% -*- latex -*-

\chapter{Working with Cells}
\label{chap:WorkingWithCells}

\index{cell|(}

In the control environment, data is defined in mesh structures that
comprise a set of finite cells. (See Section~\ref{sec:DataSets:CellSets}
starting on page~\pageref{sec:DataSets:CellSets} for information on
defining cell sets in the control environment.) When worklets that operate
on cells are scheduled, these grid structures are broken into their
independent cells, and that data is handed to the worklet. Thus, cell-based
operations in the execution environment exclusively operate on independent
cells.

Unlike some other libraries such as VTK, VTK-m does not have a cell class
that holds all the information pertaining to a cell of a particular type.
Instead, VTK-m provides tags or identifiers defining the cell shape, and
companion data like coordinate and field information are held in separate
structures. This organization is designed so a worklet may specify exactly
what information it needs, and only that information will be loaded.

\section{Cell Shape Tags and Ids}
\label{sec:CellShapeTagsIds}

\index{shape|(}
\index{cell shape|(}
\index{tag!cell shape|(}
\index{tag!shape|(}

Cell shapes can be specified with either a tag (defined with a struct with
a name like \textidentifier{CellShapeTag*}) or an enumerated identifier
(defined with a constant number with a name like
\textidentifier{CELL\_SHAPE\_*}). These shape tags and identifiers are
defined in \vtkmheader{vtkm}{CellShape.h} and declared in the \vtkm{}
namespace (because they can be used in either the control or the execution
environment). Figure~\ref{fig:CellShapes} gives both the identifier and the
tag names.

\begin{figure}
  \centering
  \small
  \begin{tabular}{@{}c@{ }c@{ }c@{}}
    \raisebox{-0.5\height}{\includegraphics{images/CellConnectionsVertex}} &
    \raisebox{-0.5\height}{\includegraphics{images/CellConnectionsLine}} &
    \raisebox{-0.5\height}{\includegraphics{images/CellConnectionsTriangle}} \\
    \vtkm{CELL\_SHAPE\_VERTEX} &
    \vtkm{CELL\_SHAPE\_LINE} &
    \vtkm{CELL\_SHAPE\_TRIANGLE} \\
    \vtkm{CellShapeTagVertex} \index{vertex} &
    \vtkm{CellShapeTagLine} \index{line} &
    \vtkm{CellShapeTagTriangle} \index{triangle} \\[2ex]
    \raisebox{-0.5\height}{\includegraphics{images/CellConnectionsPolygon}} &
    \raisebox{-0.5\height}{\includegraphics{images/CellConnectionsQuadrilateral}} &
    \raisebox{-0.5\height}{\includegraphics{images/CellConnectionsTetrahedron}} \\
    \vtkm{CELL\_SHAPE\_POLYGON} &
    \vtkm{CELL\_SHAPE\_QUAD} &
    \vtkm{CELL\_SHAPE\_TETRA} \\
    \vtkm{CellShapeTagPolygon} \index{polygon} &
    \vtkm{CellShapeTagQuad} \index{quadrilateral} &
    \vtkm{CellShapeTagTetra} \index{tetrahedron} \\[2ex]
    \raisebox{-0.5\height}{\includegraphics{images/CellConnectionsHexahedron}} &
    \raisebox{-0.5\height}{\includegraphics{images/CellConnectionsWedge}} &
    \raisebox{-0.5\height}{\includegraphics{images/CellConnectionsPyramid}} \\
    \vtkm{CELL\_SHAPE\_HEXAHEDRON} &
    \vtkm{CELL\_SHAPE\_WEDGE} &
    \vtkm{CELL\_SHAPE\_PYRAMID} \\
    \vtkm{CellShapeTagHexahedron} \index{hexahedron} &
    \vtkm{CellShapeTagWedge} \index{wedge} &
    \vtkm{CellShapeTagPyramid} \index{pyramid}
  \end{tabular}
  \caption{Basic Cell Shapes}
  \label{fig:CellShapes}
\end{figure}

In addition to the basic cell shapes, there is a special ``empty'' cell
with the identifier \vtkm{CELL\_SHAPE\_EMPTY} and tag
\vtkm{CellShapeTagEmpty}. This type of cell has no points, edges, or faces
and can be thought of as a placeholder for a null or void cell.

There is also a special cell shape ``tag'' named \vtkm{CellShapeTagGeneric}
that is used when the actual cell shape is not known at compile time.
\textidentifier{CellShapeTagGeneric} actually has a member variable named
\textcode{Id} that stores the identifier for the cell shape. There is no
equivalent identifier for a generic cell; cell shape identifiers can be
placed in a \vtkm{IdComponent} at runtime.

When using cell shapes in templated classes and functions, you can use the
\vtkmmacro{VTKM\_IS\_CELL\_SHAPE\_TAG} to ensure a type is a valid cell
shape tag. This macro takes one argument and will produce a compile error
if the argument is not a cell shape tag type.

\subsection{Converting Between Tags and Identifiers}

Every cell shape tag has a member variable named \textcode{Id} that
contains the identifier for the cell shape. This provides a convenient
mechanism for converting a cell shape tag to an identifier. Most cell shape
tags have their \textcode{Id} member as a compile-time constant, but
\textidentifier{CellShapeTagGeneric} is set at run time.

\vtkmheader{vtkm}{CellShape.h} also declares a templated class named
\vtkm{CellShapeIdToTag} that converts a cell shape identifier to a cell
shape tag. \textidentifier{CellShapeIdToTag} has a single template argument
that is the identifier. Inside the class is a type named \textcode{Tag}
that is the type of the correct tag.

\vtkmlisting{Using \textidentifier{CellShapeIdToTag}.}{CellShapeIdToTag.cxx}

However, \textidentifier{CellShapeIdToTag} is only viable if the identifier
can be resolved at compile time. In the case where a cell identifier is
stored in a variable or an array or the code is using a
\textidentifier{CellShapeTagGeneric}, the correct cell shape is not known
at run time. In this case, \vtkmmacro{vtkmGenericCellShapeMacro} can be
used to check all possible conditions. This macro is embedded in a switch
statement where the condition is the cell shape identifier.
\vtkmmacro{vtkmGenericCellShapeMacro} has a single argument, which is an
expression to be executed. Before the expression is executed, a type named
\textcode{CellShapeTag} is defined as the type of the appropriate cell
shape tag. Often this method is used to implement the condition for a
\textidentifier{CellShapeTagGeneric} in a function overloaded for cell
types. A demonstration of \vtkmmacro{vtkmGenericCellShapeMacro} is given in
Example~\ref{ex:GenericCellNormal}.

\index{tag!shape|)}
\index{tag!cell shape|)}

\subsection{Cell Traits}

\index{cell traits|(}

The \vtkmheader{vtkm}{CellTraits.h} header file contains a traits class
named \vtkm{CellTraits} that provides information about a cell. Each
specialization of \textidentifier{CellTraits} contains the following
members.

\begin{description}
\item[\textcode{TOPOLOGICAL\_DIMENSIONS}] Defines the topological
  dimensions of the cell type. This is 3 for polyhedra, 2 for polygons, 1
  for lines, and 0 for points.
\item[\textcode{TopologicalDimensionsTag}] A type set to either
  \vtkm{CellTopologicalDimensionsTag}\textcode{<3>},
  \textidentifier{CellTopologicalDimensionsTag}\textcode{<2>},
  \textidentifier{CellTopologicalDimensionsTag}\textcode{<1>}, or
  \textidentifier{CellTopologicalDimensionsTag}\textcode{<0>}. The number
  is consistent with \textcode{TOPOLOGICAL\_DIMENSIONS}. This tag is
  provided for convenience when specializing functions.
\item[\textcode{IsSizeFixed}] Set to either \vtkm{CellTraitsTagSizeFixed}
  for cell types with a fixed number of points (for example, triangle) or
  \vtkm{CellTraitsTagSizeVariable} for cell types with a variable number of
  points (for example, polygon).
\item[\textcode{NUM\_POINTS}] A \vtkm{IdComponent} set to the number of
  points in the cell. This member is only defined when there is a constant
  number of points (i.e. \textcode{IsSizeFixed} is set to
  \vtkm{CellTraitsTagSizeFixed}).
\end{description}

\vtkmlisting[ex:GenericCellNormal]{Using \textidentifier{CellTraits} to implement a polygon normal estimator.}{GenericCellNormal.cxx}

\index{cell traits|)}

\index{cell shape|)}
\index{shape|)}

\section{Parametric and World Coordinates}

\index{parametric coordinates|(}
\index{world coordinates|(}
\index{cell!parametric coordinates|(}
\index{cell!world coordinates|(}

Each cell type supports a one-to-one mapping between a set of parametric
coordinates in the unit cube (or some subset of it) and the points in 3D
space that are the locus contained in the cell. Parametric coordinates are
useful because certain features of the cell, such as vertex location and
center, are at a consistent location in parametric space irrespective of
the location and distortion of the cell in world space. Also, many field
operations are much easier with parametric coordinates.

The \vtkmheader{vtkm/exec}{ParametricCoordinates.h} header file contains the
following functions for working with parametric coordinates.

\begin{description}
\item[\vtkmexec{ParametricCoordinatesCenter}] Returns the parametric
  coordinates for the center of a given shape. It takes 4 arguments: the
  number of points in the cell, a \vtkm{Vec} of size 3 to store the
  results, a shape tag, and a worklet object (for raising errors). A second
  form of this method takes 3 arguments and returns the result as a
  \vtkm{Vec}\textcode{<}\vtkm{FloatDefault}\textcode{,3>} instead of
  passing it as a parameter.
\item[\vtkmexec{ParametricCoordinatesPoint}] Returns the parametric
  coordinates for a given point of a given shape. It takes 5 arguments: the
  number of points in the cell, the index of the point to query, a
  \vtkm{Vec} of size 3 to store the results, a shape tag, and a worklet
  object (for raising errors). A second form of this method takes 3
  arguments and returns the result as a
  \vtkm{Vec}\textcode{<}\vtkm{FloatDefault}\textcode{,3>} instead of
  passing it as a parameter.
\item[\vtkmexec{ParametricCoordinatesToWorldCoordinates}] Given a vector of
  point coordinates (usually given by a \sigtag{FieldPointIn} worklet
  argument), a \vtkm{Vec} of size 3 containing parametric coordinates, a
  shape tag, and a worklet object (for raising errors), returns the world
  coordinates.
\item[\vtkmexec{WorldCoordinatesToParametricCoordinates}] Given a vector of
  point coordinates (usually given by a \sigtag{FieldPointIn} worklet
  argument), a \vtkm{Vec} of size 3 containing world coordinates, a shape
  tag, and a worklet object (for raising errors), returns the parametric
  coordinates. This function can be slow for cell types with nonlinear
  interpolation (which is anything that is not a simplex).
\end{description}

\index{cell!world coordinates|)}
\index{cell!parametric coordinates|)}
\index{world coordinates|)}
\index{parametric coordinates|)}

\section{Interpolation}

\index{cell!interpolation|(}
\index{interpolation|(}

The shape of every cell is defined by the connections of some finite set of
points. Field values defined on those points can be interpolated to any
point within the cell to estimate a continuous field.

The \vtkmheader{vtkm/exec}{CellInterpolate.h} header contains the function
\vtkmexec{CellInterpolate} that takes a vector of point field values
(usually given by a \sigtag{FieldPointIn} worklet argument), a \vtkm{Vec}
of size 3 containing parametric coordinates, a shape tag, and a worklet
object (for raising errors). It returns the field interpolated to the
location represented by the given parametric coordinates.

\vtkmlisting{Interpolating field values to a cell's center.}{CellCenters.cxx}

\index{interpolation|)}
\index{cell!interpolation|)}

\section{Derivatives}

\index{cell!derivative|(}
\index{cell!gradient|(}
\index{derivative|(}
\index{gradient|(}

Since interpolations provide a continuous field function over a cell, it is
reasonable to consider the derivative of this function. The
\vtkmheader{vtkm/exec}{CellDerivative.h} header contains the function
\vtkmexec{CellDerivative} that takes a vector of scalar point field values
(usually given by a \sigtag{FieldPointIn} worklet argument), a \vtkm{Vec}
of size 3 containing parametric coordinates, a shape tag, and a worklet
object (for raising errors). It returns the field derivative at the
location represented by the given parametric coordinates. The derivative is
return in a \vtkm{Vec} of size 3 corresponding to the partial derivatives
in the $x$, $y$, and $z$ directions. This derivative is equivalent to the
gradient of the field.

\vtkmlisting{Computing the derivative of the field at cell centers.}{CellDerivatives.cxx}

\index{gradient|)}
\index{derivative|)}
\index{cell!gradient|)}
\index{cell!derivative|)}


\section{Edges and Faces}

As explained earlier in this chapter, a cell is defined by a collection of \index{point} \index{cell!point} \index{shape!point} points and a \index{shape} \index{cell!shape} shape identifier that describes how the points come together to form the structure of the cell.
The cell shapes supported by \VTKm are documented in Section~\ref{sec:CellShapeTagsIds}.
It contains Figure~\ref{fig:CellShapes} on page~\pageref{fig:CellShapes}, which shows how the points for each shape form the structure of the cell.

Most cell shapes can be broken into subelements. 2D and 3D cells have pairs of points that form \index{edge}\index{cell!edge}\index{shape!edge}\keyterm{edges} at the boundaries of the cell.
Likewise, 3D cells have loops of edges that form \index{face}\index{cell!face}\index{shape!face}\keyterm{faces} that encase the cell.
Figure~\ref{fig:CellConstituents} demonstrates the relationship of these constituent elements for some example cell shapes.

\begin{figure}[htb]
  \centering
  \includegraphics{images/CellConstituents}
  \caption{The constituent elements (points, edges, and faces) of cells.}
  \label{fig:CellConstituents}
\end{figure}

\index{edge|(}
\index{cell!edge|(}
\index{shape!edge|(}

The header file \vtkmheader{vtkm/exec}{CellEdge.h} contains a collection of functions to help identify the edges of a cell.
The first such function is \vtkmexec{CellEdgeNumberOfEdges}.
This function takes the number of points in the cell, the shape of the cell, and an instance of the calling worklet (for error reporting).
It returns the number of edges the cell has (as a \vtkm{IdComponent}).

The second function is \vtkmexec{CellEdgeLocalIndices}.
This function takes the number of points, the local index of the edge (0 to the number of edges in the cell), the shape of the cell, and an instance of the calling worklet.
It returns a \vtkm{Vec}\tparams{\vtkm{IdComponent},2} containing the local indices of the two points forming the edge.
These local indices are consistent with the point labels in Figure~\ref{fig:CellConstituents}.
To get the point indices relative to the data set, the edge indices should be used to reference a \sigtag{PointIndices} list.

The third function is \vtkmexec{CellEdgeCanonicalId}.
This function takes the number of points, the local index of the edge, the shape of the cell, a \Veclike containing the global id of each cell point, and an instance of the calling worklet.
It returns a \vtkm{Id2} that is globally unique to that edge.
If \textidentifier{CellEdgeCanonicalId} is called on an edge for a different cell, the two will be the same if and only if the two cells share that edge.
\textidentifier{CellEdgeCanonicalId} is useful for finding coincident components of topology.

The following example demonstrates a pair of worklets that use the cell edge functions.
As is typical for operations of this nature, one worklet counts the number of edges in each cell and another uses this count to generate the data.

\begin{didyouknow}
  Example~\ref{ex:CellEdgeFunctions} demonstrates one of many techniques for creating cell sets in a worklet.
  Chapter~\ref{chap:GeneratingCellSets} describes this and many more such techniques.
\end{didyouknow}

\vtkmlisting[ex:CellEdgeFunctions]{Using cell edge functions.}{CellEdge.cxx}

\index{shape!edge|)}
\index{cell!edge|)}
\index{edge|)}

\index{face|(}
\index{cell!face|(}
\index{shape!face|(}

The header file \vtkmheader{vtkm/exec}{CellFace.h} contains a collection of functions to help identify the faces of a cell.
The first such function is \vtkmexec{CellFaceNumberOfFaces}.
This function takes the shape of the cell and an instance of the calling worklet (for error reporting).
It returns the number of faces the cell has (as a \vtkm{IdComponent}).

The second function is \vtkmexec{CellFaceNumberOfPoints}.
This function takes the local index of the face (0 to the number of faces in the cell), the shape of the cell, and an instance of the calling worklet.
It returns the number of points the specified face has (as a \vtkm{IdComponent}).

The third function is \vtkmexec{CellFaceLocalIndices}.
This function takes the local index of the face, the shape of the cell, and an instance of the calling worklet.
It returns a \vtkm{VecCConst}\tparams{\vtkm{IdComponent}} containing the local indices of the points forming the face.
The points are given in counterclockwise order when viewing the face from the outside of the cell.
These local indices are consistent with the point labels in Figure~\ref{fig:CellConstituents}.
To get the point indices relative to the data set, the face indices should be used to reference a \sigtag{PointIndices} list.

The fourth function is \vtkmexec{CellFaceCanonicalId}.
This function takes the local index of the face, the shape of the cell, a \Veclike containing the global id of each cell point, and an instance of the calling worklet.
It returns a \vtkm{Id3} that is globally unique to that face.
If \textidentifier{CellFaceCanonicalId} is called on a face for a different cell, the two will be the same if and only if the two cells share that face.
\textidentifier{CellFaceCanonicalId} is useful for finding coincident components of topology.

The following example demonstrates a triple of worklets that use the cell face functions.
As is typical for operations of this nature, the worklets are used in steps to first count entities and then generate new entities.
In this case, the first worklet counts the number of faces and the second worklet counts the points in each face.
The third worklet generates cells for each face.

\vtkmlisting{Using cell face functions.}{CellFace.cxx}

\index{shape!face|)}
\index{cell!face|)}
\index{face|)}

\index{cell|)}
