Reference documentation for deal.II version GIT 8d72163873 2022-05-16 02:25:02+00:00
FlatManifold< dim, spacedim > Class Template Reference

#include <deal.II/grid/manifold.h>

Inheritance diagram for FlatManifold< dim, spacedim >:
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## Public Types

using FaceVertexNormals = std::array< Tensor< 1, spacedim >, GeometryInfo< dim >::vertices_per_face >

## Public Member Functions

FlatManifold (const Tensor< 1, spacedim > &periodicity=Tensor< 1, spacedim >(), const double tolerance=1e-10)

virtual std::unique_ptr< Manifold< dim, spacedim > > clone () const override

virtual Point< spacedim > get_new_point (const ArrayView< const Point< spacedim >> &surrounding_points, const ArrayView< const double > &weights) const override

virtual void get_new_points (const ArrayView< const Point< spacedim >> &surrounding_points, const Table< 2, double > &weights, ArrayView< Point< spacedim >> new_points) const override

virtual Point< spacedim > project_to_manifold (const ArrayView< const Point< spacedim >> &points, const Point< spacedim > &candidate) const override

virtual Tensor< 1, spacedim > get_tangent_vector (const Point< spacedim > &x1, const Point< spacedim > &x2) const override

virtual Tensor< 1, spacedim > normal_vector (const typename Triangulation< dim, spacedim >::face_iterator &face, const Point< spacedim > &p) const override

virtual void get_normals_at_vertices (const typename Triangulation< dim, spacedim >::face_iterator &face, typename Manifold< dim, spacedim >::FaceVertexNormals &face_vertex_normals) const override

const Tensor< 1, spacedim > & get_periodicity () const

void get_normals_at_vertices (const Triangulation< 1 >::face_iterator &, Manifold< 1, 1 >::FaceVertexNormals &) const

void get_normals_at_vertices (const Triangulation< 1, 2 >::face_iterator &, Manifold< 1, 2 >::FaceVertexNormals &) const

void get_normals_at_vertices (const Triangulation< 1, 3 >::face_iterator &, Manifold< 1, 3 >::FaceVertexNormals &) const

void get_normals_at_vertices (const Triangulation< 2 >::face_iterator &face, Manifold< 2, 2 >::FaceVertexNormals &face_vertex_normals) const

void get_normals_at_vertices (const Triangulation< 2, 3 >::face_iterator &, Manifold< 2, 3 >::FaceVertexNormals &) const

void get_normals_at_vertices (const Triangulation< 3 >::face_iterator &face, Manifold< 3, 3 >::FaceVertexNormals &face_vertex_normals) const

Tensor< 1, 1 > normal_vector (const Triangulation< 1, 1 >::face_iterator &, const Point< 1 > &) const

Tensor< 1, 2 > normal_vector (const Triangulation< 1, 2 >::face_iterator &, const Point< 2 > &) const

Tensor< 1, 3 > normal_vector (const Triangulation< 1, 3 >::face_iterator &, const Point< 3 > &) const

Tensor< 1, 2 > normal_vector (const Triangulation< 2, 2 >::face_iterator &face, const Point< 2 > &p) const

Point< 1 > get_new_point_on_quad (const Triangulation< 1, 1 >::quad_iterator &) const

Point< 2 > get_new_point_on_quad (const Triangulation< 1, 2 >::quad_iterator &) const

Point< 3 > get_new_point_on_quad (const Triangulation< 1, 3 >::quad_iterator &) const

Point< 3 > get_new_point_on_hex (const Triangulation< 3, 3 >::hex_iterator &hex) const

Point< 1 > get_new_point_on_face (const Triangulation< 1, 1 >::face_iterator &) const

Point< 2 > get_new_point_on_face (const Triangulation< 1, 2 >::face_iterator &) const

Point< 3 > get_new_point_on_face (const Triangulation< 1, 3 >::face_iterator &) const

Tensor< 1, 3 > normal_vector (const Triangulation< 3, 3 >::face_iterator &face, const Point< 3 > &p) const

void get_normals_at_vertices (const Triangulation< 2, 2 >::face_iterator &face, FaceVertexNormals &n) const

void get_normals_at_vertices (const Triangulation< 3, 3 >::face_iterator &face, FaceVertexNormals &n) const

Computing the location of points.
virtual Point< spacedim > get_intermediate_point (const Point< spacedim > &p1, const Point< spacedim > &p2, const double w) const

virtual Point< spacedim > get_new_point_on_line (const typename Triangulation< dim, spacedim >::line_iterator &line) const

virtual Point< spacedim > get_new_point_on_quad (const typename Triangulation< dim, spacedim >::quad_iterator &quad) const

virtual Point< spacedim > get_new_point_on_hex (const typename Triangulation< dim, spacedim >::hex_iterator &hex) const

Point< spacedim > get_new_point_on_face (const typename Triangulation< dim, spacedim >::face_iterator &face) const

Point< spacedim > get_new_point_on_cell (const typename Triangulation< dim, spacedim >::cell_iterator &cell) const

Computing normal vectors
virtual void get_normals_at_vertices (const typename Triangulation< dim, spacedim >::face_iterator &face, FaceVertexNormals &face_vertex_normals) const

## Static Private Member Functions

static ::ExceptionBaseExcPeriodicBox (int arg1, Point< spacedim > arg2, double arg3)

## Private Attributes

const Tensor< 1, spacedim > periodicity

const double tolerance

## Subscriptor functionality

Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.

std::atomic< unsigned int > counter

std::map< std::string, unsigned int > counter_map

std::vector< std::atomic< bool > * > validity_pointers

const std::type_info * object_info

void subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int n_subscriptions () const

template<typename StreamType >
void list_subscribers (StreamType &stream) const

void list_subscribers () const

template<class Archive >
void serialize (Archive &ar, const unsigned int version)

using map_value_type = decltype(counter_map)::value_type

using map_iterator = decltype(counter_map)::iterator

static std::mutex mutex

static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

void check_no_subscribers () const noexcept

## Detailed Description

### template<int dim, int spacedim = dim> class FlatManifold< dim, spacedim >

Specialization of Manifold<dim,spacedim>, which represent a possibly periodic Euclidean space of dimension dim embedded in the Euclidean space of spacedim dimensions. The main characteristic of this Manifold is the fact that the function FlatManifold<dim,spacedim>::project_to_manifold() is the identity function.

Definition at line 632 of file manifold.h.

## ◆ FaceVertexNormals

 using Manifold< dim, spacedim >::FaceVertexNormals = std::array, GeometryInfo::vertices_per_face>
inherited

Type keeping information about the normals at the vertices of a face of a cell. Thus, there are GeometryInfo<dim>::vertices_per_face normal vectors, that define the tangent spaces of the boundary at the vertices. Note that the vectors stored in this object are not required to be normalized, nor to actually point outward, as one often will only want to check for orthogonality to define the tangent plane; if a function requires the normals to be normalized, then it must do so itself.

For obvious reasons, this type is not useful in 1d.

Definition at line 306 of file manifold.h.

## ◆ FlatManifold()

template<int dim, int spacedim = dim>
 FlatManifold< dim, spacedim >::FlatManifold ( const Tensor< 1, spacedim > & periodicity = Tensor< 1, spacedim >(), const double tolerance = 1e-10 )

Default constructor. The optional argument can be used to specify the periodicity of the spacedim-dimensional manifold (one period per direction). A periodicity value of zero means that along that direction there is no periodicity. By default no periodicity is assumed.

Periodicity affects the way a middle point is computed. It is assumed that if two points are more than half period distant, then the distance should be computed by crossing the periodicity boundary, i.e., the average is computed by adding a full period to the sum of the two. For example, if along direction 0 we have 2*pi periodicity, then the average of (2*pi-eps) and (eps) is not pi, but 2*pi (or zero), since, on a periodic manifold, these two points are at distance 2*eps and not (2*pi- eps). Special cases are taken into account, to ensure that the behavior is always as expected. The third argument is used as a relative tolerance when computing distances.

Periodicity will be intended in the following way: the domain is considered to be the box contained in [Point<spacedim>(), periodicity) where the right extreme is excluded. If any of the components of this box has zero length, then no periodicity is assumed in that direction. Whenever a function that tries to compute averages is called, an exception will be thrown if one of the points which you are using for the average lies outside the periodicity box. The return points are guaranteed to lie in the periodicity box plus or minus tolerance*periodicity.norm().

## ◆ clone()

template<int dim, int spacedim = dim>
 virtual std::unique_ptr > FlatManifold< dim, spacedim >::clone ( ) const
overridevirtual

Return a copy of this manifold.

Implements Manifold< dim, dim >.

## ◆ get_new_point()

template<int dim, int spacedim = dim>
 virtual Point FlatManifold< dim, spacedim >::get_new_point ( const ArrayView< const Point< spacedim >> & surrounding_points, const ArrayView< const double > & weights ) const
overridevirtual

Let the new point be the average sum of surrounding vertices.

This particular implementation constructs the weighted average of the surrounding points, and then calls internally the function project_to_manifold(). The reason why we do it this way, is to allow lazy programmers to implement only the project_to_manifold() function for their own Manifold classes which are small (or trivial) perturbations of a flat manifold. This is the case whenever the coarse mesh is a decent approximation of the manifold geometry. In this case, the middle point of a cell is close to true middle point of the manifold, and a projection may suffice.

For most simple geometries, it is possible to get reasonable results by deriving your own Manifold class from FlatManifold, and write a new interface only for the project_to_manifold function. You will have good approximations also with large deformations, as long as in the coarsest mesh size you are trying to refine, the middle point is not too far from the manifold mid point, i.e., as long as the coarse mesh size is small enough.

Reimplemented from Manifold< dim, dim >.

## ◆ get_new_points()

template<int dim, int spacedim = dim>
 virtual void FlatManifold< dim, spacedim >::get_new_points ( const ArrayView< const Point< spacedim >> & surrounding_points, const Table< 2, double > & weights, ArrayView< Point< spacedim >> new_points ) const
overridevirtual

Compute a new set of points that interpolate between the given points surrounding_points. weights is a table with as many columns as surrounding_points.size(). The number of rows in weights must match the length of new_points.

For this particular implementation, the interpolation of the surrounding_points according to the weights is simply performed in Cartesian space.

Reimplemented from Manifold< dim, dim >.

## ◆ project_to_manifold()

template<int dim, int spacedim = dim>
 virtual Point FlatManifold< dim, spacedim >::project_to_manifold ( const ArrayView< const Point< spacedim >> & points, const Point< spacedim > & candidate ) const
overridevirtual

Project to FlatManifold. This is the identity function for flat, Euclidean spaces. Note however that this function can be overloaded by derived classes, which will then benefit from the logic behind the get_new_point() function which are often very similar (if not identical) to the one implemented in this class.

Reimplemented from Manifold< dim, dim >.

## ◆ get_tangent_vector()

template<int dim, int spacedim = dim>
 virtual Tensor<1, spacedim> FlatManifold< dim, spacedim >::get_tangent_vector ( const Point< spacedim > & x1, const Point< spacedim > & x2 ) const
overridevirtual

Return a vector that, at $$\mathbf x_1$$, is tangential to the geodesic that connects two points $$\mathbf x_1,\mathbf x_2$$. For the current class, we assume that the manifold is flat, so the geodesic is the straight line between the two points, and we return $$\mathbf x_2-\mathbf x_1$$. The normalization of the vector is chosen so that it fits the convention described in Manifold::get_tangent_vector().

Note
If you use this class as a stepping stone to build a manifold that only "slightly" deviates from a flat manifold, by overloading the project_to_manifold() function.
Parameters
 x1 The first point that describes the geodesic, and the one at which the "direction" is to be evaluated. x2 The second point that describes the geodesic.
Returns
A "direction" vector tangential to the geodesic. Here, this is $$\mathbf x_2-\mathbf x_1$$, possibly modified by the periodicity of the domain as set in the constructor, to use the "shortest" connection between the points through the periodic boundary as necessary.

Reimplemented from Manifold< dim, dim >.

## ◆ normal_vector() [1/6]

template<int dim, int spacedim = dim>
 virtual Tensor<1, spacedim> FlatManifold< dim, spacedim >::normal_vector ( const typename Triangulation< dim, spacedim >::face_iterator & face, const Point< spacedim > & p ) const
overridevirtual

Return the normal vector to the given face at point p taking into account that quadrilateral faces of hexahedral cells in 3d may not be planar. In those cases, the face is assumed to have a geometry described by a bilinear function, and the normal vector is computed by embedding this bilinear form into a Cartesian space with a flat metric.

Reimplemented from Manifold< dim, dim >.

## ◆ get_normals_at_vertices() [1/10]

template<int dim, int spacedim = dim>
 virtual void FlatManifold< dim, spacedim >::get_normals_at_vertices ( const typename Triangulation< dim, spacedim >::face_iterator & face, typename Manifold< dim, spacedim >::FaceVertexNormals & face_vertex_normals ) const
overridevirtual

Compute the normal vectors to the boundary at each vertex of the given face taking into account that quadrilateral faces of hexahedral cells in 3d may not be planar. In those cases, the face is assumed to have a geometry described by a bilinear function, and the normal vector is computed by embedding this bilinear form into a Cartesian space with a flat metric.

## ◆ get_periodicity()

template<int dim, int spacedim = dim>
 const Tensor<1, spacedim>& FlatManifold< dim, spacedim >::get_periodicity ( ) const

Return the periodicity of this Manifold.

## ◆ get_normals_at_vertices() [2/10]

 void FlatManifold< 1 >::get_normals_at_vertices ( const Triangulation< 1 >::face_iterator & , Manifold< 1, 1 >::FaceVertexNormals & ) const

Definition at line 727 of file manifold.cc.

## ◆ get_normals_at_vertices() [3/10]

 void FlatManifold< 1, 2 >::get_normals_at_vertices ( const Triangulation< 1, 2 >::face_iterator & , Manifold< 1, 2 >::FaceVertexNormals & ) const

Definition at line 738 of file manifold.cc.

## ◆ get_normals_at_vertices() [4/10]

 void FlatManifold< 1, 3 >::get_normals_at_vertices ( const Triangulation< 1, 3 >::face_iterator & , Manifold< 1, 3 >::FaceVertexNormals & ) const

Definition at line 749 of file manifold.cc.

## ◆ get_normals_at_vertices() [5/10]

 void FlatManifold< 2 >::get_normals_at_vertices ( const Triangulation< 2 >::face_iterator & face, Manifold< 2, 2 >::FaceVertexNormals & face_vertex_normals ) const

Definition at line 760 of file manifold.cc.

## ◆ get_normals_at_vertices() [6/10]

 void FlatManifold< 2, 3 >::get_normals_at_vertices ( const Triangulation< 2, 3 >::face_iterator & , Manifold< 2, 3 >::FaceVertexNormals & ) const

Definition at line 775 of file manifold.cc.

## ◆ get_normals_at_vertices() [7/10]

 void FlatManifold< 3 >::get_normals_at_vertices ( const Triangulation< 3 >::face_iterator & face, Manifold< 3, 3 >::FaceVertexNormals & face_vertex_normals ) const

Definition at line 786 of file manifold.cc.

## ◆ normal_vector() [2/6]

 Tensor< 1, 1 > FlatManifold< 1, 1 >::normal_vector ( const Triangulation< 1, 1 >::face_iterator & , const Point< 1 > & ) const

Definition at line 814 of file manifold.cc.

## ◆ normal_vector() [3/6]

 Tensor< 1, 2 > FlatManifold< 1, 2 >::normal_vector ( const Triangulation< 1, 2 >::face_iterator & , const Point< 2 > & ) const

Definition at line 825 of file manifold.cc.

## ◆ normal_vector() [4/6]

 Tensor< 1, 3 > FlatManifold< 1, 3 >::normal_vector ( const Triangulation< 1, 3 >::face_iterator & , const Point< 3 > & ) const

Definition at line 836 of file manifold.cc.

## ◆ normal_vector() [5/6]

 Tensor< 1, 2 > FlatManifold< 2, 2 >::normal_vector ( const Triangulation< 2, 2 >::face_iterator & face, const Point< 2 > & p ) const

Definition at line 847 of file manifold.cc.

## ◆ get_intermediate_point()

 Point< spacedim > Manifold< dim, spacedim >::get_intermediate_point ( const Point< spacedim > & p1, const Point< spacedim > & p2, const double w ) const
virtualinherited

Return an intermediate point between two given points. Overloading this function allows the default pair-wise reduction implementation of the method get_new_point() that takes a Quadrature object as input to work properly.

An implementation of this function should returns a parametric curve on the manifold, joining the points p1 and p2, with parameter w in the interval [0,1]. In particular get_intermediate_point(p1, p2, 0.0) should return p1 and get_intermediate_point(p1, p2, 1.0) should return p2.

In its default implementation, this function calls the project_to_manifold() method with the convex combination of p1 and p2. User classes can get away by simply implementing the project_to_manifold() method.

Reimplemented in SphericalManifold< dim, spacedim >, and ChartManifold< dim, spacedim, chartdim >.

Definition at line 347 of file manifold.cc.

## ◆ get_new_point_on_line()

 Point< spacedim > Manifold< dim, spacedim >::get_new_point_on_line ( const typename Triangulation< dim, spacedim >::line_iterator & line ) const
virtualinherited

Backward compatibility interface. Return the point which shall become the new middle vertex of the two children of a regular line. In 2D, this line is a line at the boundary, while in 3d, it is bounding a face at the boundary (the lines therefore is also on the boundary).

The default implementation of this function passes its argument to the Manifolds::get_default_points_and_weights() function, and then calls the Manifold<dim,spacedim>::get_new_point() function. User derived classes can overload Manifold<dim,spacedim>::get_new_point() or Manifold<dim,spacedim>::project_to_manifold(), which is called by the default implementation of Manifold<dim,spacedim>::get_new_point().

Definition at line 427 of file manifold.cc.

## ◆ get_new_point_on_quad() [1/4]

 Point< spacedim > Manifold< dim, spacedim >::get_new_point_on_quad ( const typename Triangulation< dim, spacedim >::quad_iterator & quad ) const
virtualinherited

Backward compatibility interface. Return the point which shall become the common point of the four children of a quad at the boundary in three or more spatial dimensions. This function therefore is only useful in at least three dimensions and should not be called for lower dimensions.

This function is called after the four lines bounding the given quad are refined, so you may want to use the information provided by quad->line(i)->child(j), i=0...3, j=0,1.

The default implementation of this function passes its argument to the Manifolds::get_default_points_and_weights() function, and then calls the Manifold<dim,spacedim>::get_new_point() function. User derived classes can overload Manifold<dim,spacedim>::get_new_point() or Manifold<dim,spacedim>::project_to_manifold(), which is called by the default implementation of Manifold<dim,spacedim>::get_new_point().

Definition at line 448 of file manifold.cc.

## ◆ get_new_point_on_quad() [2/4]

 Point< 1 > Manifold< 1, 1 >::get_new_point_on_quad ( const Triangulation< 1, 1 >::quad_iterator & ) const
inherited

Definition at line 418 of file manifold.cc.

## ◆ get_new_point_on_quad() [3/4]

 Point< 2 > Manifold< 1, 2 >::get_new_point_on_quad ( const Triangulation< 1, 2 >::quad_iterator & ) const
inherited

Definition at line 429 of file manifold.cc.

## ◆ get_new_point_on_quad() [4/4]

 Point< 3 > Manifold< 1, 3 >::get_new_point_on_quad ( const Triangulation< 1, 3 >::quad_iterator & ) const
inherited

Definition at line 440 of file manifold.cc.

## ◆ get_new_point_on_hex() [1/2]

 Point< spacedim > Manifold< dim, spacedim >::get_new_point_on_hex ( const typename Triangulation< dim, spacedim >::hex_iterator & hex ) const
virtualinherited

Backward compatibility interface. Return the point which shall become the common point of the eight children of a hex in three or spatial dimensions. This function therefore is only useful in at least three dimensions and should not be called for lower dimensions.

This function is called after the all the bounding objects of the given hex are refined, so you may want to use the information provided by hex->quad(i)->line(j)->child(k), i=0...5, j=0...3, k=0,1.

The default implementation of this function passes its argument to the Manifolds::get_default_points_and_weights() function, and then calls the Manifold<dim,spacedim>::get_new_point() function. User derived classes can overload Manifold<dim,spacedim>::get_new_point() or Manifold<dim,spacedim>::project_to_manifold(), which is called by the default implementation of Manifold<dim,spacedim>::get_new_point().

Definition at line 470 of file manifold.cc.

## ◆ get_new_point_on_hex() [2/2]

 Point< 3 > Manifold< 3, 3 >::get_new_point_on_hex ( const Triangulation< 3, 3 >::hex_iterator & hex ) const
inherited

Definition at line 462 of file manifold.cc.

## ◆ get_new_point_on_face() [1/4]

 Point< spacedim > Manifold< dim, spacedim >::get_new_point_on_face ( const typename Triangulation< dim, spacedim >::face_iterator & face ) const
inherited

Backward compatibility interface. Depending on dim=2 or dim=3 this function calls the get_new_point_on_line or the get_new_point_on_quad function. It throws an exception for dim=1. This wrapper allows dimension independent programming.

Definition at line 481 of file manifold.cc.

## ◆ get_new_point_on_face() [2/4]

 Point< 1 > Manifold< 1, 1 >::get_new_point_on_face ( const Triangulation< 1, 1 >::face_iterator & ) const
inherited

Definition at line 385 of file manifold.cc.

## ◆ get_new_point_on_face() [3/4]

 Point< 2 > Manifold< 1, 2 >::get_new_point_on_face ( const Triangulation< 1, 2 >::face_iterator & ) const
inherited

Definition at line 396 of file manifold.cc.

## ◆ get_new_point_on_face() [4/4]

 Point< 3 > Manifold< 1, 3 >::get_new_point_on_face ( const Triangulation< 1, 3 >::face_iterator & ) const
inherited

Definition at line 407 of file manifold.cc.

## ◆ get_new_point_on_cell()

 Point< spacedim > Manifold< dim, spacedim >::get_new_point_on_cell ( const typename Triangulation< dim, spacedim >::cell_iterator & cell ) const
inherited

Backward compatibility interface. Depending on dim=1, dim=2 or dim=3 this function calls the get_new_point_on_line, get_new_point_on_quad or the get_new_point_on_hex function. This wrapper allows dimension independent programming.

Definition at line 492 of file manifold.cc.

## ◆ normal_vector() [6/6]

 Tensor< 1, 3 > Manifold< 3, 3 >::normal_vector ( const Triangulation< 3, 3 >::face_iterator & face, const Point< 3 > & p ) const
inherited

Definition at line 165 of file manifold.cc.

## ◆ get_normals_at_vertices() [8/10]

 void Manifold< dim, spacedim >::get_normals_at_vertices ( const typename Triangulation< dim, spacedim >::face_iterator & face, FaceVertexNormals & face_vertex_normals ) const
virtualinherited

Compute the normal vectors to the boundary at each vertex of the given face embedded in the Manifold. It is not required that the normal vectors be normed somehow. Neither is it required that the normals actually point outward.

This function is needed to compute data for C1 mappings. The default implementation calls normal_vector() on each vertex.

Note that when computing normal vectors at a vertex where the boundary is not differentiable, you have to make sure that you compute the one-sided limits, i.e. limit with respect to points inside the given face.

Definition at line 614 of file manifold.cc.

## ◆ get_normals_at_vertices() [9/10]

 void Manifold< 2, 2 >::get_normals_at_vertices ( const Triangulation< 2, 2 >::face_iterator & face, FaceVertexNormals & n ) const
inherited

Definition at line 250 of file manifold.cc.

## ◆ get_normals_at_vertices() [10/10]

 void Manifold< 3, 3 >::get_normals_at_vertices ( const Triangulation< 3, 3 >::face_iterator & face, FaceVertexNormals & n ) const
inherited

Definition at line 272 of file manifold.cc.

## ◆ periodicity

template<int dim, int spacedim = dim>
 const Tensor<1, spacedim> FlatManifold< dim, spacedim >::periodicity
private

The periodicity of this Manifold. Periodicity affects the way a middle point is computed. It is assumed that if two points are more than half period distant, then the distance should be computed by crossing the periodicity boundary, i.e., the average is computed by adding a full period to the sum of the two. For example, if along direction 0 we have 2*pi periodicity, then the average of (2*pi-eps) and (eps) is not pi, but 2*pi (or zero), since, on a periodic manifold, these two points are at distance 2*eps and not (2*pi-eps).

A periodicity 0 along one direction means no periodicity. This is the default value for all directions.

Definition at line 794 of file manifold.h.

## ◆ tolerance

template<int dim, int spacedim = dim>
 const double FlatManifold< dim, spacedim >::tolerance
private

Relative tolerance. This tolerance is used to compute distances in double precision.

Definition at line 808 of file manifold.h.

The documentation for this class was generated from the following file: