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