% !TEX root = thesis.tex

\chapter{A short dictionary of dimension-based GIS terms}
\label{ch:dictionary}

\begin{multicols}{2}

\begin{description}
\footnotesize

\item[ambient space]
the space in which objects are embedded.

\item[area]
1.\ measure of the 2D extent of an object;
2.\ 2D combinatorial element;
{\color{gray} 3.\ 2D geometric object}.

\item[ball]
1.\ ($\rightarrow$\ same as 3-\emph{ball}) topological definition of the space within a 2-sphere (\ie\ a sphere), often defined as the space within a certain distance (the radius) from a point in $\mathbb{R}^3$ (the centre);
2.\ ($\rightarrow$\ often $n$-\emph{ball}) topological definition of the space within an $(n-1)$-sphere, often defined as the space within a certain distance (the radius) from a point in $\mathbb{R}^n$ (the centre);
3.\ geometric definition of a perfectly round filled 3D object.

\item[body]
$n$-dimensional combinatorial element in an $n$D context.

\item[box]
1.\ cuboid;
2.\ orthotope of any dimension.

\item[cavity]
3D hole.

\item[cell]
1.\ ($\rightarrow$\ sometimes $n$-\emph{cell}) $n$D combinatorial element;
{\color{gray} 2.\ 3D combinatorial element}.

\item[cell complex]
topological space formed by a set of cells glued along common faces, all faces of a cell in the complex should be in the complex.

\item[circle]
1.\ topological definition of the space at a certain distance (the radius) from a point in $\mathbb{R}^2$ (the centre);
2.\ geometric definition of a perfectly round hollow 2D object.

\item[closed]
1.\ topological definition of an object that includes its boundary;
2.\ geometric definition of an object in $n$D, usually defined using boundary representation, that encloses an $n$D subspace.

\item[congruent]
having the same shape and size.

\item[cube]
1.\ ($\rightarrow$\ same as $3$-\emph{cube}) 3D polyhedron with 6 square facets;
2.\ ($\rightarrow$\ usually $n$-\emph{cube}) $n$-orthotope with identical $(n-1)$-cube facets, akin to a square in 2D and a cube in 3D.

\item[cuboid]
1.\ box-shaped 3D polyhedron with 6 rectangular facets, its opposite facets are congruent and parallel;
{\color{gray} 2.\ parallelotope}.

\item[curve]
1.\ 1D geometric object, often with a non-linear geometry;
2.\ 1D combinatorial element;
3.\ 1-manifold.

\item[cut-line]
degenerate part of a 2D shape forming a curve and protruding inwards from its boundary.

\item[disk]
topological definition of the space within a circle, often defined as the space within a certain distance (the radius) from a point in $\mathbb{R}^2$ (the centre).

\item[edge]
1D combinatorial element.

\item[face]
1.\ 2D combinatorial element;
2.\ ($\rightarrow$\ sometimes \emph{$n$-face of $X$}) $n$D combinatorial element on the boundary of $X$;
{\color{gray} 3.\ ($\rightarrow$\ usually \emph{face of $X$}) $(n-1)$D combinatorial element on the boundary of an $n$D combinatorial element $X$}.

\item[facet]
1.\ $(n-1)$D combinatorial element in an $n$D context;
2.\ ($\rightarrow$\ often \emph{facet of $X$}) $(n-1)$D combinatorial element on the boundary of an $n$D combinatorial element $X$;
{\color{gray} 3.\ 2D combinatorial element}.

\item[flat]
unbounded geometric object with linear geometry of any dimension, such as a point, line or plane.

\item[hole]
1.\ 2D void region;
2.\ $n$D void region.

\item[hyperball]
higher-dimensional ball.

\item[hypercell]
1.\ higher-dimensional combinatorial element;
2.\ 4D combinatorial element.

\item[hypercube]
1.\ higher-dimensional cube;
2.\ 4-cube.

\item[hyperplane]
1.\ $(n-1)$D linear subspace in an $n$D context;
{\color{gray} 2.\ plane in a higher-dimensional context.}

\item[hyperrectangle]
higher-dimensional orthotope.

\item[hypersphere]
higher-dimensional sphere.

\item[hypersurface]
$(n-1)$D subspace in an $n$D context, often curved;

\item[interval]
topological definition of the space between two points in $\mathbb{R}$ (the endpoints).

\item[Lebesgue measure]
measure of the $n$D extent of an object, akin to length in 1D, area in 2D and volume in 3D.

\item[length]
measure of the 1D extent of an object.

\item[line segment]
1D geometric object.

\item[manifold]
($\rightarrow$\ sometimes $n$-\emph{manifold}) topological space that resembles $\mathbb{R}^n$ at every point.

\item[manifold with boundary]
($\rightarrow$\ sometimes $n$-\emph{manifold with boundary}) topological space that resembles $\mathbb{R}^n$ at every point in its interior.

\item[node]
0D combinatorial element.

\item[open]
1.\ topological definition of an object that does not include its boundary;
2.\ geometric definition of an object in $n$D, usually defined using boundary representation, that does not enclose any $n$D subspace.

\item[orthotope]
($\rightarrow$\ sometimes $n$-\emph{orthotope}) $n$-polytope with congruent parallel facets forming right angles to each other, akin to a rectangle in 2D and a cuboid in 3D.

\item[parallelepiped]
3D polyhedron bounded by 6 parallelogram-shaped 2D faces.

\item[parallelogram]
polygon bounded by 4 parallel and have the same shape.

\item[parallelotope]
a polytope whose opposite facets are parallel and have the same shape, akin to a parallelogram in 2D and a parallelepiped in 3D.

\item[peak]
1.\ $(n-3)$D combinatorial element in an $n$D context;
2.\ ($\rightarrow$\ \emph{peak of $X$}) $(n-3)$D combinatorial element on the boundary of an $n$D combinatorial element $X$;

\item[plane]
2D unbounded linear subspace.

\item[point]
1.\ 0D geometric element;
2.\ topological definition of the space at one location.

\item[polychoron]
4D geometric object with linear geometry.

\item[polygon]
2D geometric object with linear geometry, possibly with holes.

\item[polygonal curve]
curve formed by a sequence of line segments joined at their endpoints.

\item[polyhedron]
1.\ 3D geometric object with linear geometry, possibly with holes;
{\color{gray} 2.\ an $n$D geometric object with linear geometry}.

\item[polyline]
curve formed by a sequence of line segments joined at their endpoints.

\item[polyteron]
5D geometric object with linear geometry.

\item[polytope]
$n$D geometric object with linear geometry.

\item[puncture]
0D (point) hole, often formed from a degenerate polyline or polygon.

\item[ridge]
1.\ $(n-2)$D combinatorial element in an $n$D context;
2.\ ($\rightarrow$\ \emph{ridge of $X$}) $(n-2)$D combinatorial element on the boundary of an $n$D combinatorial element $X$.

\item[line string]
curve formed by joining a sequence of vertices by straight line segments, represented by these vertices.

\item[linear ring]
2D geometric object with linear geometry represented as a sequence of vertices.

\item[ring]
2D combinatorial element represented by its 1D boundary.

\item[subfacet]
1.\ $(n-2)$D combinatorial element in an $n$D context;
2.\ ($\rightarrow$\ \emph{subfacet of $X$}) $(n-2)$D combinatorial element on the boundary of an $n$D combinatorial element $X$.

\item[shell]
3D geometric object with linear geometry without 3D holes, usually defined as the volume enclosed by its 2D boundary, often represented as a set of 2D faces.

\item[simplex]
1.\ ($\rightarrow$\ sometimes \emph{$n$-simplex}) combinatorial element with $n+1$ vertices and $n+1$ facets;
2.\ ($\rightarrow$\ sometimes \emph{$n$-simplex}) geometric object based on $n+1$ affinely independent vertices, akin to a point in 0D, a line segment in 1D, a triangle in 2D, or a tetrahedron in 3D.

\item[simplicial complex]
1.\ ($\rightarrow$\ sometimes \emph{abstract simplicial complex}) topological space formed by a set of simplices glued along common faces, all faces of a simplex in the complex should be in the complex;
2.\ ($\rightarrow$\ sometimes \emph{geometric simplicial complex}) abstract simplicial complex where simplices are embedded into Euclidean space, their interiors should not intersect geometrically.

\item[solid]
1.\ 3D geometric object with linear geometry, possibly with 3D holes, often represented as a set of shells;
2.\ object defined based on solid modelling, often as opposed one based on boundary representation;
3.\ an object containing its interior (as opposed to \emph{hollow}).

\item[spike]
degenerate part of a 2D shape forming a curve, often protruding outwards from its boundary.

\item[sphere]
1.\ ($\rightarrow$\ same as $2$-\emph{sphere}) topological definition of the space at a certain distance (the radius) from a point in $\mathbb{R}^3$ (the centre);
2.\ ($\rightarrow$\ often $n$-\emph{sphere}) topological definition of the space at a certain distance (the radius) from a point in $\mathbb{R}^{n+1}$ (the centre);
3.\ geometric definition of a perfectly round hollow 3D object.

\item[surface]
1.\ 2-manifold;
2.\ 2D combinatorial element.

\item[tesseract]
4-cube: polychoron with 8 cubical facets.

\item[vertex]
0D combinatorial element.

\item[volume]
1.\ measure of the 3D extent of an object;
2.\ 3D combinatorial element.

\item[wire]
2D combinatorial element represented by its 1D boundary.

\end{description}

\end{multicols}

% \begin{table}
% \scriptsize
% \begin{tabular}{ccccccc}
% \toprule
% \multicolumn{7}{c}{dimension} \\
% $n$ & 0 & 1 & 2 & 3 & 4 \\
% \midrule
% \multicolumn{7}{c}{topological} \\
% $n$-simplex & vertex & edge & face & volume \\
% $n$-cell & vertex & edge & face & volume \\
% $n$-sphere & pair of points & circle & sphere \\
% $n$-ball & & interval & disk & ball \\
% open $n$-ball & & open interval & open disk & open ball \\
% $n$-manifold & discrete space & curve & surface \\
% \midrule
% \multicolumn{7}{c}{geometric} \\
% $n$-simplex & point & line segment & triangle & tetrahedron & pentachoron \\
% $n$-cube & point & square & cube & tesseract \\
% $n$-orthotope & point & line segment & rectangle & cuboid/box \\
% $n$-parallelotope & point & line segment & parallelogram & parallelepiped \\
% prismatic $n$-polytope & & line segment & rectangle & prism & prismatic polychoron \\
% $n$-polytope & point & line segment & polygon & polyhedron & polychoron \\
% \midrule
% \multicolumn{7}{c}{other} \\
% $n$-th Lebesgue measure & length & area & volume \\
% & & collinear & coplanar \\
% \bottomrule
% \end{tabular}
% \end{table}
