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