Эхлэл Бүгдийг харах Тусламж
Нэвтрэх
javascript
/
three.js
-ын хуулбар https://github.com/tazdij/three.js.git
2
0
Салаа 0
Файлууд Асуудлууд 0 Мэдлэгийн сан
Эх сурвалжийг харах

moved NURBS utility function to NURBSUtils and code clean-up

renej-github 12 жил өмнө
parent
e154333d2b
commit
ca803d97c0
5 өөрчлөгдсөн 405 нэмэгдсэн , 410 устгасан
Өөрчлөллтийг нэгтгэж харах Өөрчлөлтийн статистик харах
  1. 1 0
      examples/canvas_geometry_nurbs.html
  2. 7 6
      examples/webgl_geometry_nurbs.html
  3. 2 346
      src/extras/core/NURBSCurve.js
  4. 1 58
      src/extras/core/NURBSSurface.js
  5. 394 0
      src/extras/core/NURBSUtils.js

+ 1 - 0
examples/canvas_geometry_nurbs.html
Файл харах 

@@ -28,6 +28,7 @@
 
 
 
 		<script src="../build/three.min.js"></script>
 
 		<script src="../build/three.min.js"></script>
 
 		<script src="js/libs/stats.min.js"></script>
 
 		<script src="js/libs/stats.min.js"></script>
 
+    		<script src="../src/extras/core/NURBSUtils.js"></script>
 
     		<script src="../src/extras/core/NURBSCurve.js"></script>
 
     		<script src="../src/extras/core/NURBSCurve.js"></script>
 
 
 
 		<script>
 
 		<script>

+ 7 - 6
examples/webgl_geometry_nurbs.html
Файл харах 

@@ -1,6 +1,6 @@
 
 <!DOCTYPE html>
 
 <!DOCTYPE html>
 
 <html lang="en">
 
 <html lang="en">
 
-	<!-- based on webgl_geometry_shapes.html -->
 
+	<!-- based on webgl_geometry_shapes.html and webgl_geometries2.html -->
 
 	<head>
 
 	<head>
 
 		<title>three.js webgl - geometry - NURBS</title>
 
 		<title>three.js webgl - geometry - NURBS</title>
 
 		<meta charset="utf-8">
 
 		<meta charset="utf-8">
 
@@ -24,9 +24,10 @@
 
 	<body>
 
 	<body>
 
 		<canvas id="debug" style="position:absolute; left:100px"></canvas>
 
 		<canvas id="debug" style="position:absolute; left:100px"></canvas>
 
 
 
-		<div id="info"><a href="http://threejs.org" target="_blank">three.js</a> - NURBS curve example</div>
 
+		<div id="info"><a href="http://threejs.org" target="_blank">three.js</a> - NURBS curve and surface example</div>
 
 
 
 		<script src="../build/three.min.js"></script>
 
 		<script src="../build/three.min.js"></script>
 
+		<script src="../src/extras/core/NURBSUtils.js"></script>
 
 		<script src="../src/extras/core/NURBSCurve.js"></script>
 
 		<script src="../src/extras/core/NURBSCurve.js"></script>
 
 		<script src="../src/extras/core/NURBSSurface.js"></script>
 
 		<script src="../src/extras/core/NURBSSurface.js"></script>
 
 
 
@@ -124,7 +125,7 @@
 
 				parent.add( nurbsLine );
 
 				parent.add( nurbsLine );
 
 				parent.add( nurbsControlPointsLine );
 
 				parent.add( nurbsControlPointsLine );
 
 
 
-				// NURBS Surface
 
+				// NURBS surface
 
 
 
 				var nsControlPoints = [
 
 				var nsControlPoints = [
 
 					[
 
 					[
 
@@ -135,8 +136,8 @@
 
 					],
 
 					],
 
 					[
 
 					[
 
 						new THREE.Vector4 ( 0, -200, 0, 1 ),
 
 						new THREE.Vector4 ( 0, -200, 0, 1 ),
 
-						new THREE.Vector4 ( 0, -100, -100, 1 ),
 
-						new THREE.Vector4 ( 0, 100, 150, 1 ),
 
+						new THREE.Vector4 ( 0, -100, -100, 5 ),
 
+						new THREE.Vector4 ( 0, 100, 150, 5 ),
 
 						new THREE.Vector4 ( 0, 200, 0, 1 )
 
 						new THREE.Vector4 ( 0, 200, 0, 1 )
 
 					],
 
 					],
 
 					[
 
 					[
 
@@ -165,7 +166,7 @@
 
 					return nurbsSurface.getPoint(u, v);
 
 					return nurbsSurface.getPoint(u, v);
 
 				};
 
 				};
 
 
 
-				var geo = new THREE.ParametricGeometry( getSurfacePoint, 50, 50 );
 
+				var geo = new THREE.ParametricGeometry( getSurfacePoint, 20, 20 );
 
 				var object = THREE.SceneUtils.createMultiMaterialObject( geo, materials );
 
 				var object = THREE.SceneUtils.createMultiMaterialObject( geo, materials );
 
 				object.position.set( 0, 100, 0 );
 
 				object.position.set( 0, 100, 0 );
 
 				object.scale.multiplyScalar( 1 );
 
 				object.scale.multiplyScalar( 1 );

+ 2 - 346
src/extras/core/NURBSCurve.js
Файл харах 

@@ -34,7 +34,7 @@ THREE.NURBSCurve.prototype.getPoint = function ( t ) {
 
 	var u = this.knots[0] + t * (this.knots[this.knots.length - 1] - this.knots[0]); // linear mapping t->u
 
 	var u = this.knots[0] + t * (this.knots[this.knots.length - 1] - this.knots[0]); // linear mapping t->u
 
 
 
 	// following results in (wx, wy, wz, w) homogeneous point
 
 	// following results in (wx, wy, wz, w) homogeneous point
 
-	var hpoint = THREE.NURBSCurve.Utils.calcBSplinePoint(this.degree, this.knots, this.controlPoints, u);
 
+	var hpoint = THREE.NURBSUtils.calcBSplinePoint(this.degree, this.knots, this.controlPoints, u);
 
 
 
 	if (hpoint.w != 1.0) { // project to 3D space: (wx, wy, wz, w) -> (x, y, z, 1)
 
 	if (hpoint.w != 1.0) { // project to 3D space: (wx, wy, wz, w) -> (x, y, z, 1)
 
 		hpoint.divideScalar(hpoint.w);
 
 		hpoint.divideScalar(hpoint.w);
 
@@ -47,354 +47,10 @@ THREE.NURBSCurve.prototype.getPoint = function ( t ) {
 
 THREE.NURBSCurve.prototype.getTangent = function ( t ) {
 
 THREE.NURBSCurve.prototype.getTangent = function ( t ) {
 
 
 
 	var u = this.knots[0] + t * (this.knots[this.knots.length - 1] - this.knots[0]);
 
 	var u = this.knots[0] + t * (this.knots[this.knots.length - 1] - this.knots[0]);
 
-	var ders = THREE.NURBSCurve.Utils.calcNURBSDerivatives(this.degree, this.knots, this.controlPoints, u, 1);
 
+	var ders = THREE.NURBSUtils.calcNURBSDerivatives(this.degree, this.knots, this.controlPoints, u, 1);
 
 	var tangent = ders[1].clone();
 
 	var tangent = ders[1].clone();
 
 	tangent.normalize();
 
 	tangent.normalize();
 
 
 
 	return tangent;
 
 	return tangent;
 
 };
 
 };
 
 
 
-
 
-/**************************************************************
 
- *	Utils
 
- **************************************************************/
 
-
 
-THREE.NURBSCurve.Utils = {
 
-
 
-	/*
 
-	Finds knot vector span.
 
-
 
-	p : degree
 
-	u : parametric value
 
-	U : knot vector
 
-	
-	returns the span
 
-	*/
 
-	findSpan: function( p,  u,  U ) {
 
-		var n = U.length - p - 1;
 
-
 
-		if (u >= U[n]) {
 
-			return n - 1;
 
-		}
 
-
 
-		if (u <= U[p]) {
 
-			return p;
 
-		}
 
-
 
-		var low = p;
 
-		var high = n;
 
-		var mid = Math.floor((low + high) / 2);
 
-
 
-		while (u < U[mid] || u >= U[mid + 1]) {
 
-		  
-			if (u < U[mid]) {
 
-				high = mid;
 
-			} else {
 
-				low = mid;
 
-			}
 
-
 
-			mid = Math.floor((low + high) / 2);
 
-		}
 
-
 
-		return mid;
 
-	},
 
-    
-		
-	/*
 
-	Calculate basis functions. See The NURBS Book, page 70, algorithm A2.2
 
-   
-	span : span in which u lies
 
-	u    : parametric point
 
-	p    : degree
 
-	U    : knot vector
 
-	
-	returns array[p+1] with basis functions values.
 
-	*/
 
-	calcBasisFunctions: function( span, u, p, U ) {
 
-		var N = [];
 
-		var left = [];
 
-		var right = [];
 
-		N[0] = 1.0;
 
-
 
-		for (var j = 1; j <= p; ++j) {
 
-	   
-			left[j] = u - U[span + 1 - j];
 
-			right[j] = U[span + j] - u;
 
-
 
-			var saved = 0.0;
 
-
 
-			for (var r = 0; r < j; ++r) {
 
-
 
-				var rv = right[r + 1];
 
-				var lv = left[j - r];
 
-				var temp = N[r] / (rv + lv);
 
-				N[r] = saved + rv * temp;
 
-				saved = lv * temp;
 
-			 }
 
-
 
-			 N[j] = saved;
 
-		 }
 
-
 
-		 return N;
 
-	},
 
-
 
-	/*
 
-	Calculate B-Spline curve points. See The NURBS Book, page 82, algorithm A3.1.
 
- 
-	p : degree of B-Spline
 
-	U : knot vector
 
-	P : control points (x, y, z, w)
 
-	u : parametric point
 
-
 
-	returns point for given u
 
-	*/
 
-	calcBSplinePoint: function( p, U, P, u ) {
 
-		var span = this.findSpan(p, u, U);
 
-		var N = this.calcBasisFunctions(span, u, p, U);
 
-		var C = new THREE.Vector4(0, 0, 0, 0);
 
-
 
-		for (var j = 0; j <= p; ++j) {
 
-			var point = P[span - p + j];
 
-			var Nj = N[j];
 
-			var wNj = point.w * Nj;
 
-			C.x += point.x * wNj;
 
-			C.y += point.y * wNj;
 
-			C.z += point.z * wNj;
 
-			C.w += point.w * Nj;
 
-		}
 
-
 
-		return C;
 
-	},
 
-
 
-	/*
 
-	Calculate basis functions derivatives. See The NURBS Book, page 72, algorithm A2.3.
 
-
 
-	span : span in which u lies
 
-	u    : parametric point
 
-	p    : degree
 
-	n    : number of derivatives to calculate
 
-	U    : knot vector
 
-
 
-	returns array[n+1][p+1] with basis functions derivatives
 
-	*/
 
-	calcBasisFunctionDerivatives: function( span,  u,  p,  n,  U ) {
 
-
 
-		var zeroArr = [];
 
-		for (var i = 0; i <= p; ++i)
 
-			zeroArr[i] = 0.0;
 
-
 
-		var ders = [];
 
-		for (var i = 0; i <= n; ++i)
 
-			ders[i] = zeroArr.slice(0);
 
-
 
-		var ndu = [];
 
-		for (var i = 0; i <= p; ++i)
 
-			ndu[i] = zeroArr.slice(0);
 
-
 
-		ndu[0][0] = 1.0;
 
-
 
-		var left = zeroArr.slice(0);
 
-		var right = zeroArr.slice(0);
 
-
 
-		for (var j = 1; j <= p; ++j) {
 
-			left[j] = u - U[span + 1 - j];
 
-			right[j] = U[span + j] - u;
 
-
 
-			var saved = 0.0;
 
-
 
-			for (var r = 0; r < j; ++r) {
 
-				var rv = right[r + 1];
 
-				var lv = left[j - r];
 
-				ndu[j][r] = rv + lv;
 
-
 
-				var temp = ndu[r][j - 1] / ndu[j][r];
 
-				ndu[r][j] = saved + rv * temp;
 
-				saved = lv * temp;
 
-			}
 
-
 
-			ndu[j][j] = saved;
 
-		}
 
-
 
-		for (var j = 0; j <= p; ++j) {
 
-			ders[0][j] = ndu[j][p];
 
-		}
 
-
 
-		for (var r = 0; r <= p; ++r) {
 
-			var s1 = 0;
 
-			var s2 = 1;
 
-
 
-			var a = [];
 
-			for (var i = 0; i <= p; ++i) {
 
-				a[i] = zeroArr.slice(0);
 
-			}
 
-			a[0][0] = 1.0;
 
-
 
-			for (var k = 1; k <= n; ++k) {
 
-				var d = 0.0;
 
-				var rk = r - k;
 
-				var pk = p - k;
 
-
 
-				if (r >= k) {
 
-					a[s2][0] = a[s1][0] / ndu[pk + 1][rk];
 
-					d = a[s2][0] * ndu[rk][pk];
 
-				}
 
-
 
-				var j1 = (rk >= -1) ? 1 : -rk;
 
-				var j2 = (r - 1 <= pk) ? k - 1 :  p - r;
 
-
 
-				for (var j = j1; j <= j2; ++j) {
 
-					a[s2][j] = (a[s1][j] - a[s1][j - 1]) / ndu[pk + 1][rk + j];
 
-					d += a[s2][j] * ndu[rk + j][pk];
 
-				}
 
-
 
-				if (r <= pk) {
 
-					a[s2][k] = -a[s1][k - 1] / ndu[pk + 1][r];
 
-					d += a[s2][k] * ndu[r][pk];
 
-				}
 
-
 
-				ders[k][r] = d;
 
-
 
-				var j = s1;
 
-				s1 = s2;
 
-				s2 = j;
 
-			}
 
-		}
 
-
 
-		var r = p;
 
-
 
-		for (var k = 1; k <= n; ++k) {
 
-			for (var j = 0; j <= p; ++j) {
 
-				ders[k][j] *= r;
 
-			}
 
-			r *= p - k;
 
-		}
 
-
 
-		return ders;
 
-	},
 
-
 
- 	/*
 
-	Calculate derivatives of a B-Spline. See The NURBS Book, page 93, algorithm A3.2.
 
-
 
-	p  : degree
 
-	U  : knot vector
 
-	P  : control points
 
-	u  : Parametric points
 
-	nd : number of derivatives
 
-
 
-	returns array[d+1] with derivatives
 
-	*/
 
-	calcBSplineDerivatives: function( p,  U,  P,  u,  nd ) {
 
-		var du = nd < p ? nd : p;
 
-		var CK = [];
 
-		var span = this.findSpan(p, u, U);
 
-		var nders = this.calcBasisFunctionDerivatives(span, u, p, du, U);
 
-		var Pw = [];
 
-
 
-		for (var i = 0; i < P.length; ++i) {
 
-			var point = P[i].clone();
 
-			var w = point.w;
 
-
 
-			point.x *= w;
 
-			point.y *= w;
 
-			point.z *= w;
 
-
 
-			Pw[i] = point;
 
-		}
 
-		for (var k = 0; k <= du; ++k) {
 
-			var point = Pw[span - p].clone().multiplyScalar(nders[k][0]);
 
-
 
-			for (var j = 1; j <= p; ++j) {
 
-				point.add(Pw[span - p + j].clone().multiplyScalar(nders[k][j]));
 
-			}
 
-
 
-			CK[k] = point;
 
-		}
 
-
 
-		for (var k = du + 1; k <= nd + 1; ++k) {
 
-			CK[k] = new THREE.Vector4(0, 0, 0);
 
-		}
 
-
 
-		return CK;
 
-	},
 
-
 
-
 
-	/*
 
-	Calculate "K over I"
 
-
 
-	returns k!/(i!(k-i)!)
 
-	*/
 
-	calcKoverI: function( k, i ) {
 
-		var nom = 1;
 
-
 
-		for (var j = 2; j <= k; ++j) {
 
-			nom *= j;
 
-		}
 
-
 
-		var denom = 1;
 
-
 
-		for (var j = 2; j <= i; ++j) {
 
-			denom *= j;
 
-		}
 
-
 
-		for (var j = 2; j <= k - i; ++j) {
 
-			denom *= j;
 
-		}
 
-
 
-		return nom / denom;
 
-	},
 
-
 
-
 
-	/*
 
-	Calculate derivatives (0-nd) of rational curve. See The NURBS Book, page 127, algorithm A4.2.
 
-
 
-	Pders : result of function calcBSplineDerivatives
 
-
 
-	returns array with derivatives for rational curve.
 
-	*/
 
-	calcRationalCurveDerivatives: function ( Pders ) {
 
-		var nd = Pders.length;
 
-		var Aders = [];
 
-		var wders = [];
 
-
 
-		for (var i = 0; i < nd; ++i) {
 
-			var point = Pders[i];
 
-			Aders[i] = new THREE.Vector3(point.x, point.y, point.z);
 
-			wders[i] = point.w;
 
-		}
 
-
 
-		var CK = [];
 
-
 
-		for (var k = 0; k < nd; ++k) {
 
-			var v = Aders[k].clone();
 
-
 
-			for (var i = 1; i <= k; ++i) {
 
-				v.sub(CK[k - i].clone().multiplyScalar(this.calcKoverI(k,i) * wders[i]));
 
-			}
 
-
 
-			CK[k] = v.divideScalar(wders[0]);
 
-		}
 
-
 
-		return CK;
 
-	},
 
-
 
-
 
-	/*
 
-	Calculate NURBS curve derivatives. See The NURBS Book, page 127, algorithm A4.2.
 
-
 
-	p  : degree
 
-	U  : knot vector
 
-	P  : control points in homogeneous space
 
-	u  : parametric points
 
-	nd : number of derivatives
 
-
 
-	returns array with derivatives.
 
-	*/
 
-	calcNURBSDerivatives: function( p,  U,  P,  u,  nd ) {
 
-		var Pders = this.calcBSplineDerivatives(p, U, P, u, nd);
 
-		return this.calcRationalCurveDerivatives(Pders);
 
-	}
 
-
 
-};
 
-
 
-
 
-

+ 1 - 58
src/extras/core/NURBSSurface.js
Файл харах 

@@ -2,8 +2,6 @@
 
  * @author renej
 
  * @author renej
 
  * NURBS surface object
 
  * NURBS surface object
 
  *
 
  *
 
- * Derives from Curve, overriding getPoint and getTangent.
 
- *
 
  * Implementation is based on (x, y [, z=0 [, w=1]]) control points with w=weight.
 
  * Implementation is based on (x, y [, z=0 [, w=1]]) control points with w=weight.
 
  *
 
  *
 
  **/
 
  **/
 
@@ -24,14 +22,10 @@ THREE.NURBSSurface = function ( degree1, degree2, knots1, knots2 /* arrays of re
 
 	var len1 = knots1.length - degree1 - 1;
 
 	var len1 = knots1.length - degree1 - 1;
 
 	var len2 = knots2.length - degree2 - 1;
 
 	var len2 = knots2.length - degree2 - 1;
 
 
 
-	//console.log("controlPoints: " + controlPoints);
 
-	//console.log("controlPoints[0][0].x: " + controlPoints[0][0].x);
 
-
 
 	// ensure Vector4 for control points
 
 	// ensure Vector4 for control points
 
 	for (var i = 0; i < len1; ++i) {
 
 	for (var i = 0; i < len1; ++i) {
 
 		this.controlPoints[i] = []
 
 		this.controlPoints[i] = []
 
 		for (var j = 0; j < len2; ++j) {
 
 		for (var j = 0; j < len2; ++j) {
 
-			//console.log("i=" + i + " j=" + j);
 
 			var point = controlPoints[i][j];
 
 			var point = controlPoints[i][j];
 
 			this.controlPoints[i][j] = new THREE.Vector4(point.x, point.y, point.z, point.w);
 
 			this.controlPoints[i][j] = new THREE.Vector4(point.x, point.y, point.z, point.w);
 
 		}
 
 		}
 
@@ -45,62 +39,11 @@ THREE.NURBSSurface.prototype = {
 
 
 
 	getPoint: function ( t1, t2 ) {
 
 	getPoint: function ( t1, t2 ) {
 
 
 
-		//console.log("this.knots1=" + this.knots1);
 
-		//console.log("this.knots2=" + this.knots2);
 
-		//console.log("this.controlPoints=" + this.controlPoints);
 
 		var u = this.knots1[0] + t1 * (this.knots1[this.knots1.length - 1] - this.knots1[0]); // linear mapping t1->u
 
 		var u = this.knots1[0] + t1 * (this.knots1[this.knots1.length - 1] - this.knots1[0]); // linear mapping t1->u
 
 		var v = this.knots2[0] + t2 * (this.knots2[this.knots2.length - 1] - this.knots2[0]); // linear mapping t2->u
 
 		var v = this.knots2[0] + t2 * (this.knots2[this.knots2.length - 1] - this.knots2[0]); // linear mapping t2->u
 
 
 
-		var point = THREE.NURBSSurface.Utils.calcSurfacePoint(this.degree1, this.degree2, this.knots1, this.knots2, this.controlPoints, u, v);
 
-		//console.log("u=" + u + " v=" + v + " point=" + point.x + ", " + point.y + ", " + point.z);
 
-		return point;
 
+		return THREE.NURBSUtils.calcSurfacePoint(this.degree1, this.degree2, this.knots1, this.knots2, this.controlPoints, u, v);
 
 	}
 
 	}
 
 };
 
 };
 
 
 
 
 
-/**************************************************************
 
- *	Utils
 
- **************************************************************/
 
-
 
-THREE.NURBSSurface.Utils = {
 
-
 
-	/*
 
-	Calculate rational B-Spline surface point. See The NURBS Book, page 134, algorithm A4.3.
 
- 
-	p1, p2 : degrees of B-Spline surface
 
-	U1, U2 : knot vectors
 
-	P      : control points (x, y, z, w)
 
-	u, v   : parametric values
 
-
 
-	returns point for given (u, v)
 
-	*/
 
-	calcSurfacePoint: function( p, q, U, V, P, u, v ) {
 
-		var uspan = THREE.NURBSCurve.Utils.findSpan(p, u, U);
 
-		var vspan = THREE.NURBSCurve.Utils.findSpan(q, v, V);
 
-		var Nu = THREE.NURBSCurve.Utils.calcBasisFunctions(uspan, u, p, U);
 
-		var Nv = THREE.NURBSCurve.Utils.calcBasisFunctions(vspan, v, q, V);
 
-		var temp = [];
 
-
 
-		for (var l = 0; l <= q; ++l) {
 
-			temp[l] = new THREE.Vector4(0, 0, 0, 0);
 
-			for (var k = 0; k <= p; ++k) {
 
-				var point = P[uspan - p + k][vspan - q + l].clone();
 
-				var w = point.w;
 
-				point.x *= w;
 
-				point.y *= w;
 
-				point.z *= w;
 
-				temp[l].add(point.multiplyScalar(Nu[k]));
 
-			}
 
-		}
 
-
 
-		var Sw = new THREE.Vector4(0, 0, 0, 0);
 
-		for (var l = 0; l <= q; ++l) {
 
-			Sw.add(temp[l].multiplyScalar(Nv[l]));
 
-		}
 
-
 
-		Sw.divideScalar(Sw.w);
 
-		return new THREE.Vector3(Sw.x, Sw.y, Sw.z);
 
-	}
 
-
 
-};
 
-

+ 394 - 0
src/extras/core/NURBSUtils.js
Файл харах 

@@ -0,0 +1,394 @@
 
+/**
 
+ * @author renej
 
+ * NURBS utils
 
+ *
 
+ * See NURBSCurve and NURBSSurface.
 
+ *
 
+ **/
 
+
 
+
 
+/**************************************************************
 
+ *	NURBS Utils
 
+ **************************************************************/
 
+
 
+THREE.NURBSUtils = {
 
+
 
+	/*
 
+	Finds knot vector span.
 
+
 
+	p : degree
 
+	u : parametric value
 
+	U : knot vector
 
+	
+	returns the span
 
+	*/
 
+	findSpan: function( p,  u,  U ) {
 
+		var n = U.length - p - 1;
 
+
 
+		if (u >= U[n]) {
 
+			return n - 1;
 
+		}
 
+
 
+		if (u <= U[p]) {
 
+			return p;
 
+		}
 
+
 
+		var low = p;
 
+		var high = n;
 
+		var mid = Math.floor((low + high) / 2);
 
+
 
+		while (u < U[mid] || u >= U[mid + 1]) {
 
+		  
+			if (u < U[mid]) {
 
+				high = mid;
 
+			} else {
 
+				low = mid;
 
+			}
 
+
 
+			mid = Math.floor((low + high) / 2);
 
+		}
 
+
 
+		return mid;
 
+	},
 
+    
+		
+	/*
 
+	Calculate basis functions. See The NURBS Book, page 70, algorithm A2.2
 
+   
+	span : span in which u lies
 
+	u    : parametric point
 
+	p    : degree
 
+	U    : knot vector
 
+	
+	returns array[p+1] with basis functions values.
 
+	*/
 
+	calcBasisFunctions: function( span, u, p, U ) {
 
+		var N = [];
 
+		var left = [];
 
+		var right = [];
 
+		N[0] = 1.0;
 
+
 
+		for (var j = 1; j <= p; ++j) {
 
+	   
+			left[j] = u - U[span + 1 - j];
 
+			right[j] = U[span + j] - u;
 
+
 
+			var saved = 0.0;
 
+
 
+			for (var r = 0; r < j; ++r) {
 
+
 
+				var rv = right[r + 1];
 
+				var lv = left[j - r];
 
+				var temp = N[r] / (rv + lv);
 
+				N[r] = saved + rv * temp;
 
+				saved = lv * temp;
 
+			 }
 
+
 
+			 N[j] = saved;
 
+		 }
 
+
 
+		 return N;
 
+	},
 
+
 
+
 
+	/*
 
+	Calculate B-Spline curve points. See The NURBS Book, page 82, algorithm A3.1.
 
+ 
+	p : degree of B-Spline
 
+	U : knot vector
 
+	P : control points (x, y, z, w)
 
+	u : parametric point
 
+
 
+	returns point for given u
 
+	*/
 
+	calcBSplinePoint: function( p, U, P, u ) {
 
+		var span = this.findSpan(p, u, U);
 
+		var N = this.calcBasisFunctions(span, u, p, U);
 
+		var C = new THREE.Vector4(0, 0, 0, 0);
 
+
 
+		for (var j = 0; j <= p; ++j) {
 
+			var point = P[span - p + j];
 
+			var Nj = N[j];
 
+			var wNj = point.w * Nj;
 
+			C.x += point.x * wNj;
 
+			C.y += point.y * wNj;
 
+			C.z += point.z * wNj;
 
+			C.w += point.w * Nj;
 
+		}
 
+
 
+		return C;
 
+	},
 
+
 
+
 
+	/*
 
+	Calculate basis functions derivatives. See The NURBS Book, page 72, algorithm A2.3.
 
+
 
+	span : span in which u lies
 
+	u    : parametric point
 
+	p    : degree
 
+	n    : number of derivatives to calculate
 
+	U    : knot vector
 
+
 
+	returns array[n+1][p+1] with basis functions derivatives
 
+	*/
 
+	calcBasisFunctionDerivatives: function( span,  u,  p,  n,  U ) {
 
+
 
+		var zeroArr = [];
 
+		for (var i = 0; i <= p; ++i)
 
+			zeroArr[i] = 0.0;
 
+
 
+		var ders = [];
 
+		for (var i = 0; i <= n; ++i)
 
+			ders[i] = zeroArr.slice(0);
 
+
 
+		var ndu = [];
 
+		for (var i = 0; i <= p; ++i)
 
+			ndu[i] = zeroArr.slice(0);
 
+
 
+		ndu[0][0] = 1.0;
 
+
 
+		var left = zeroArr.slice(0);
 
+		var right = zeroArr.slice(0);
 
+
 
+		for (var j = 1; j <= p; ++j) {
 
+			left[j] = u - U[span + 1 - j];
 
+			right[j] = U[span + j] - u;
 
+
 
+			var saved = 0.0;
 
+
 
+			for (var r = 0; r < j; ++r) {
 
+				var rv = right[r + 1];
 
+				var lv = left[j - r];
 
+				ndu[j][r] = rv + lv;
 
+
 
+				var temp = ndu[r][j - 1] / ndu[j][r];
 
+				ndu[r][j] = saved + rv * temp;
 
+				saved = lv * temp;
 
+			}
 
+
 
+			ndu[j][j] = saved;
 
+		}
 
+
 
+		for (var j = 0; j <= p; ++j) {
 
+			ders[0][j] = ndu[j][p];
 
+		}
 
+
 
+		for (var r = 0; r <= p; ++r) {
 
+			var s1 = 0;
 
+			var s2 = 1;
 
+
 
+			var a = [];
 
+			for (var i = 0; i <= p; ++i) {
 
+				a[i] = zeroArr.slice(0);
 
+			}
 
+			a[0][0] = 1.0;
 
+
 
+			for (var k = 1; k <= n; ++k) {
 
+				var d = 0.0;
 
+				var rk = r - k;
 
+				var pk = p - k;
 
+
 
+				if (r >= k) {
 
+					a[s2][0] = a[s1][0] / ndu[pk + 1][rk];
 
+					d = a[s2][0] * ndu[rk][pk];
 
+				}
 
+
 
+				var j1 = (rk >= -1) ? 1 : -rk;
 
+				var j2 = (r - 1 <= pk) ? k - 1 :  p - r;
 
+
 
+				for (var j = j1; j <= j2; ++j) {
 
+					a[s2][j] = (a[s1][j] - a[s1][j - 1]) / ndu[pk + 1][rk + j];
 
+					d += a[s2][j] * ndu[rk + j][pk];
 
+				}
 
+
 
+				if (r <= pk) {
 
+					a[s2][k] = -a[s1][k - 1] / ndu[pk + 1][r];
 
+					d += a[s2][k] * ndu[r][pk];
 
+				}
 
+
 
+				ders[k][r] = d;
 
+
 
+				var j = s1;
 
+				s1 = s2;
 
+				s2 = j;
 
+			}
 
+		}
 
+
 
+		var r = p;
 
+
 
+		for (var k = 1; k <= n; ++k) {
 
+			for (var j = 0; j <= p; ++j) {
 
+				ders[k][j] *= r;
 
+			}
 
+			r *= p - k;
 
+		}
 
+
 
+		return ders;
 
+	},
 
+
 
+
 
+ 	/*
 
+	Calculate derivatives of a B-Spline. See The NURBS Book, page 93, algorithm A3.2.
 
+
 
+	p  : degree
 
+	U  : knot vector
 
+	P  : control points
 
+	u  : Parametric points
 
+	nd : number of derivatives
 
+
 
+	returns array[d+1] with derivatives
 
+	*/
 
+	calcBSplineDerivatives: function( p,  U,  P,  u,  nd ) {
 
+		var du = nd < p ? nd : p;
 
+		var CK = [];
 
+		var span = this.findSpan(p, u, U);
 
+		var nders = this.calcBasisFunctionDerivatives(span, u, p, du, U);
 
+		var Pw = [];
 
+
 
+		for (var i = 0; i < P.length; ++i) {
 
+			var point = P[i].clone();
 
+			var w = point.w;
 
+
 
+			point.x *= w;
 
+			point.y *= w;
 
+			point.z *= w;
 
+
 
+			Pw[i] = point;
 
+		}
 
+		for (var k = 0; k <= du; ++k) {
 
+			var point = Pw[span - p].clone().multiplyScalar(nders[k][0]);
 
+
 
+			for (var j = 1; j <= p; ++j) {
 
+				point.add(Pw[span - p + j].clone().multiplyScalar(nders[k][j]));
 
+			}
 
+
 
+			CK[k] = point;
 
+		}
 
+
 
+		for (var k = du + 1; k <= nd + 1; ++k) {
 
+			CK[k] = new THREE.Vector4(0, 0, 0);
 
+		}
 
+
 
+		return CK;
 
+	},
 
+
 
+
 
+	/*
 
+	Calculate "K over I"
 
+
 
+	returns k!/(i!(k-i)!)
 
+	*/
 
+	calcKoverI: function( k, i ) {
 
+		var nom = 1;
 
+
 
+		for (var j = 2; j <= k; ++j) {
 
+			nom *= j;
 
+		}
 
+
 
+		var denom = 1;
 
+
 
+		for (var j = 2; j <= i; ++j) {
 
+			denom *= j;
 
+		}
 
+
 
+		for (var j = 2; j <= k - i; ++j) {
 
+			denom *= j;
 
+		}
 
+
 
+		return nom / denom;
 
+	},
 
+
 
+
 
+	/*
 
+	Calculate derivatives (0-nd) of rational curve. See The NURBS Book, page 127, algorithm A4.2.
 
+
 
+	Pders : result of function calcBSplineDerivatives
 
+
 
+	returns array with derivatives for rational curve.
 
+	*/
 
+	calcRationalCurveDerivatives: function ( Pders ) {
 
+		var nd = Pders.length;
 
+		var Aders = [];
 
+		var wders = [];
 
+
 
+		for (var i = 0; i < nd; ++i) {
 
+			var point = Pders[i];
 
+			Aders[i] = new THREE.Vector3(point.x, point.y, point.z);
 
+			wders[i] = point.w;
 
+		}
 
+
 
+		var CK = [];
 
+
 
+		for (var k = 0; k < nd; ++k) {
 
+			var v = Aders[k].clone();
 
+
 
+			for (var i = 1; i <= k; ++i) {
 
+				v.sub(CK[k - i].clone().multiplyScalar(this.calcKoverI(k,i) * wders[i]));
 
+			}
 
+
 
+			CK[k] = v.divideScalar(wders[0]);
 
+		}
 
+
 
+		return CK;
 
+	},
 
+
 
+
 
+	/*
 
+	Calculate NURBS curve derivatives. See The NURBS Book, page 127, algorithm A4.2.
 
+
 
+	p  : degree
 
+	U  : knot vector
 
+	P  : control points in homogeneous space
 
+	u  : parametric points
 
+	nd : number of derivatives
 
+
 
+	returns array with derivatives.
 
+	*/
 
+	calcNURBSDerivatives: function( p,  U,  P,  u,  nd ) {
 
+		var Pders = this.calcBSplineDerivatives(p, U, P, u, nd);
 
+		return this.calcRationalCurveDerivatives(Pders);
 
+	},
 
+
 
+
 
+	/*
 
+	Calculate rational B-Spline surface point. See The NURBS Book, page 134, algorithm A4.3.
 
+ 
+	p1, p2 : degrees of B-Spline surface
 
+	U1, U2 : knot vectors
 
+	P      : control points (x, y, z, w)
 
+	u, v   : parametric values
 
+
 
+	returns point for given (u, v)
 
+	*/
 
+	calcSurfacePoint: function( p, q, U, V, P, u, v ) {
 
+		var uspan = this.findSpan(p, u, U);
 
+		var vspan = this.findSpan(q, v, V);
 
+		var Nu = this.calcBasisFunctions(uspan, u, p, U);
 
+		var Nv = this.calcBasisFunctions(vspan, v, q, V);
 
+		var temp = [];
 
+
 
+		for (var l = 0; l <= q; ++l) {
 
+			temp[l] = new THREE.Vector4(0, 0, 0, 0);
 
+			for (var k = 0; k <= p; ++k) {
 
+				var point = P[uspan - p + k][vspan - q + l].clone();
 
+				var w = point.w;
 
+				point.x *= w;
 
+				point.y *= w;
 
+				point.z *= w;
 
+				temp[l].add(point.multiplyScalar(Nu[k]));
 
+			}
 
+		}
 
+
 
+		var Sw = new THREE.Vector4(0, 0, 0, 0);
 
+		for (var l = 0; l <= q; ++l) {
 
+			Sw.add(temp[l].multiplyScalar(Nv[l]));
 
+		}
 
+
 
+		Sw.divideScalar(Sw.w);
 
+		return new THREE.Vector3(Sw.x, Sw.y, Sw.z);
 
+	}
 
+
 
+};
 
+
 
+
 
+