384 lines
11 KiB
C
384 lines
11 KiB
C
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/****************************************************************************
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* VCGLib o o *
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* Visual and Computer Graphics Library o o *
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* _ O _ *
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* Copyright(C) 2005 \/)\/ *
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* Visual Computing Lab /\/| *
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* ISTI - Italian National Research Council | *
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* \ *
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* All rights reserved. *
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* *
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* This program is free software; you can redistribute it and/or modify *
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* it under the terms of the GNU General Public License as published by *
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* the Free Software Foundation; either version 2 of the License, or *
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* (at your option) any later version. *
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* *
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* This program is distributed in the hope that it will be useful, *
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* but WITHOUT ANY WARRANTY; without even the implied warranty of *
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *
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* GNU General Public License (http://www.gnu.org/licenses/gpl.txt) *
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* for more details. *
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* *
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****************************************************************************/
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/****************************************************************************
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History
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$Log: not supported by cvs2svn $
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Revision 1.3 2006/03/29 10:12:08 corsini
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Add cast to avoid warning
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Revision 1.2 2005/12/12 12:08:30 cignoni
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First working version
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Revision 1.1 2005/11/21 15:58:12 cignoni
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First Release (not working!)
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Revision 1.13 2005/11/17 00:42:03 cignoni
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****************************************************************************/
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#ifndef _VCG_INERTIA_
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#define _VCG_INERTIA_
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/*
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The algorithm is based on a three step reduction of the volume integrals
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to successively simpler integrals. The algorithm is designed to minimize
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the numerical errors that can result from poorly conditioned alignment of
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polyhedral faces. It is also designed for efficiency. All required volume
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integrals of a polyhedron are computed together during a single walk over
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the boundary of the polyhedron; exploiting common subexpressions reduces
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floating point operations.
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For more information, check out:
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Brian Mirtich,
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``Fast and Accurate Computation of Polyhedral Mass Properties,''
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journal of graphics tools, volume 1, number 2, 1996
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*/
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#include <vcg/math/matrix33.h>
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#include <vcg/math/lin_algebra.h>
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#include <vcg/complex/trimesh/update/normal.h>
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namespace vcg
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{
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namespace tri
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{
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template <class InertiaMeshType>
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class Inertia
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{
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typedef InertiaMeshType MeshType;
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typedef typename MeshType::VertexType VertexType;
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typedef typename MeshType::VertexPointer VertexPointer;
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typedef typename MeshType::VertexIterator VertexIterator;
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typedef typename MeshType::ScalarType ScalarType;
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typedef typename MeshType::FaceType FaceType;
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typedef typename MeshType::FacePointer FacePointer;
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typedef typename MeshType::FaceIterator FaceIterator;
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typedef typename MeshType::ConstFaceIterator ConstFaceIterator;
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typedef typename MeshType::FaceContainer FaceContainer;
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typedef typename MeshType::CoordType CoordType;
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private :
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enum {X=0,Y=1,Z=2};
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inline ScalarType SQR(ScalarType &x) const { return x*x;}
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inline ScalarType CUBE(ScalarType &x) const { return x*x*x;}
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int A; /* alpha */
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int B; /* beta */
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int C; /* gamma */
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/* projection integrals */
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double P1, Pa, Pb, Paa, Pab, Pbb, Paaa, Paab, Pabb, Pbbb;
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/* face integrals */
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double Fa, Fb, Fc, Faa, Fbb, Fcc, Faaa, Fbbb, Fccc, Faab, Fbbc, Fcca;
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/* volume integrals */
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double T0, T1[3], T2[3], TP[3];
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public:
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/* compute various integrations over projection of face */
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void compProjectionIntegrals(FaceType &f)
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{
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double a0, a1, da;
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double b0, b1, db;
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double a0_2, a0_3, a0_4, b0_2, b0_3, b0_4;
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double a1_2, a1_3, b1_2, b1_3;
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double C1, Ca, Caa, Caaa, Cb, Cbb, Cbbb;
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double Cab, Kab, Caab, Kaab, Cabb, Kabb;
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int i;
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P1 = Pa = Pb = Paa = Pab = Pbb = Paaa = Paab = Pabb = Pbbb = 0.0;
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for (i = 0; i < 3; i++) {
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a0 = f.V(i)->P()[A];
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b0 = f.V(i)->P()[B];
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a1 = f.V1(i)->P()[A];
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b1 = f.V1(i)->P()[B];
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da = a1 - a0;
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db = b1 - b0;
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a0_2 = a0 * a0; a0_3 = a0_2 * a0; a0_4 = a0_3 * a0;
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b0_2 = b0 * b0; b0_3 = b0_2 * b0; b0_4 = b0_3 * b0;
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a1_2 = a1 * a1; a1_3 = a1_2 * a1;
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b1_2 = b1 * b1; b1_3 = b1_2 * b1;
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C1 = a1 + a0;
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Ca = a1*C1 + a0_2; Caa = a1*Ca + a0_3; Caaa = a1*Caa + a0_4;
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Cb = b1*(b1 + b0) + b0_2; Cbb = b1*Cb + b0_3; Cbbb = b1*Cbb + b0_4;
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Cab = 3*a1_2 + 2*a1*a0 + a0_2; Kab = a1_2 + 2*a1*a0 + 3*a0_2;
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Caab = a0*Cab + 4*a1_3; Kaab = a1*Kab + 4*a0_3;
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Cabb = 4*b1_3 + 3*b1_2*b0 + 2*b1*b0_2 + b0_3;
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Kabb = b1_3 + 2*b1_2*b0 + 3*b1*b0_2 + 4*b0_3;
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P1 += db*C1;
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Pa += db*Ca;
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Paa += db*Caa;
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Paaa += db*Caaa;
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Pb += da*Cb;
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Pbb += da*Cbb;
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Pbbb += da*Cbbb;
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Pab += db*(b1*Cab + b0*Kab);
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Paab += db*(b1*Caab + b0*Kaab);
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Pabb += da*(a1*Cabb + a0*Kabb);
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}
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P1 /= 2.0;
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Pa /= 6.0;
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Paa /= 12.0;
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Paaa /= 20.0;
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Pb /= -6.0;
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Pbb /= -12.0;
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Pbbb /= -20.0;
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Pab /= 24.0;
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Paab /= 60.0;
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Pabb /= -60.0;
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}
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void CompFaceIntegrals(FaceType &f)
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{
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Point3<ScalarType> n;
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ScalarType w;
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double k1, k2, k3, k4;
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compProjectionIntegrals(f);
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n = f.N();
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w = -f.V(0)->P()*n;
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k1 = 1 / n[C]; k2 = k1 * k1; k3 = k2 * k1; k4 = k3 * k1;
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Fa = k1 * Pa;
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Fb = k1 * Pb;
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Fc = -k2 * (n[A]*Pa + n[B]*Pb + w*P1);
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Faa = k1 * Paa;
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Fbb = k1 * Pbb;
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Fcc = k3 * (SQR(n[A])*Paa + 2*n[A]*n[B]*Pab + SQR(n[B])*Pbb
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+ w*(2*(n[A]*Pa + n[B]*Pb) + w*P1));
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Faaa = k1 * Paaa;
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Fbbb = k1 * Pbbb;
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Fccc = -k4 * (CUBE(n[A])*Paaa + 3*SQR(n[A])*n[B]*Paab
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+ 3*n[A]*SQR(n[B])*Pabb + CUBE(n[B])*Pbbb
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+ 3*w*(SQR(n[A])*Paa + 2*n[A]*n[B]*Pab + SQR(n[B])*Pbb)
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+ w*w*(3*(n[A]*Pa + n[B]*Pb) + w*P1));
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Faab = k1 * Paab;
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Fbbc = -k2 * (n[A]*Pabb + n[B]*Pbbb + w*Pbb);
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Fcca = k3 * (SQR(n[A])*Paaa + 2*n[A]*n[B]*Paab + SQR(n[B])*Pabb
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+ w*(2*(n[A]*Paa + n[B]*Pab) + w*Pa));
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}
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void Compute(MeshType &m)
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{
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tri::UpdateNormals<MeshType>::PerFaceNormalized(m);
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double nx, ny, nz;
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T0 = T1[X] = T1[Y] = T1[Z]
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= T2[X] = T2[Y] = T2[Z]
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= TP[X] = TP[Y] = TP[Z] = 0;
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FaceIterator fi;
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for (fi=m.face.begin(); fi!=m.face.end();++fi) if(!(*fi).IsD()) {
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FaceType &f=(*fi);
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nx = fabs(f.N()[0]);
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ny = fabs(f.N()[1]);
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nz = fabs(f.N()[2]);
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if (nx > ny && nx > nz) C = X;
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else C = (ny > nz) ? Y : Z;
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A = (C + 1) % 3;
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B = (A + 1) % 3;
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CompFaceIntegrals(f);
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T0 += f.N()[X] * ((A == X) ? Fa : ((B == X) ? Fb : Fc));
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T1[A] += f.N()[A] * Faa;
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T1[B] += f.N()[B] * Fbb;
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T1[C] += f.N()[C] * Fcc;
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T2[A] += f.N()[A] * Faaa;
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T2[B] += f.N()[B] * Fbbb;
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T2[C] += f.N()[C] * Fccc;
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TP[A] += f.N()[A] * Faab;
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TP[B] += f.N()[B] * Fbbc;
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TP[C] += f.N()[C] * Fcca;
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}
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T1[X] /= 2; T1[Y] /= 2; T1[Z] /= 2;
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T2[X] /= 3; T2[Y] /= 3; T2[Z] /= 3;
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TP[X] /= 2; TP[Y] /= 2; TP[Z] /= 2;
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}
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ScalarType Mass()
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{
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return static_cast<ScalarType>(T0);
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}
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Point3<ScalarType> CenterOfMass()
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{
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Point3<ScalarType> r;
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r[X] = T1[X] / T0;
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r[Y] = T1[Y] / T0;
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r[Z] = T1[Z] / T0;
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return r;
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}
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void InertiaTensor(Matrix33<ScalarType> &J ){
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Point3<ScalarType> r;
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r[X] = T1[X] / T0;
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r[Y] = T1[Y] / T0;
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r[Z] = T1[Z] / T0;
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/* compute inertia tensor */
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J[X][X] = (T2[Y] + T2[Z]);
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J[Y][Y] = (T2[Z] + T2[X]);
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J[Z][Z] = (T2[X] + T2[Y]);
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J[X][Y] = J[Y][X] = - TP[X];
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J[Y][Z] = J[Z][Y] = - TP[Y];
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J[Z][X] = J[X][Z] = - TP[Z];
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J[X][X] -= T0 * (r[Y]*r[Y] + r[Z]*r[Z]);
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J[Y][Y] -= T0 * (r[Z]*r[Z] + r[X]*r[X]);
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J[Z][Z] -= T0 * (r[X]*r[X] + r[Y]*r[Y]);
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J[X][Y] = J[Y][X] += T0 * r[X] * r[Y];
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J[Y][Z] = J[Z][Y] += T0 * r[Y] * r[Z];
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J[Z][X] = J[X][Z] += T0 * r[Z] * r[X];
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}
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void InertiaTensor(Matrix44<ScalarType> &J )
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{
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J.SetIdentity();
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Point3<ScalarType> r;
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r[X] = T1[X] / T0;
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r[Y] = T1[Y] / T0;
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r[Z] = T1[Z] / T0;
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/* compute inertia tensor */
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J[X][X] = (T2[Y] + T2[Z]);
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J[Y][Y] = (T2[Z] + T2[X]);
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J[Z][Z] = (T2[X] + T2[Y]);
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J[X][Y] = J[Y][X] = - TP[X];
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J[Y][Z] = J[Z][Y] = - TP[Y];
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J[Z][X] = J[X][Z] = - TP[Z];
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J[X][X] -= T0 * (r[Y]*r[Y] + r[Z]*r[Z]);
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J[Y][Y] -= T0 * (r[Z]*r[Z] + r[X]*r[X]);
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J[Z][Z] -= T0 * (r[X]*r[X] + r[Y]*r[Y]);
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J[X][Y] = J[Y][X] += T0 * r[X] * r[Y];
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J[Y][Z] = J[Z][Y] += T0 * r[Y] * r[Z];
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J[Z][X] = J[X][Z] += T0 * r[Z] * r[X];
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}
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/** Compute eigenvalues and eigenvectors of inertia tensor.
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The eigenvectors make a rotation matrix that aligns the mesh along the axes of min/max inertia
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*/
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void InertiaTensorEigen(Matrix44<ScalarType> &EV, Point4<ScalarType> &ev )
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{
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Matrix44<ScalarType> it;
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InertiaTensor(it);
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Matrix44d EVd,ITd;ITd.Import(it);
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Point4d evd;
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int n;
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Jacobi(ITd,evd,EVd,n);
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EV.Import(EVd);
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ev.Import(evd);
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}
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/** Compute covariance matrix of a mesh, i.e. the integral
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int_{M} { (x-b)(x-b)^T }dx where b is the barycenter and x spans over the mesh M
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*/
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static void Covariance(const MeshType & m, vcg::Point3<ScalarType> & bary, vcg::Matrix33<ScalarType> &C){
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// find the barycenter
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ConstFaceIterator fi;
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ScalarType area = 0.0;
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bary.SetZero();
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for(fi = m.face.begin(); fi != m.face.end(); ++fi)
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if(!(*fi).IsD())
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{
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bary += vcg::Barycenter( *fi )* vcg::DoubleArea(*fi);
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area+=vcg::DoubleArea(*fi);
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}
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bary/=area;
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C.SetZero();
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// C as covariance of triangle (0,0,0)(1,0,0)(0,1,0)
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vcg::Matrix33<ScalarType> C0;
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||
|
|
C0.SetZero();
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||
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|
C0[0][0] = C0[1][1] = 2.0;
|
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|
|
C0[0][1] = C0[1][0] = 1.0;
|
||
|
|
C0*=1/24.0;
|
||
|
|
|
||
|
|
// integral of (x,y,0) in the same triangle
|
||
|
|
CoordType X(1/6.0,1/6.0,0);
|
||
|
|
vcg::Matrix33<ScalarType> A, // matrix that bring the vertices to (v1-v0,v2-v0,n)
|
||
|
|
DC;
|
||
|
|
for(fi = m.face.begin(); fi != m.face.end(); ++fi)
|
||
|
|
if(!(*fi).IsD())
|
||
|
|
{
|
||
|
|
const CoordType &P0 = (*fi).P(0);
|
||
|
|
const CoordType &P1 = (*fi).P(1);
|
||
|
|
const CoordType &P2 = (*fi).P(2);
|
||
|
|
CoordType n = ((P1-P0)^(P2-P0));
|
||
|
|
const float da = n.Norm();
|
||
|
|
n/=da*da;
|
||
|
|
|
||
|
|
#ifndef VCG_USE_EIGEN
|
||
|
|
A.SetColumn(0, P1-P0);
|
||
|
|
A.SetColumn(1, P2-P0);
|
||
|
|
A.SetColumn(2, n);
|
||
|
|
#else
|
||
|
|
A.col(0) = P1-P0;
|
||
|
|
A.col(1) = P2-P0;
|
||
|
|
A.col(2) = n;
|
||
|
|
#endif
|
||
|
|
CoordType delta = P0 - bary;
|
||
|
|
|
||
|
|
/* DC is calculated as integral of (A*x+delta) * (A*x+delta)^T over the triangle,
|
||
|
|
where delta = v0-bary
|
||
|
|
*/
|
||
|
|
|
||
|
|
DC.SetZero();
|
||
|
|
DC+= A*C0*A.transpose();
|
||
|
|
vcg::Matrix33<ScalarType> tmp;
|
||
|
|
tmp.OuterProduct(A*X,delta);
|
||
|
|
DC += tmp + tmp.transpose();
|
||
|
|
DC+= tmp;
|
||
|
|
tmp.OuterProduct(delta,delta);
|
||
|
|
DC+=tmp*0.5;
|
||
|
|
// DC*=fabs(A.Determinant()); // the determinant of A is the jacobian of the change of variables A*x+delta
|
||
|
|
DC*=da; // the determinant of A is also the double area of *fi
|
||
|
|
C+=DC;
|
||
|
|
}
|
||
|
|
|
||
|
|
}
|
||
|
|
}; // end class Inertia
|
||
|
|
|
||
|
|
} // end namespace tri
|
||
|
|
} // end namespace vcg
|
||
|
|
|
||
|
|
|
||
|
|
#endif
|