From cc1795e9a6645ec64d9cf5bda6866e27104f4054 Mon Sep 17 00:00:00 2001
From: Oliver Sander <sander@igpm.rwth-aachen.de>
Date: Sun, 30 Dec 2012 11:35:59 +0000
Subject: [PATCH] New TargetSpace implementation for points in hyperbolic
half-spaces of any dimension
This is just an initial commit. It is a copy of the corresponding code for
unit vectors, with just a few names exchanged. The actual functionality
will come in subsequent commits.
[[Imported from SVN: r9076]]
---
dune/gfe/Makefile.am | 1 +
dune/gfe/hyperbolichalfspacepoint.hh | 374 +++++++++++++++++++++++++++
2 files changed, 375 insertions(+)
create mode 100644 dune/gfe/hyperbolichalfspacepoint.hh
diff --git a/dune/gfe/Makefile.am b/dune/gfe/Makefile.am
index 1768b26a..47e42cc0 100644
--- a/dune/gfe/Makefile.am
+++ b/dune/gfe/Makefile.am
@@ -15,6 +15,7 @@ srcinclude_HEADERS = averagedistanceassembler.hh \
globalgfetestfunctionbasis.hh \
globalgfetestfunction.hh \
harmonicenergystiffness.hh \
+ hyperbolichalfspacepoint.hh \
linearalgebra.hh \
localgeodesicfefunction.hh \
localgeodesicfestiffness.hh \
diff --git a/dune/gfe/hyperbolichalfspacepoint.hh b/dune/gfe/hyperbolichalfspacepoint.hh
new file mode 100644
index 00000000..18cd0775
--- /dev/null
+++ b/dune/gfe/hyperbolichalfspacepoint.hh
@@ -0,0 +1,374 @@
+#ifndef HYPERBOLIC_HALF_SPACE_POINT_HH
+#define HYPERBOLIC_HALF_SPACE_POINT_HH
+
+#include <dune/common/fvector.hh>
+#include <dune/common/fmatrix.hh>
+
+#include <dune/gfe/tensor3.hh>
+
+/** \brief A point in the hyperbolic half-space H^N
+
+ \tparam N Dimension of the hyperbolic space space
+ \tparam T The type used for individual coordinates
+*/
+template <class T, int N>
+class HyperbolicHalfspacePoint
+{
+ dune_static_assert(N>=2, "A hyperbolic half-space needs to be at least two-dimensional!");
+
+ /** \brief Computes sin(x) / x without getting unstable for small x */
+ static T sinc(const T& x) {
+ return (x < 1e-4) ? 1 - (x*x/6) : std::sin(x)/x;
+ }
+
+ /** \brief Compute the derivative of arccos^2 without getting unstable for x close to 1 */
+ static T derivativeOfArcCosSquared(const T& x) {
+ const T eps = 1e-4;
+ if (x > 1-eps) { // regular expression is unstable, use the series expansion instead
+ return -2 + 2*(x-1)/3 - 4/15*(x-1)*(x-1);
+ } else if (x < -1+eps) { // The function is not differentiable
+ DUNE_THROW(Dune::Exception, "arccos^2 is not differentiable at x==-1!");
+ } else
+ return -2*std::acos(x) / std::sqrt(1-x*x);
+ }
+
+ /** \brief Compute the second derivative of arccos^2 without getting unstable for x close to 1 */
+ static T secondDerivativeOfArcCosSquared(const T& x) {
+ const T eps = 1e-4;
+ if (x > 1-eps) { // regular expression is unstable, use the series expansion instead
+ return 2.0/3 - 8*(x-1)/15;
+ } else if (x < -1+eps) { // The function is not differentiable
+ DUNE_THROW(Dune::Exception, "arccos^2 is not differentiable at x==-1!");
+ } else
+ return 2/(1-x*x) - 2*x*std::acos(x) / std::pow(1-x*x,1.5);
+ }
+
+ /** \brief Compute the third derivative of arccos^2 without getting unstable for x close to 1 */
+ static T thirdDerivativeOfArcCosSquared(const T& x) {
+ const T eps = 1e-4;
+ if (x > 1-eps) { // regular expression is unstable, use the series expansion instead
+ return -8.0/15 + 24*(x-1)/35;
+ } else if (x < -1+eps) { // The function is not differentiable
+ DUNE_THROW(Dune::Exception, "arccos^2 is not differentiable at x==-1!");
+ } else {
+ T d = 1-x*x;
+ return 6*x/(d*d) - 6*x*x*std::acos(x)/(d*d*std::sqrt(d)) - 2*std::acos(x)/(d*std::sqrt(d));
+ }
+ }
+
+public:
+
+ /** \brief The type used for coordinates */
+ typedef T ctype;
+
+ /** \brief The type used for global coordinates */
+ typedef Dune::FieldVector<T,N> CoordinateType;
+
+ /** \brief Dimension of the manifold formed by unit vectors */
+ static const int dim = N-1;
+
+ /** \brief Dimension of the Euclidean space the manifold is embedded in */
+ static const int embeddedDim = N;
+
+ typedef Dune::FieldVector<T,N-1> TangentVector;
+
+ typedef Dune::FieldVector<T,N> EmbeddedTangentVector;
+
+ /** \brief Default constructor */
+ HyperbolicHalfspacePoint()
+ {}
+
+ /** \brief Constructor from a vector. The vector gets normalized */
+ HyperbolicHalfspacePoint(const Dune::FieldVector<T,N>& vector)
+ : data_(vector)
+ {
+ data_ /= data_.two_norm();
+ }
+
+ /** \brief Constructor from an array. The array gets normalized */
+ HyperbolicHalfspacePoint(const Dune::array<T,N>& vector)
+ {
+ for (int i=0; i<N; i++)
+ data_[i] = vector[i];
+ data_ /= data_.two_norm();
+ }
+
+ UnitVector<T,N>& operator=(const Dune::FieldVector<T,N>& vector)
+ {
+ data_ = vector;
+ data_ /= data_.two_norm();
+ return *this;
+ }
+
+ /** \brief The exponential map */
+ static HyperbolicHalfspacePoint exp(const HyperbolicHalfspacePoint& p, const TangentVector& v) {
+
+ Dune::FieldMatrix<T,N-1,N> frame = p.orthonormalFrame();
+
+ EmbeddedTangentVector ev;
+ frame.mtv(v,ev);
+
+ return exp(p,ev);
+ }
+
+ /** \brief The exponential map */
+ static HyperbolicHalfspacePoint exp(const HyperbolicHalfspacePoint& p, const EmbeddedTangentVector& v) {
+
+ assert( std::abs(p.data_*v) < 1e-5 );
+
+ const T norm = v.two_norm();
+ HyperbolicHalfspacePoint result = p;
+ result.data_ *= std::cos(norm);
+ result.data_.axpy(sinc(norm), v);
+ return result;
+ }
+
+ /** \brief Length of the great arc connecting the two points */
+ static T distance(const HyperbolicHalfspacePoint& a, const HyperbolicHalfspacePoint& b) {
+
+ // Not nice: we are in a class for unit vectors, but the class is actually
+ // supposed to handle perturbations of unit vectors as well. Therefore
+ // we normalize here.
+ T x = a.data_ * b.data_/a.data_.two_norm()/b.data_.two_norm();
+
+ // paranoia: if the argument is just eps larger than 1 acos returns NaN
+ x = std::min(x,1.0);
+
+ return std::acos(x);
+ }
+
+ /** \brief Compute the gradient of the squared distance function keeping the first argument fixed
+
+ Unlike the distance itself the squared distance is differentiable at zero
+ */
+ static EmbeddedTangentVector derivativeOfDistanceSquaredWRTSecondArgument(const HyperbolicHalfspacePoint& a, const HyperbolicHalfspacePoint& b) {
+ T x = a.data_ * b.data_;
+
+ EmbeddedTangentVector result = a.data_;
+
+ result *= derivativeOfArcCosSquared(x);
+
+ // Project gradient onto the tangent plane at b in order to obtain the surface gradient
+ result = b.projectOntoTangentSpace(result);
+
+ // Gradient must be a tangent vector at b, in other words, orthogonal to it
+ assert( std::abs(b.data_ * result) < 1e-5);
+
+ 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
+ */
+ static Dune::FieldMatrix<T,N,N> secondDerivativeOfDistanceSquaredWRTSecondArgument(const HyperbolicHalfspacePoint& p, const HyperbolicHalfspacePoint& q) {
+
+ T sp = p.data_ * q.data_;
+
+ Dune::FieldVector<T,N> pProjected = q.projectOntoTangentSpace(p.globalCoordinates());
+
+ Dune::FieldMatrix<T,N,N> A;
+ for (int i=0; i<N; i++)
+ for (int j=0; j<N; j++)
+ A[i][j] = pProjected[i]*pProjected[j];
+
+ A *= secondDerivativeOfArcCosSquared(sp);
+
+ // Compute matrix B (see notes)
+ Dune::FieldMatrix<T,N,N> Pq;
+ for (int i=0; i<N; i++)
+ for (int j=0; j<N; j++)
+ Pq[i][j] = (i==j) - q.data_[i]*q.data_[j];
+
+ // Bring it all together
+ A.axpy(-1*derivativeOfArcCosSquared(sp)*sp, Pq);
+
+ return A;
+ }
+
+ /** \brief Compute the mixed second derivate \partial d^2 / \partial da db
+
+ Unlike the distance itself the squared distance is differentiable at zero
+ */
+ static Dune::FieldMatrix<T,N,N> secondDerivativeOfDistanceSquaredWRTFirstAndSecondArgument(const HyperbolicHalfspacePoint& a, const HyperbolicHalfspacePoint& b) {
+
+ T sp = a.data_ * b.data_;
+
+ // Compute vector A (see notes)
+ Dune::FieldMatrix<T,1,N> row;
+ row[0] = b.projectOntoTangentSpace(a.globalCoordinates());
+
+ Dune::FieldVector<T,N> tmp = a.projectOntoTangentSpace(b.globalCoordinates());
+ Dune::FieldMatrix<T,N,1> column;
+ for (int i=0; i<N; i++) // turn row vector into column vector
+ column[i] = tmp[i];
+
+ Dune::FieldMatrix<T,N,N> A;
+ // A = row * column
+ Dune::FMatrixHelp::multMatrix(column,row,A);
+ A *= secondDerivativeOfArcCosSquared(sp);
+
+ // Compute matrix B (see notes)
+ Dune::FieldMatrix<T,N,N> Pp, Pq;
+ for (int i=0; i<N; i++)
+ for (int j=0; j<N; j++) {
+ Pp[i][j] = (i==j) - a.data_[i]*a.data_[j];
+ Pq[i][j] = (i==j) - b.data_[i]*b.data_[j];
+ }
+
+ Dune::FieldMatrix<T,N,N> B;
+ Dune::FMatrixHelp::multMatrix(Pp,Pq,B);
+
+ // Bring it all together
+ A.axpy(derivativeOfArcCosSquared(sp), B);
+
+ return A;
+ }
+
+
+ /** \brief Compute the third derivative \partial d^3 / \partial dq^3
+
+ Unlike the distance itself the squared distance is differentiable at zero
+ */
+ static Tensor3<T,N,N,N> thirdDerivativeOfDistanceSquaredWRTSecondArgument(const HyperbolicHalfspacePoint& p, const HyperbolicHalfspacePoint& q) {
+
+ Tensor3<T,N,N,N> result;
+
+ T sp = p.data_ * q.data_;
+
+ // The projection matrix onto the tangent space at p and q
+ Dune::FieldMatrix<T,N,N> Pq;
+ for (int i=0; i<N; i++)
+ for (int j=0; j<N; j++)
+ Pq[i][j] = (i==j) - q.globalCoordinates()[i]*q.globalCoordinates()[j];
+
+ Dune::FieldVector<T,N> pProjected = q.projectOntoTangentSpace(p.globalCoordinates());
+
+ for (int i=0; i<N; i++)
+ for (int j=0; j<N; j++)
+ for (int k=0; k<N; k++) {
+
+ result[i][j][k] = thirdDerivativeOfArcCosSquared(sp) * pProjected[i] * pProjected[j] * pProjected[k]
+ - secondDerivativeOfArcCosSquared(sp) * ((i==j)*sp + p.globalCoordinates()[i]*q.globalCoordinates()[j])*pProjected[k]
+ - secondDerivativeOfArcCosSquared(sp) * ((i==k)*sp + p.globalCoordinates()[i]*q.globalCoordinates()[k])*pProjected[j]
+ - secondDerivativeOfArcCosSquared(sp) * pProjected[i] * Pq[j][k] * sp
+ + derivativeOfArcCosSquared(sp) * ((i==j)*q.globalCoordinates()[k] + (i==k)*q.globalCoordinates()[j]) * sp
+ - derivativeOfArcCosSquared(sp) * p.globalCoordinates()[i] * Pq[j][k];
+ }
+
+ result = Pq * result;
+
+ return result;
+ }
+
+ /** \brief Compute the mixed third derivative \partial d^3 / \partial da db^2
+
+ Unlike the distance itself the squared distance is differentiable at zero
+ */
+ static Tensor3<T,N,N,N> thirdDerivativeOfDistanceSquaredWRTFirst1AndSecond2Argument(const HyperbolicHalfspacePoint& p, const HyperbolicHalfspacePoint& q) {
+
+ Tensor3<T,N,N,N> result;
+
+ T sp = p.data_ * q.data_;
+
+ // The projection matrix onto the tangent space at p and q
+ Dune::FieldMatrix<T,N,N> Pp, Pq;
+ for (int i=0; i<N; i++)
+ for (int j=0; j<N; j++) {
+ Pp[i][j] = (i==j) - p.globalCoordinates()[i]*p.globalCoordinates()[j];
+ Pq[i][j] = (i==j) - q.globalCoordinates()[i]*q.globalCoordinates()[j];
+ }
+
+ Dune::FieldVector<T,N> pProjected = q.projectOntoTangentSpace(p.globalCoordinates());
+ Dune::FieldVector<T,N> qProjected = p.projectOntoTangentSpace(q.globalCoordinates());
+
+ Tensor3<T,N,N,N> derivativeOfPqOTimesPq;
+ for (int i=0; i<N; i++)
+ for (int j=0; j<N; j++)
+ for (int k=0; k<N; k++) {
+ derivativeOfPqOTimesPq[i][j][k] = 0;
+ for (int l=0; l<N; l++)
+ derivativeOfPqOTimesPq[i][j][k] += Pp[i][l] * (Pq[j][l]*pProjected[k] + pProjected[j]*Pq[k][l]);
+ }
+
+ result = thirdDerivativeOfArcCosSquared(sp) * Tensor3<T,N,N,N>::product(qProjected,pProjected,pProjected)
+ + secondDerivativeOfArcCosSquared(sp) * derivativeOfPqOTimesPq
+ - secondDerivativeOfArcCosSquared(sp) * sp * Tensor3<T,N,N,N>::product(qProjected,Pq)
+ - derivativeOfArcCosSquared(sp) * Tensor3<T,N,N,N>::product(qProjected,Pq);
+
+ return result;
+ }
+
+
+ /** \brief Project tangent vector of R^n onto the tangent space */
+ EmbeddedTangentVector projectOntoTangentSpace(const EmbeddedTangentVector& v) const {
+ EmbeddedTangentVector result = v;
+ result.axpy(-1*(data_*result), data_);
+ return result;
+ }
+
+ /** \brief The global coordinates, if you really want them */
+ const CoordinateType& globalCoordinates() const {
+ return data_;
+ }
+
+ /** \brief Compute an orthonormal basis of the tangent space of S^n.
+
+ This basis is of course not globally continuous.
+ */
+ Dune::FieldMatrix<T,N-1,N> orthonormalFrame() const {
+
+ Dune::FieldMatrix<T,N-1,N> result;
+
+ // Coordinates of the stereographic projection
+ Dune::FieldVector<T,N-1> X;
+
+ if (data_[N-1] <= 0) {
+
+ // Stereographic projection from the north pole onto R^{N-1}
+ for (size_t i=0; i<N-1; i++)
+ X[i] = data_[i] / (1-data_[N-1]);
+
+ } else {
+
+ // Stereographic projection from the south pole onto R^{N-1}
+ for (size_t i=0; i<N-1; i++)
+ X[i] = data_[i] / (1+data_[N-1]);
+
+ }
+
+ T RSquared = X.two_norm2();
+
+ for (size_t i=0; i<N-1; i++)
+ for (size_t j=0; j<N-1; j++)
+ // Note: the matrix is the transpose of the one in the paper
+ result[j][i] = 2*(i==j)*(1+RSquared) - 4*X[i]*X[j];
+
+ for (size_t j=0; j<N-1; j++)
+ result[j][N-1] = 4*X[j];
+
+ // Upper hemisphere: adapt formulas so it is the stereographic projection from the south pole
+ if (data_[N-1] > 0)
+ for (size_t j=0; j<N-1; j++)
+ result[j][N-1] *= -1;
+
+ // normalize the rows to make the orthogonal basis orthonormal
+ for (size_t i=0; i<N-1; i++)
+ result[i] /= result[i].two_norm();
+
+ return result;
+ }
+
+ /** \brief Write unit vector object to output stream */
+ friend std::ostream& operator<< (std::ostream& s, const HyperbolicHalfspacePoint& unitVector)
+ {
+ return s << unitVector.data_;
+ }
+
+
+private:
+
+ Dune::FieldVector<T,N> data_;
+};
+
+#endif
--
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