4b665fa82b
URDF import demo: add COLLADA .dae file support add FiniteElementMethod demo, extracted from the OpenTissue library (under the zlib license) don't crash if loading an invalid STL file add comparison with Assimp for COLLADA file loading (disabled by default, to avoid library dependency) Gwen: disable some flags that make the build incompatible
223 lines
9.9 KiB
C++
223 lines
9.9 KiB
C++
#ifndef OPENTISSUE_DYNAMICS_FEM_FEM_UPDATE_ORIENTATION_H
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#define OPENTISSUE_DYNAMICS_FEM_FEM_UPDATE_ORIENTATION_H
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//
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// OpenTissue Template Library
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// - A generic toolbox for physics-based modeling and simulation.
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// Copyright (C) 2008 Department of Computer Science, University of Copenhagen.
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//
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// OTTL is licensed under zlib: http://opensource.org/licenses/zlib-license.php
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//
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#include <OpenTissue/configuration.h>
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namespace OpenTissue
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{
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namespace fem
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{
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namespace detail
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{
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/**
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*
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*/
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template<typename tetrahedron_type>
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inline void update_orientation_single(tetrahedron_type* T)
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{
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typedef typename tetrahedron_type::real_type real_type;
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typedef typename tetrahedron_type::vector3_type vector3_type;
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typedef typename tetrahedron_type::matrix3x3_type matrix3x3_type;
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{
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real_type div6V = 1.0 / T->m_V*6.0;
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//--- The derivation in the orignal paper on stiffness warping were stated as
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//---
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//--- | n1^T |
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//--- N = | n2^T | : WCS -> U
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//--- | n3^T |
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//---
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//--- | n1'^T |
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//--- N' = | n2'^T | : WCS -> D
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//--- | n3'^T |
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//---
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//--- From which we have
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//---
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//--- R = N' N^T
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//---
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//--- This is valid under the assumption that the n-vectors form a orthonormal basis. In that
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//--- paper the n-vectors were determined using a heuristic approach. Besides
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//--- the rotation were computed on a per vertex basis not on a per-tetrahedron
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//--- bases as we have outline above.
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//---
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//---
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//--- Later Muller et. al. used barycentric coordinates to find the transform. We
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//--- will here go into details on this method. Let the deformed corners be q0, q1,
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//--- q2, and q3 and the undeformed corners p0, p1, p2,and p3. Looking at some
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//--- point p = [x y z 1]^T inside the undeformed tetrahedron this can be written
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//---
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//--- | w0 |
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//--- p = | p0 p1 p2 p3 | | w1 | = P w (*1)
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//--- | 1 1 1 1 | | w2 |
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//--- | w3 |
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//---
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//--- The same point in the deformed tetrahedron has the same barycentric coordinates
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//--- which mean
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//---
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//--- | w0 |
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//--- q = | q0 q1 q2 q3 | | w1 | = Q w (*2)
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//--- | 1 1 1 1 | | w2 |
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//--- | w3 |
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//---
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//--- We can now use (*1) to solve for w and insert this into (*2), this yields
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//---
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//--- q = Q P^{-1} p
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//---
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//--- The matirx Q P^{-1} transforms p into q. Due to P and Q having their fourth rows
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//--- equal to 1 it can be shown that this matrix have the block structure
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//---
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//--- Q P^{-1} = | R t | (*3)
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//--- | 0^T 1 |
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//---
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//--- Which we recognize as a transformation matrix of homegeneous coordinates. The t vector
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//--- gives the translation, the R matrix includes, scaling, shearing and rotation.
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//---
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//--- We thus need to extract the rotational information from this R matrix. In their
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//--- paper Mueller et. al. suggest using Polar Decompostions (cite Shoemake and Duff; Etzmuss).
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//--- However, a simpler although more imprecise approach would simply be to apply a Grahram-Schimdt
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//--- orthonormalization to transform R into an orthonormal matrix. This seems to work quite well
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//--- in practice. We have not observed any visual difference in using polar decomposition or
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//--- orthonormalization.
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//---
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//--- Now for some optimizations. Since we are only interested in the R part of (*3) we can
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//--- compute this more efficiently exploiting the fact that barycentric coordinates sum
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//--- up to one. If we substitute w0 = 1 - w1 - w2 - w3 in (*1) and (*2) we can throw away
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//--- the fourth rows since they simply state 1=1, which is trivially true.
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//---
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//--- | 1-w1-w2-w3 |
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//--- p = | p0 p1 p2 p3 | | w1 |
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//--- | w2 |
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//--- | w3 |
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//---
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//--- Which is
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//---
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//--- | 1 |
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//--- p = | p0 (p1-p0) (p2-p0) (p3-p0) | | w1 |
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//--- | w2 |
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//--- | w3 |
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//---
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//--- We can move the first column over on the left hand side
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//---
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//--- | w1 |
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//--- (p-p0) = | (p1-p0) (p2-p0) (p3-p0) | | w2 |
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//--- | w3 |
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//---
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//--- Introducing e10 = p1-p0, e20 = p2-p0, e30 = p3-p0, and E = [e10 e20 e30 ] we have
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//---
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//--- w1
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//--- (p-p0) = E w2 (*5)
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//--- w3
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//---
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//--- Similar for *(2) we have
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//---
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//--- w1
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//--- (q-q0) = E' w2 (*6)
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//--- w3
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//---
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//--- Where E' = [e10' e20' e3'] and e10' = q1-q0, e20' = q2-q0, e30' = q3-q0. Now
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//--- inverting (*5) and insertion into (*6) yields
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//---
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//--- (q-q0) = E' E^{-1} (p-p0)
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//---
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//--- By comparison with (*3) we see that
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//---
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//--- R = E' E^{-1}
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//---
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//--- Using Cramers rule the inverse of E can be written as
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//---
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//--- | (e2 x e3)^T |
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//--- E^{-1} = 1/6V | (e3 x e1)^T |
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//--- | (e1 x e2)^T |
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//---
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//--- This can easily be confirmed by straigthforward computation
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//---
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//--- | e1 \cdot (e2 x e3) e2 \cdot (e2 x e3) e3 \cdot (e2 x e3) |
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//--- E^{-1} E = 1/6v | e1 \cdot (e3 x e1) e2 \cdot (e3 x e1) e3 \cdot (e3 x e1) |
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//--- | e1 \cdot (e1 x e2) e2 \cdot (e1 x e2) e3 \cdot (e1 x e2) |
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//---
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//--- | 6V 0 0 |
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//--- = 1/6V | 0 6V 0 | = I
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//--- | 0 0 6V |
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//---
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//--- Using the notation
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//---
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//--- n1 = e2 x e3 / 6V
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//--- n2 = e3 x e1 / 6V
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//--- n3 = e1 x e2 / 6V
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//---
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//--- we write
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//---
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//--- | n1^T |
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//--- E^{-1} = | n2^T |
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//--- | n3^T |
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//---
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//--- And we end up with
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//---
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//--- | n1^T |
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//--- R = [e10' e20' e30'] | n2^T |
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//--- | n3^T |
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//---
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//--- Observe that E' is very in-expensive to compute and all non-primed quantities (n1, n2, n3) can
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//--- be precomputed and stored on a per tetrahedron basis if there is enough memory available. Even
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//--- in case where memory is not available n1, n2 and n3 are quite cheap to compute.
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//---
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real_type e1x = T->m_e10(0);
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real_type e1y = T->m_e10(1);
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real_type e1z = T->m_e10(2);
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real_type e2x = T->m_e20(0);
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real_type e2y = T->m_e20(1);
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real_type e2z = T->m_e20(2);
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real_type e3x = T->m_e30(0);
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real_type e3y = T->m_e30(1);
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real_type e3z = T->m_e30(2);
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real_type n1x = (e2y * e3z - e3y * e2z) * div6V;
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real_type n1y = (e3x * e2z - e2x * e3z) * div6V;
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real_type n1z = (e2x * e3y - e3x * e2y) * div6V;
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real_type n2x = (e1z * e3y - e1y * e3z) * div6V;
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real_type n2y = (e1x * e3z - e1z * e3x) * div6V;
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real_type n2z = (e1y * e3x - e1x * e3y) * div6V;
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real_type n3x = (e1y * e2z - e1z * e2y) * div6V;
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real_type n3y = (e1z * e2x - e1x * e2z) * div6V;
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real_type n3z = (e1x * e2y - e1y * e2x) * div6V;
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const vector3_type & p0 = T->m_owner->m_nodes[T->m_nodes[0]].m_coord;
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const vector3_type & p1 = T->m_owner->m_nodes[T->m_nodes[1]].m_coord;
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const vector3_type & p2 = T->m_owner->m_nodes[T->m_nodes[2]].m_coord;
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const vector3_type & p3 = T->m_owner->m_nodes[T->m_nodes[3]].m_coord;
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e1x = p1(0) - p0(0);
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e1y = p1(1) - p0(1);
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e1z = p1(2) - p0(2);
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e2x = p2(0) - p0(0);
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e2y = p2(1) - p0(1);
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e2z = p2(2) - p0(2);
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e3x = p3(0) - p0(0);
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e3y = p3(1) - p0(1);
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e3z = p3(2) - p0(2);
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T->m_Re(0,0) = e1x * n1x + e2x * n2x + e3x * n3x; T->m_Re(0,1) = e1x * n1y + e2x * n2y + e3x * n3y; T->m_Re(0,2) = e1x * n1z + e2x * n2z + e3x * n3z;
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T->m_Re(1,0) = e1y * n1x + e2y * n2x + e3y * n3x; T->m_Re(1,1) = e1y * n1y + e2y * n2y + e3y * n3y; T->m_Re(1,2) = e1y * n1z + e2y * n2z + e3y * n3z;
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T->m_Re(2,0) = e1z * n1x + e2z * n2x + e3z * n3x; T->m_Re(2,1) = e1z * n1y + e2z * n2y + e3z * n3y; T->m_Re(2,2) = e1z * n1z + e2z * n2z + e3z * n3z;
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T->m_Re = ortonormalize(T->m_Re);
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//matrix3x3_type M = T->m_Re,S;
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//OpenTissue::PolarDecomposition3x3 decomp;
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//decomp.eigen(M, T->m_Re, S);//--- Etzmuss-style
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//decomp.newton(M, T->m_Re );//--- Shoemake-Duff-style
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}
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}
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} // namespace detail
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} // namespace fem
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} // namespace OpenTissue
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//OPENTISSUE_DYNAMICS_FEM_FEM_UPDATE_ORIENTATION_H
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#endif
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