From cf075276f49568ec88eda70a1789bc9ddf0d8fb1 Mon Sep 17 00:00:00 2001
From: Oliver Sander <sander@igpm.rwth-aachen.de>
Date: Thu, 12 Jan 2012 10:01:10 +0000
Subject: [PATCH] Reimplement method
secondDerivativeOfDistanceSquaredWRTSecondArgument
Properly taking into account the two-sheet covering structure
[[Imported from SVN: r8370]]
---
dune/gfe/rotation.hh | 35 +++++++++++++++++++++++------------
1 file changed, 23 insertions(+), 12 deletions(-)
diff --git a/dune/gfe/rotation.hh b/dune/gfe/rotation.hh
index 9d2196f7..56ae8b5b 100644
--- a/dune/gfe/rotation.hh
+++ b/dune/gfe/rotation.hh
@@ -643,19 +643,30 @@ public:
return result;
}
- /** \brief Compute the Hessian of the squared distance function keeping the first argument fixed
-
- Unlike the distance itself the squared distance is differentiable at zero
- */
+ /** \brief Compute the Hessian of the squared distance function keeping the first argument fixed */
static Dune::FieldMatrix<T,4,4> secondDerivativeOfDistanceSquaredWRTSecondArgument(const Rotation<T,3>& p, const Rotation<T,3>& q) {
- // use the functionality from the unitvector class
- Dune::FieldMatrix<T,4,4> result = UnitVector<T,4>::secondDerivativeOfDistanceSquaredWRTSecondArgument(p.globalCoordinates(),
- q.globalCoordinates());
- // The unit quaternions form a double cover of SO(3). That means going once around the unit sphere (2\pi)
- // means going twice around SO(3) (4\pi). Hence there is a factor 2, which in addition we need to square,
- // because we are considering the squared distance.
- result *= 4;
- return result;
+
+ T sp = p.globalCoordinates() * q.globalCoordinates();
+
+ EmbeddedTangentVector pProjected = q.projectOntoTangentSpace(p.globalCoordinates());
+
+ Dune::FieldMatrix<T,4,4> A;
+ for (int i=0; i<4; i++)
+ for (int j=0; j<4; j++)
+ A[i][j] = pProjected[i]*pProjected[j];
+
+ A *= 4*UnitVector<T,4>::secondDerivativeOfArcCosSquared(std::fabs(sp));
+
+ // Compute matrix B (see notes)
+ Dune::FieldMatrix<T,4,4> Pq;
+ for (int i=0; i<4; i++)
+ for (int j=0; j<4; j++)
+ Pq[i][j] = (i==j) - q.globalCoordinates()[i]*q.globalCoordinates()[j];
+
+ // Bring it all together
+ A.axpy(-4* ((sp<0) ? -1 : 1) * derivativeOfArcCosSquared(std::fabs(sp))*sp, Pq);
+
+ return A;
}
/** \brief Compute the mixed second derivate \partial d^2 / \partial da db
--
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