three.js/examples/js/curves/NURBSUtils.js

/**
 * @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);
	}

};