为的伯恩斯坦多项式i,j,k>=0 和 i+j+k=n

或在其他情况下为 0.r,s,t 在重心坐标中的定义域(其中 r,s,t 是来自 [0, 1] 和 r+s+t=1 的实数)，其中 r=(r=1, s=t=0), s=(s=1, r=t=0), t=(t=1, r=s=0)看起来如下(黑点 - 我们将每个三角形的边分成5 个部分 - 但我们可以将其更改为任意数量)

我们在方法 barycentricCoords(n) 中计算黑域点的规则位置，并在 的方法 genTrianglesIndexes(n) 中定义哪个点创建哪些三角形几何类.但是，您可以将此点的位置和密度更改为任何(三角形内)以获得不同的表面三角形分割.下面是在 2D 中显示域的片段

let pp= ((s='.myCanvas',c=document.querySelector(s),ctx=c.getContext('2d'),id=ctx.createImageData(1,1)) => (x,y,r=0,g=0,b=0,a=255)=>(id.data.set([r,g,b,a]),ctx.putImageData(id, x, y),c))()cr=[255,0,0,255];cg=[0,255,0,255];cb=[0,0,255,255];w=400；h=400；const p1=[0,h-1];const p2=[w-1,h-1];const p3=[w/2,0];mainTriangle=[p1,p2,p3];//mainTriangle.map(p => pp(...p,...cr));让 n=5;让点=[];函数 calcPoint(p1,p2,p3,r,s,t) {const px=p1[0]*r + p2[0]*s + p3[0]*t;const py=p1[1]*r + p2[1]*s + p3[1]*t;返回 [px,py];}//三角形中点的重心坐标r,s,t//从三角形底部到顶部逐行给出的点//第一行有 n+1 个点，第二行有 n 个，第三行有 n-1 个//坐标具有属性 r+s+t=1函数重心坐标(n){让 rst=[];for(let i=0; i<=n; i++) for(let j=0; j<=n-i; j++) {s=(j/n);t=(i/n);r=1-s-t；rst.push([r,s,t]);}返回第一；}//每个三角形的过程计算索引来自//点列表(以 barycentricCoords(n) 返回的格式)函数 genTrianglesIndexes(n) {让 st=0;让 m=n;让三角形=[];for(let j=n; j>0; j--) {for(let i=0; ix?f(x-1)*x:1//小数 f(4)=1*2*3*4=24返回 r**i * s**j * t**k * f(n)/(f(i)*f(j)*f(k));}//drawTriangle(...mainTriangle,cr);//绘制主三角形让 bar=barycentricCoords(n);//每个域点的重心坐标让 ti=genTrianglesIndexes(n);//每个三角形的柱线索引//三角形计算到笛卡尔坐标系让三角形 = ti.map(tr=> tr.map(x=>calcPoint(...mainTriangle,...bar[x]) ) );三角形.map(t => drawTriangle(...t, cg));//计算到笛卡尔坐标系的域点(用于绘制)让 dp = bar.map(x=> calcPoint(...mainTriangle,...x) );//绘制黑点(每个点 4 个像素)dp.map(x=> pp(x[0],x[1]))dp.map(x=> pp(x[0],x[1]-1))dp.map(x=> pp(x[0]-1,x[1]))dp.map(x=> pp(x[0]-1,x[1]-1))

下面是带有 3D 贝塞尔三次三角形的最终片段(算法从 Geometry 类中的方法 genTrianglesForCubicBezierTriangle(n, controlPoints) 开始) - (注意:它是奇怪，但在第一次运行后的 SO 片段中，您将看不到线条，您需要重新加载页面并再次运行以查看三角形线条)

////////////////////////////////////////////////////////这部分/类用于算法和计算///////////////////////////////////////////////////////类几何{构造函数() { this.init();}初始化(n){这个.pts = [{ x:-16, y: -8, z:0, color:0xcc0000 },//p003 红色{ x:8, y:-12, z:0, 颜色:0x888888 },//p201{ x:-8, y:-12, z:0, 颜色:0x999999 },//p102{ x:16, y:-8, z:0, color:0x00cc00 },//p300 绿色{ x:12, y:-6, z:-8, 颜色:0x777777 },//p210{ x:8, y:6, z:-8, 颜色:0x666666 },//p120{ x:0, y:12, z:0, color:0x0000cc },//p030 蓝色{ x:-8, y:6, z:-8, 颜色:0x555555 },//p021{ x:-12, y:-6, z:-8, 颜色:0x444444 },//p012{ x:0, y:0, z:8, color:0xffff00 },//p111 YELLOW(平面控制点)];this.mainTriangle = [this.pts[0],this.pts[3],this.pts[6]];this.bezierCurvesPoints = [[ this.pts[0], this.pts[2], this.pts[1], this.pts[3] ],[ this.pts[3], this.pts[4], this.pts[5], this.pts[6] ],[ this.pts[6], this.pts[7], this.pts[8], this.pts[0] ]];//this.triangles = [//{ points: [this.pts[0], this.pts[1], this.pts[2]], color: null },//线框//{ 点: [this.pts[1], this.pts[2], this.pts[3]], color: 0xffff00 }//黄色//]this.triangles = this.genTrianglesForCubicBezierTriangle(25, this.pts);}//n = 每条三角形边的三角形数genTrianglesForCubicBezierTriangle(n, controlPoints) {让 bar= this.barycentricCoords(n);//重心坐标域让 ti = this.genTrianglesIndexes(n);//三角形的索引(在条形数组中)让 val= bar.map(x => this.calcCubicBezierTriangleValue(controlPoints,...x));//计算每个域(条)点的贝塞尔三角形顶点让 tv= ti.map(tr=> tr.map(x=>val[x]) );//使用它们的索引 (ti) 和 val 生成三角形返回 tv.map(t=>({points: t, color: null}));//将三角形映射到正确的格式(color=null 给出线框)//生成域三角形//let td= ti.map(tr=> tr.map(x=>this.calcPointFromBar(...this.mainTriangle,...bar[x]) ) );//this.trianglesDomain = td.map(t=> ({ points: t, color: null}) );}//更多:https://www.mdpi.com/2073-8994/8/3/13/pdf//具有 G2 跨边界连续性的贝塞尔三角形//Chang-Ki Lee、Hae-Do Hwang 和 Seung-Hyun YooncalcCubicBezierTriangleValue(controlPoints, r,s,t ) {让 p = controlPoints, b=[];b[0]= this.bp(0,0,3,r,s,t);//p[0]=p003b[1]= this.bp(2,0,1,r,s,t);//p[1]=p201b[2]= this.bp(1,0,2,r,s,t);//p[2]=p102b[3]= this.bp(3,0,0,r,s,t);//p[3]=p300b[4]​​= this.bp(2,1,0,r,s,t);//p[4]=p210b[5]= this.bp(1,2,0,r,s,t);//p[5]=p120b[6]= this.bp(0,3,0,r,s,t);//p[6]=p030b[7]= this.bp(0,2,1,r,s,t);//p[7]=p021b[8]= this.bp(0,1,2,r,s,t);//p[8]=p012b[9]= this.bp(1,1,1,r,s,t);//p[9]=p111让 x=0, y=0, z=0;for(让 i=0; i<=9; i++) {x+=p[i].x*b[i];y+=p[i].y*b[i];z+=p[i].z*b[i];}返回 { x:x, y:y, z:z };}//伯恩斯坦多项式次数 n, i+j+k=nbp(i,j,k, r,s,t, n=3) {const f=x=>x?f(x-1)*x:1//小数 f(4)=1*2*3*4=24返回 r**i * s**j * t**k * f(n)/(f(i)*f(j)*f(k));}coordArrToObj(p) { return { x:p[0], y:p[1], z:p[2] } }//来自重心坐标系的 Calc 笛卡尔点calcPointFromBar(p1,p2,p3,r,s,t) {const px=p1.x*r + p2.x*s + p3.x*t;const py=p1.y*r + p2.y*s + p3.y*t;const pz=p1.z*r + p2.z*s + p3.z*t;返回 { x:px, y:py, z:pz};}//三角形中点的重心坐标r,s,t//从三角形底部到顶部逐行给出的点//第一行有 n+1 个点，第二行有 n 个，第三行有 n-1 个//坐标具有属性 r+s+t=1重心坐标(n){让 rst=[];for(let i=0; i<=n; i++) for(let j=0; j<=n-i; j++) {让 s=(j/n);让 t=(i/n);让 r=1-s-t;rst.push([r,s,t]);}返回第一；}//每个三角形的过程计算索引来自//点列表(以 barycentricCoords(n) 返回的格式)genTrianglesIndexes(n) {让 st=0;让 m=n;让三角形=[];for(let j=n; j>0; j--) {for(let i=0; ithis.createPoint(p));this.geometry.getTriangles().forEach(t=> this.createTriangle(t));this.geometry.getBezierCurves().forEach(c=> this.createEdge(...c));}初始化(几何){this.geometry = 几何；这个.W = 480,这个.H = 400,this.DISTANCE = 100 ;this.PI = Math.PI，this.renderer = new THREE.WebGLRenderer({画布:document.querySelector('canvas'),抗锯齿:真的，阿尔法:真}),this.camera = new THREE.PerspectiveCamera(25, this.W/this.H),this.scene = new THREE.Scene(),this.center = new THREE.Vector3(0, 0, 0),this.pts = [] ;this.renderer.setClearColor(0x000000, 0) ;this.renderer.setSize(this.W, this.H) ;//camera.position.set(-48, 32, 80) ;this.camera.position.set(0, 0, this.DISTANCE) ;this.camera.lookAt(this.center) ;this.initGeom();this.azimut = 0;this.pitch = 90;this.isDown = false;this.prevEv = null;this.renderer.domElement.onmousedown = e =>this.down(e) ;window.onmousemove = e =>this.move(e) ;window.onmouseup = e =>this.up(e) ;this.renderer.render(this.scene, this.camera) ;}创建点(p){让 {x, y, z, 颜色} = p;让 pt = 新的 THREE.Mesh(新三.SphereGeometry(1, 10, 10),new THREE.MeshBasicMaterial({颜色}));pt.position.set(x, y, z) ;pt.x = x ;pt.y = y ;pt.z = z ;this.pts.push(pt) ;this.scene.add(pt) ;}createTriangle(t) {var geom = new THREE.Geometry();var v1 = new THREE.Vector3(t.points[0].x, t.points[0].y, t.points[0].z);var v2 = new THREE.Vector3(t.points[1].x, t.points[1].y, t.points[1].z);var v3 = new THREE.Vector3(t.points[2].x, t.points[2].y, t.points[2].z);geom.vertices.push(v1);geom.vertices.push(v2);geom.vertices.push(v3);let material = new THREE.MeshNormalMaterial({wireframe: true,})if(t.color != null) material = new THREE.MeshBasicMaterial( {颜色:t.color，边:三.DoubleSide，});geom.faces.push(new THREE.Face3(0, 1, 2));geom.computeFaceNormals();var mesh= new THREE.Mesh(geom, material);this.scene.add(mesh) ;}创建边缘(pt1，pt2，pt3，pt4){让曲线 = 新的 THREE.CubicBezierCurve3(新三.Vector3(pt1.x, pt1.y, pt1.z),新三.Vector3(pt2.x, pt2.y, pt2.z),新三.Vector3(pt3.x, pt3.y, pt3.z),新三.Vector3(pt4.x, pt4.y, pt4.z),),网格 = 新的 THREE.Mesh(新的 THREE.TubeGeometry(curve, 8, 0.5, 8, false),新三.MeshBasicMaterial({颜色:0x203040}));this.scene.add(mesh) ;}向下(德){this.prevEv = de ;this.isDown = 真;}移动(我){如果 (!this.isDown) 返回；this.azimut -= (me.clientX - this.prevEv.clientX) * 0.5 ;this.azimut %= 360 ;if (this.azimut <0) this.azimut = 360 - this.azimut ;this.pitch -= (me.clientY - this.prevEv.clientY) * 0.5 ;如果 (this.pitch <1) this.pitch = 1 ;如果 (this.pitch > 180) this.pitch = 180 ;this.prevEv = 我；让 theta = this.pitch/180 * this.PI，phi = this.azimut/180 * this.PI,半径 = this.DISTANCE ;this.camera.position.set(半径 * Math.sin(theta) * Math.sin(phi),半径 * Math.cos(theta),半径 * Math.sin(theta) * Math.cos(phi),);this.camera.lookAt(this.center) ;this.renderer.render(this.scene, this.camera) ;}向上(ue){this.isDown = false ;}}//系统设置让 geom= new Geometry();let draw = new Draw(geom);

body {显示:弹性；弹性方向:行；对齐内容:居中；对齐项目:居中；高度:100vh；边距:0；背景:#1c2228；溢出:隐藏；}

<script src="https://cdnjs.cloudflare.com/ajax/libs/three.js/101/three.min.js"></script><canvas></canvas>

小提琴版本是这里.我在评论中添加了信息，但算法很复杂，如果您有问题 - 作为评论提出 - 我会回答.

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 current implementation :

var edges = this.getEdges(), // An edge is a line following 4 dots as a bezier curve.
    dots = self.getDotsFromEdges(edges), // Get all dots in order for building the surface.

    ctrlPoints = [ // Is generated only once before, but copy-pasted here for this sample code.
        [
            new THREE.Vector4(0, 0, 0, 1),
            new THREE.Vector4(0, 0, 0, 1),
            new THREE.Vector4(0, 0, 0, 1),
            new THREE.Vector4(0, 0, 0, 1)
        ],
        [
            new THREE.Vector4(0, 0, 0, 1),
            new THREE.Vector4(0, 0, 0, 1),
            new THREE.Vector4(0, 0, 0, 1),
            new THREE.Vector4(0, 0, 0, 1)
        ],
        [
            new THREE.Vector4(0, 0, 0, 1),
            new THREE.Vector4(0, 0, 0, 1),
            new THREE.Vector4(0, 0, 0, 1),
            new THREE.Vector4(0, 0, 0, 1)
        ]
    ],

    nc,
    deg1 = ctrlPoints.length - 1,
    knots1 = [],
    deg2 = 3,                           // Cubic bezier
    knots2 = [0, 0, 0, 0, 1, 1, 1, 1],  // <-
    cpts,
    nurbs ;

nc = ctrlPoints.length ;
while (nc-- > 0) knots1.push(0) ;
nc = ctrlPoints.length ;
while (nc-- > 0) knots1.push(1) ;


// The following seems to be the problem... :

cpts = ctrlPoints[0] ;
cpts[0].set(dots[0].x, dots[0].y, dots[0].z, 1) ;
cpts[1].set(dots[1].x, dots[1].y, dots[1].z, 1) ;
cpts[2].set(dots[2].x, dots[2].y, dots[2].z, 1) ;
cpts[3].set(dots[3].x, dots[3].y, dots[3].z, 1) ;

cpts = ctrlPoints[1] ;
cpts[0].set(dots[6].x, dots[6].y, dots[6].z, 1) ;
cpts[1].set(dots[5].x, dots[5].y, dots[5].z, 1) ;
cpts[2].set(dots[4].x, dots[4].y, dots[4].z, 1) ;
cpts[3].set(dots[3].x, dots[3].y, dots[3].z, 1) ;

cpts = ctrlPoints[2] ;
cpts[0].set(dots[6].x, dots[6].y, dots[6].z, 1) ;
cpts[1].set(dots[7].x, dots[7].y, dots[7].z, 1) ;
cpts[2].set(dots[8].x, dots[8].y, dots[8].z, 1) ;
cpts[3].set(dots[0].x, dots[0].y, dots[0].z, 1) ;



nurbs = new THREE.NURBSSurface(deg1, deg2, knots1, knots2, ctrlPoints) ;

this.mesh.geometry.dispose() ;
this.mesh.geometry = new THREE.ParametricBufferGeometry(function(u, v, target) {
    return nurbs.getPoint(u, v, target) ;
}, 10, 10) ;

And here is the result:

I tried many different settings but can't find any working well.

Note: The white points are the edges ends ; The red points are the bezier curve middle points.
Note 2: dots[0] refers to the point 0 in the sample picture, and so on.

Here is working snippet (and fiddle version here)

const
		PI = Math.PI,
    sin = Math.sin,
    cos = Math.cos,
		W = 480,
    H = 400,
    log = console.log,
    DISTANCE = 100 ;

let renderer = new THREE.WebGLRenderer({
      canvas : document.querySelector('canvas'),
      antialias : true,
      alpha : true
    }),
    camera = new THREE.PerspectiveCamera(25, W/H),
    scene = new THREE.Scene(),
    center = new THREE.Vector3(0, 0, 0),

		pts = [] ;

renderer.setClearColor(0x000000, 0) ;

renderer.setSize(W, H) ;
// camera.position.set(-48, 32, 80) ;
camera.position.set(0, 0, DISTANCE) ;
camera.lookAt(center) ;

function createPoint(x, y, z, color) {
		let pt = new THREE.Mesh(
      new THREE.SphereGeometry(1, 10, 10),
      new THREE.MeshBasicMaterial({ color })
    ) ;
    pt.position.set(x, y, z) ;
    pt.x = x ;
    pt.y = y ;
    pt.z = z ;
    pts.push(pt) ;
    
    scene.add(pt) ;
}

function createEdge(pt1, pt2, pt3, pt4) {
		let curve = new THREE.CubicBezierCurve3(
          pt1.position,
          pt2.position,
          pt3.position,
          pt4.position
        ),
    		mesh = new THREE.Mesh(
          new THREE.TubeGeometry(curve, 8, 0.5, 8, false),
          new THREE.MeshBasicMaterial({
            color : 0x203040
          })
        ) ;
        
    scene.add(mesh) ;
}

///////////////////////////////////////////////

// POINTS //
createPoint(-16, -8, 0, 0xcc0000) ; // RED
createPoint(-8, -12, 0, 0x999999) ;
createPoint(8, -12, 0, 0x888888) ;
createPoint(16, -8, 0, 0x00cc00) ; // GREEN
createPoint(12, -6, -8, 0x777777) ;
createPoint(8, 6, -8, 0x666666) ;
createPoint(0, 12, 0, 0x0000cc) ; // BLUE
createPoint(-8, 6, -8, 0x555555) ;
createPoint(-12, -6, -8, 0x444444) ;

// EDGES //
createEdge(pts[0], pts[1], pts[2], pts[3]) ;
createEdge(pts[3], pts[4], pts[5], pts[6]) ;
createEdge(pts[6], pts[7], pts[8], pts[0]) ;

// SURFACE //
let ctrlPoints = [
        [
            new THREE.Vector4(pts[0].x, pts[0].y, pts[0].z, 1),
            new THREE.Vector4(pts[1].x, pts[1].y, pts[1].z, 1),
            new THREE.Vector4(pts[2].x, pts[2].y, pts[2].z, 1),
            new THREE.Vector4(pts[3].x, pts[3].y, pts[3].z, 1)
        ],
        [
            new THREE.Vector4(pts[6].x, pts[6].y, pts[6].z, 1),
            new THREE.Vector4(pts[5].x, pts[5].y, pts[5].z, 1),
            new THREE.Vector4(pts[4].x, pts[4].y, pts[4].z, 1),
            new THREE.Vector4(pts[3].x, pts[3].y, pts[3].z, 1)
        ],
        [
            new THREE.Vector4(pts[6].x, pts[6].y, pts[6].z, 1),
            new THREE.Vector4(pts[7].x, pts[7].y, pts[7].z, 1),
            new THREE.Vector4(pts[8].x, pts[8].y, pts[8].z, 1),
            new THREE.Vector4(pts[0].x, pts[0].y, pts[0].z, 1)
        ]
    ],

    nc,
    deg1 = ctrlPoints.length - 1,
    knots1 = [],
    deg2 = 3,                           // Cubic bezier
    knots2 = [0, 0, 0, 0, 1, 1, 1, 1],  // <-
    cpts,
    nurbs ;

nc = ctrlPoints.length ;
while (nc-- > 0) knots1.push(0) ;
nc = ctrlPoints.length ;
while (nc-- > 0) knots1.push(1) ;

nurbs = new THREE.NURBSSurface(deg1, deg2, knots1, knots2, ctrlPoints) ;

let surfaceMesh = new THREE.Mesh(
    new THREE.ParametricBufferGeometry(function(u, v, target) {
        return nurbs.getPoint(u, v, target) ;
    }, 10, 10),
    new THREE.MeshBasicMaterial({
        side : THREE.DoubleSide,
        opacity : 0.9,
        transparent : true,
        color : 0x405060
    })
) ;

scene.add(surfaceMesh) ;



///////////////////////////////////////////////

let azimut = 0,
	  pitch = 90,
    isDown = false,
    prevEv ;

function down(de) {
		prevEv = de ;
    isDown = true ;
}

function move(me) {
		if (!isDown) return ;
    
		azimut -= (me.clientX - prevEv.clientX) * 0.5 ;
    azimut %= 360 ;
    if (azimut < 0) azimut = 360 - azimut ;
    
		pitch -= (me.clientY - prevEv.clientY) * 0.5 ;
    if (pitch < 1) pitch = 1 ;
    if (pitch > 180) pitch = 180 ;
    
    prevEv = me ;
    
    let theta = pitch / 180 * PI,
        phi = azimut / 180 * PI,
        radius = DISTANCE ;
    
    camera.position.set(
      	radius * sin(theta) * sin(phi),
      	radius * cos(theta),
      	radius * sin(theta) * cos(phi),
    ) ;
  	camera.lookAt(center) ;
    
    renderer.render(scene, camera) ;
}

function up(ue) {
		isDown = false ;
}

renderer.domElement.onmousedown = down ;
window.onmousemove = move ;
window.onmouseup = up ;

renderer.render(scene, camera) ;

body {
  display: flex;
  flex-direction: row;
  justify-content: center;
  align-items: center;
  height: 100vh;
  margin: 0;
  background: #1c2228;
  overflow: hidden;
}

<script src="https://cdnjs.cloudflare.com/ajax/libs/three.js/101/three.min.js"></script>
<script src="https://threejs.org/examples/js/curves/NURBSUtils.js"></script>
<script src="https://threejs.org/examples/js/curves/NURBSCurve.js"></script>
<script src="https://threejs.org/examples/js/curves/NURBSSurface.js"></script>

<canvas></canvas>

解决方案

Here is the way how you can draw Bezier Triangle (snippet below) - algorithm is in Geometry class. Number of triangles in one side of the triangle you set in constructor. In code I made hard separation between algorithm/calculations (Geometry class) and drawing code (Draw class).

For bezier triangle we need to use 10 control points (9 for edges and one for "plane") like in below picture (src here ):

In this code, we don't use normals, and b points names are changed to p (eg. b003 to p003). We use following formula (for cubic Bezier triangles n=3)

Where p_ijk is control point (for n=3 above sum has 10 elements so we have 10 control points), and where B^n_ijk(r,s,t) are Bernstein polynomials defined for i,j,k>=0 and i+j+k=n

or 0 in other case. The domain of r,s,t in barycentric coordinates (where r,s,t are real numbers from [0, 1] and r+s+t=1) and where r=(r=1, s=t=0), s=(s=1, r=t=0), t=(t=1, r=s=0) looks as follows (the black points - we divide each triangle side to 5 parts - but we can change it to any number)

We calculate this reqular positions for black domain dots in method barycentricCoords(n) and we define which point create which triangles in method genTrianglesIndexes(n) in Geometry class. However you can change this points positions and density to any (inside triangle) to get different surface-triangle division. Below is snippet which shows domain in 2D

let pp= ((s='.myCanvas',c=document.querySelector(s),ctx=c.getContext('2d'),id=ctx.createImageData(1,1)) => (x,y,r=0,g=0,b=0,a=255)=>(id.data.set([r,g,b,a]),ctx.putImageData(id, x, y),c))()


cr=[255,0,0,255];
cg=[0,255,0,255];
cb=[0,0,255,255];

w=400;
h=400;

const p1=[0,h-1];
const p2=[w-1,h-1];
const p3=[w/2,0];

mainTriangle=[p1,p2,p3];
//mainTriangle.map(p => pp(...p,...cr));

let n=5;
let points=[];

function calcPoint(p1,p2,p3,r,s,t) {
  const px=p1[0]*r + p2[0]*s + p3[0]*t;
  const py=p1[1]*r + p2[1]*s + p3[1]*t;
  return [px,py];
}

// barycentric coordinates r,s,t of point in triangle
// the points given from triangle bottom to top line by line
// first line has n+1 pojnts, second has n, third n-1
// coordinates has property r+s+t=1
function barycentricCoords(n) {
  let rst=[];
  for(let i=0; i<=n; i++) for(let j=0; j<=n-i; j++) {
    s=(j/n);
    t=(i/n);    
    r=1-s-t;
    rst.push([r,s,t]);    
  }
  return rst;
}

// Procedure calc indexes for each triangle from 
// points list (in format returned by barycentricCoords(n) )
function genTrianglesIndexes(n) {
  let st=0; 
  let m=n;  
  let triangles=[];

  for(let j=n; j>0; j--) {    
    for(let i=0; i<m; i++) {    
      triangles.push([st+i, st+i+1, st+m+i+1]);
      if(i<m-1) triangles.push([st+i+1, st+m+i+2, st+m+i+1 ]);
    }
    m--;
    st+=j+1;  
  }
  
  return triangles;
}

function drawLine(p1,p2,c) {
  let n=Math.max(Math.abs(p1[0]-p2[0]),Math.abs(p1[1]-p2[1]))/2;
	for(let i=0; i<=n; i++) {
  	let s=i/n;
    let x=p1[0]*s + p2[0]*(1-s);
    let y=p1[1]*s + p2[1]*(1-s);
    pp(x,y,...c);
  }
}

function drawTriangle(p1,p2,p3,c) {
	drawLine(p1,p2,c);
  drawLine(p2,p3,c);
  drawLine(p3,p1,c);
}

// Bernstein Polynomial, i+j+k=n
function bp(n,i,j,k, r,s,t) {
  const f=x=>x?f(x-1)*x:1 // number fractional f(4)=1*2*3*4=24
  
  return r**i * s**j * t**k * f(n) / (f(i)*f(j)*f(k));  
}

//drawTriangle(...mainTriangle,cr); // draw main triangle

let bar=barycentricCoords(n);  // each domain point barycentric coordinates

let ti=genTrianglesIndexes(n); // indexes in bar for each triangle

// triangles calculated to cartesian coordinate system
let triangles = ti.map(tr=> tr.map(x=>calcPoint(...mainTriangle,...bar[x]) ) ); 

triangles.map(t => drawTriangle(...t, cg));

// domain points calculated to cartesian coordinate system (for draw)
let dp = bar.map(x=> calcPoint(...mainTriangle,...x) );

// draw black dots (4 pixels for each dot)
dp.map(x=> pp(x[0],x[1]) )
dp.map(x=> pp(x[0],x[1]-1) )
dp.map(x=> pp(x[0]-1,x[1]) )
dp.map(x=> pp(x[0]-1,x[1]-1) )

<canvas class="myCanvas" width=400 height=400 style="background: white"></canvas>

Below is final snippet with 3D bezier cubic triangle ( algorithm starts in method genTrianglesForCubicBezierTriangle(n, controlPoints) in Geometry class) - (caution: It is strange, but in SO snippets after first run you will NOT see lines, and you need reload page and run it again to see triangles-lines)

///////////////////////////////////////////////////////
// THIS PART/CLASS IS FOR ALGORITHMS AND CALCULATIONS
///////////////////////////////////////////////////////
class Geometry {

  constructor() { this.init(); } 

  init(n) {
    this.pts = [
      { x:-16, y: -8, z:0,  color:0xcc0000 }, // p003 RED
      { x:8,   y:-12, z:0,  color:0x888888 }, // p201
      { x:-8,  y:-12, z:0,  color:0x999999 }, // p102    
      { x:16,  y:-8,  z:0,  color:0x00cc00 }, // p300 GREEN
      { x:12,  y:-6,  z:-8, color:0x777777 }, // p210
      { x:8,   y:6,   z:-8, color:0x666666 }, // p120
      { x:0,   y:12,  z:0,  color:0x0000cc }, // p030 BLUE
      { x:-8,  y:6,   z:-8, color:0x555555 }, // p021
      { x:-12, y:-6,  z:-8, color:0x444444 }, // p012
      { x:0,   y:0,   z:8,  color:0xffff00 }, // p111 YELLOW (plane control point)
    ];
    
    this.mainTriangle = [this.pts[0],this.pts[3],this.pts[6]];
    
    this.bezierCurvesPoints = [
    	[ this.pts[0], this.pts[2], this.pts[1], this.pts[3] ],
        [ this.pts[3], this.pts[4], this.pts[5], this.pts[6] ],
        [ this.pts[6], this.pts[7], this.pts[8], this.pts[0] ]
    ];
    
    //this.triangles = [
    // { points: [this.pts[0], this.pts[1], this.pts[2]], color: null }, // wireframe
    // { points: [this.pts[1], this.pts[2], this.pts[3]], color: 0xffff00 } // yellow
    //]
    
    this.triangles = this.genTrianglesForCubicBezierTriangle(25, this.pts);
  }
  
  // n = number of triangles per triangle side
  genTrianglesForCubicBezierTriangle(n, controlPoints) {
    let bar= this.barycentricCoords(n);     // domain in barycentric coordinats   
    let ti = this.genTrianglesIndexes(n);   // indexes of triangles (in bar array)
        
    let val= bar.map(x => this.calcCubicBezierTriangleValue(controlPoints,...x));  // Calc Bezier triangle vertex for each domain (bar) point    
    let tv= ti.map(tr=> tr.map(x=>val[x]) );         // generate triangles using their indexes (ti) and val    
    return tv.map(t=> ({ points: t, color: null}) ); // map triangles to proper format (color=null gives wireframe)
    
    
    // Generate domain triangles
    //let td= ti.map(tr=> tr.map(x=>this.calcPointFromBar(...this.mainTriangle,...bar[x]) ) );     
    //this.trianglesDomain = td.map(t=> ({ points: t, color: null}) );
  }
  
  // more: https://www.mdpi.com/2073-8994/8/3/13/pdf
  // Bézier Triangles with G2 Continuity across Boundaries
  // Chang-Ki Lee, Hae-Do Hwang and Seung-Hyun Yoon
  calcCubicBezierTriangleValue(controlPoints, r,s,t ) {
    let p = controlPoints, b=[];  
    b[0]= this.bp(0,0,3,r,s,t); // p[0]=p003
    b[1]= this.bp(2,0,1,r,s,t); // p[1]=p201 
    b[2]= this.bp(1,0,2,r,s,t); // p[2]=p102
    b[3]= this.bp(3,0,0,r,s,t); // p[3]=p300
    b[4]= this.bp(2,1,0,r,s,t); // p[4]=p210
    b[5]= this.bp(1,2,0,r,s,t); // p[5]=p120
    b[6]= this.bp(0,3,0,r,s,t); // p[6]=p030
    b[7]= this.bp(0,2,1,r,s,t); // p[7]=p021
    b[8]= this.bp(0,1,2,r,s,t); // p[8]=p012
    b[9]= this.bp(1,1,1,r,s,t); // p[9]=p111
    
    let x=0, y=0, z=0;
    for(let i=0; i<=9; i++) {
      x+=p[i].x*b[i];
      y+=p[i].y*b[i];
      z+=p[i].z*b[i];
    }
    return { x:x, y:y, z:z };
  }
  
  // Bernstein Polynomial degree n, i+j+k=n
  bp(i,j,k, r,s,t, n=3) {
    const f=x=>x?f(x-1)*x:1 // number fractional f(4)=1*2*3*4=24    
    return r**i * s**j * t**k * f(n) / (f(i)*f(j)*f(k));  
  }
  
  coordArrToObj(p) { return { x:p[0], y:p[1], z:p[2] } } 
  
  // Calc cartesian point from barycentric coords system
  calcPointFromBar(p1,p2,p3,r,s,t) {  
    const px=p1.x*r + p2.x*s + p3.x*t;
    const py=p1.y*r + p2.y*s + p3.y*t;
    const pz=p1.z*r + p2.z*s + p3.z*t;       
    return { x:px, y:py,  z:pz};
  }

  // barycentric coordinates r,s,t of point in triangle
  // the points given from triangle bottom to top line by line
  // first line has n+1 pojnts, second has n, third n-1
  // coordinates has property r+s+t=1
  barycentricCoords(n) {
    let rst=[];
    for(let i=0; i<=n; i++) for(let j=0; j<=n-i; j++) {
      let s=(j/n);
      let t=(i/n);    
      let r=1-s-t;
      rst.push([r,s,t]);    
    }
    return rst;
  }

  // Procedure calc indexes for each triangle from 
  // points list (in format returned by barycentricCoords(n) )
  genTrianglesIndexes(n) {
    let st=0; 
    let m=n;  
    let triangles=[];

    for(let j=n; j>0; j--) {    
      for(let i=0; i<m; i++) {    
        triangles.push([st+i, st+i+1, st+m+i+1]);
        if(i<m-1) triangles.push([st+i+1, st+m+i+2, st+m+i+1 ]);
      }
      m--;
      st+=j+1;  
    }

    return triangles;
  }
  
  // This procedures are interface for Draw class 
  getPoints() { return this.pts }
  getTriangles() { return this.triangles }
  getBezierCurves() { return this.bezierCurvesPoints; }
}


///////////////////////////////////////////////
// THIS PART IS FOR DRAWING
///////////////////////////////////////////////

// init tree js and draw geometry objects
class Draw {

  constructor(geometry) { this.init(geometry); }
  
  initGeom() {
  	this.geometry.getPoints().forEach(p=> this.createPoint(p));
    this.geometry.getTriangles().forEach(t=> this.createTriangle(t));
    
    this.geometry.getBezierCurves().forEach(c=> this.createEdge(...c));
  }

  init(geometry) {
    this.geometry = geometry;
    this.W = 480,
    this.H = 400,
    this.DISTANCE = 100 ;
    this.PI = Math.PI,
    
  
    this.renderer = new THREE.WebGLRenderer({
      canvas : document.querySelector('canvas'),
      antialias : true,
      alpha : true
    }),
    this.camera = new THREE.PerspectiveCamera(25, this.W/this.H),
    this.scene = new THREE.Scene(),
    this.center = new THREE.Vector3(0, 0, 0),

		this.pts = [] ;
    
    this.renderer.setClearColor(0x000000, 0) ;

    this.renderer.setSize(this.W, this.H) ;
    // camera.position.set(-48, 32, 80) ;
    this.camera.position.set(0, 0, this.DISTANCE) ;
    this.camera.lookAt(this.center) ;
    
    this.initGeom();
    
    this.azimut = 0;
    this.pitch = 90;
    this.isDown = false;
    this.prevEv = null;

    

    this.renderer.domElement.onmousedown = e => this.down(e) ;
    window.onmousemove = e => this.move(e) ;
    window.onmouseup = e => this.up(e) ;

    this.renderer.render(this.scene, this.camera) ;
    
  }
    
  createPoint(p) {
    let {x, y, z, color} = p;
		let pt = new THREE.Mesh(
      new THREE.SphereGeometry(1, 10, 10),
      new THREE.MeshBasicMaterial({ color })
    ) ;
    pt.position.set(x, y, z) ;
    pt.x = x ;
    pt.y = y ;
    pt.z = z ;
    this.pts.push(pt) ;
    
    this.scene.add(pt) ;
	}
  
  createTriangle(t) {    
    var geom = new THREE.Geometry();
    var v1 = new THREE.Vector3(t.points[0].x, t.points[0].y, t.points[0].z);
    var v2 = new THREE.Vector3(t.points[1].x, t.points[1].y, t.points[1].z);
    var v3 = new THREE.Vector3(t.points[2].x, t.points[2].y, t.points[2].z);

    geom.vertices.push(v1);
    geom.vertices.push(v2);
    geom.vertices.push(v3);
        
    let material = new THREE.MeshNormalMaterial({wireframe: true,}) 
    if(t.color != null) material = new THREE.MeshBasicMaterial( { 
    	color: t.color,
      side: THREE.DoubleSide,
      } );
    

    geom.faces.push( new THREE.Face3( 0, 1, 2 ) );
    geom.computeFaceNormals();

    var mesh= new THREE.Mesh( geom, material);
    this.scene.add(mesh) ;
  }
  
  createEdge(pt1, pt2, pt3, pt4) {
 
		let curve = new THREE.CubicBezierCurve3(
          new THREE.Vector3(pt1.x, pt1.y, pt1.z),
          new THREE.Vector3(pt2.x, pt2.y, pt2.z),
          new THREE.Vector3(pt3.x, pt3.y, pt3.z),
          new THREE.Vector3(pt4.x, pt4.y, pt4.z),
        ),
    		mesh = new THREE.Mesh(
          new THREE.TubeGeometry(curve, 8, 0.5, 8, false),
          new THREE.MeshBasicMaterial({
            color : 0x203040
          })
        ) ;
        
    this.scene.add(mesh) ;
}
  
  down(de) {
        this.prevEv = de ;
        this.isDown = true ;
    }

  move(me) {
    if (!this.isDown) return ;

    this.azimut -= (me.clientX - this.prevEv.clientX) * 0.5 ;
    this.azimut %= 360 ;
    if (this.azimut < 0) this.azimut = 360 - this.azimut ;

    this.pitch -= (me.clientY - this.prevEv.clientY) * 0.5 ;
    if (this.pitch < 1) this.pitch = 1 ;
    if (this.pitch > 180) this.pitch = 180 ;

    this.prevEv = me ;

    let theta = this.pitch / 180 * this.PI,
        phi = this.azimut / 180 * this.PI,
        radius = this.DISTANCE ;

    this.camera.position.set(
      radius * Math.sin(theta) * Math.sin(phi),
      radius * Math.cos(theta),
      radius * Math.sin(theta) * Math.cos(phi),
    ) ;
    this.camera.lookAt(this.center) ;

    this.renderer.render(this.scene, this.camera) ;
  }

  up(ue) {
    this.isDown = false ;
  }
}

// SYSTEM SET UP
let geom= new Geometry();
let draw = new Draw(geom);

body {
  display: flex;
  flex-direction: row;
  justify-content: center;
  align-items: center;
  height: 100vh;
  margin: 0;
  background: #1c2228;
  overflow: hidden;
}

<script src="https://cdnjs.cloudflare.com/ajax/libs/three.js/101/three.min.js"></script>

<canvas></canvas>

Fiddle version is here . I put info in comments but algorithm is complicated and if you have questions - ask them as comments - I will answer.

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